Active Calculus 2.0, 2017a
Active Calculus 2.0, 2017a
Active Calculus 2.0, 2017a
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s t <br />
s(t) 64 − 16(t − 1) 2 <br />
y s(t) 0 ≤ t ≤ 3 <br />
<br />
<br />
<br />
0
t < 1 1 < t < 3 t 1<br />
<br />
s(1) − s(0.5)<br />
AV [0.5,1] .<br />
1 − 0.5<br />
AV [0.5,1] <br />
<br />
AV [0.5,1] <br />
<br />
<br />
<br />
s(t) <br />
s(t) <br />
t s <br />
<br />
<br />
<br />
<br />
<br />
<br />
AV [0.5,1] <br />
[0.5, 1] <br />
<br />
<br />
t s(t) <br />
t a t b AV [a,b] <br />
AV [a,b] <br />
s(b) − s(a)<br />
.<br />
b − a<br />
AV [a,b] s t <br />
<br />
s(t) <br />
64 − 16(t − 1) 2 <br />
<br />
[0.4, 0.8] [0.7, 0.8] [0.79, 0.8] [0.799, 0.8] [0.8, 1.2] [0.8, 0.9] [0.8, 0.81] [0.8, 0.801]
A (0.4, s(0.4))<br />
B (0.8, s(0.8)) <br />
<br />
<br />
<br />
<br />
<br />
<br />
s(t) 64 − 16(t − 1) 2 <br />
t <br />
0.8 <br />
(0.8, s(0.8)) <br />
<br />
<br />
<br />
<br />
t <br />
0.8 <br />
0.4 0.8 1.2<br />
64<br />
56<br />
48<br />
feet<br />
A<br />
B<br />
s<br />
sec<br />
s(t) 64 −<br />
16(t − 1) 2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t a <br />
<br />
t a
s(t) 64−16(t −1) 2 <br />
<br />
[1.5, 2] <br />
[0, 0.5]<br />
<br />
t 1.5 t 2<br />
<br />
<br />
<br />
<br />
<br />
t 1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
t a <br />
s [a, b]<br />
b <br />
b a + h h <br />
[a, a + h] <br />
s(a + h) − s(a)<br />
AV [a,a+h] ,<br />
h<br />
h (a + h) − a h h<br />
h <br />
<br />
t a t a h < 0 AV [a,a+h] <br />
[a + h, a]<br />
t a <br />
h <br />
<br />
s(t) 16 − 16t 2 s t <br />
[0.5, 0.5+<br />
h] −0.5 < h < 0.5 h 0
[0.5, 0.75] [0.4, 0.5] <br />
t 0.5<br />
−0.5 < h < 0.5 h 0 h <br />
<br />
0 ≤ t ≤ 1 <br />
<br />
AV [0.5,0.5+h] <br />
s(0.5 + h) − s(0.5)<br />
(0.5 + h) − 0.5 .<br />
s(0.5 + h) <br />
s s(t) 16 − 16t 2 <br />
s(0.5 + h) 16 − 16(0.5 + h) 2<br />
16 − 16(0.25 + h + h 2 )<br />
16 − 4 − 16h − 16h 2<br />
12 − 16h − 16h 2 .<br />
<br />
s(0.5 + h) − s(0.5)<br />
AV [0.5,0.5+h] <br />
(0.5 + h) − 0.5<br />
(12 − 16h − 16h2 ) − (16 − 16(0.5) 2 )<br />
0.5 + h − 0.5<br />
12 − 16h − 16h2 − 12<br />
h<br />
−16h − 16h2<br />
.<br />
h<br />
<br />
h h <br />
h <br />
h <br />
AV [0.5,0.5+h] −16 − 16h.<br />
h <br />
[0.5, 0.75] h 0.25 <br />
−16−16(0.25) −20 [0.4, 0.5] h −0.1 <br />
−16 − 16(−0.1) −14.4 <br />
AV [0.5,0.5+h] h h −16h<br />
<br />
t 0.5 −16
s(t) 64 − 16(t − 1) 2 <br />
<br />
[2, 2 + h] [1.5, 2] <br />
t 2 <br />
<br />
<br />
[a, b] <br />
(a, s(a)) (b, s(b)) y s(t) <br />
s<br />
(a,s(a))<br />
m = s(b)−s(a)<br />
b−a<br />
(b,s(b))<br />
t<br />
<br />
t s <br />
<br />
[a, b] AV [a,b] s(b)−s(a)<br />
b−a<br />
<br />
[a, b] <br />
[a, a + h] <br />
<br />
AV [a,a+h] s(a+h)−s(a)<br />
h<br />
<br />
<br />
<br />
<br />
<br />
<br />
s (a, s(a)) <br />
(b, s(b)) m s(b)−s(a)<br />
b−a<br />
s <br />
[a, b]<br />
<br />
s <br />
t <br />
s <br />
0 ≤ t ≤ 0.2<br />
<br />
<br />
t 0.2
f (x) 36 − x 2 <br />
<br />
f (4)− f (0)<br />
<br />
<br />
4−0<br />
f (6)− f (4)<br />
6−4<br />
f (6)− f (0)<br />
6−0<br />
<br />
<br />
<br />
<br />
<br />
f (4)− f (0)<br />
4−0<br />
f (6)− f (4)<br />
6−4<br />
f (6)− f (0)<br />
6−0
f (x) (x) <br />
f (x) (x) <br />
<br />
<br />
0 ≤ x ≤ 2.2 <br />
5.2 ≤ x ≤ 6.1 <br />
5.2 ≤ x ≤ 8 <br />
2.2 ≤ x ≤ 4 <br />
2.2 ≤ x ≤ 6.1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
t 0 h <br />
t <br />
<br />
s(t) 100 cos(0.75t) · e −0.2t + 100<br />
t 0 t 15
[0, 15]<br />
[0, 2] [1, 6] [8, 10] <br />
200<br />
150<br />
100<br />
50<br />
s<br />
5 10 15 20<br />
t<br />
<br />
<br />
<br />
<br />
t 5 <br />
<br />
<br />
<br />
<br />
<br />
<br />
t 0 <br />
t 1.1 t 2.45<br />
<br />
t 7 <br />
t 13<br />
s(t) <br />
t −3.5 <br />
s(t) <br />
<br />
s<br />
v<br />
2 4 6 8 10 12<br />
t<br />
2 4 6 8 10 12<br />
t<br />
s(t) <br />
<br />
v(t)
t 2.45<br />
t 7 [2.45, 7]<br />
v(t) t<br />
s(t) <br />
<br />
<br />
v(t) <br />
<br />
v(t) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f f [a, b] <br />
<br />
f (b) − f (a)<br />
.<br />
b − a<br />
<br />
[a, b]<br />
f (b)− f (a)<br />
b−a<br />
y f (x) <br />
P(t) t t <br />
P <br />
t 0 t 20 <br />
<br />
<br />
P(t) 181843(1.04) t/10 .<br />
<br />
[5, 10] [5, 9] [5, 8] [5, 7] [5, 6]
lim x→a f (x) L<br />
<br />
<br />
<br />
<br />
<br />
<br />
x <br />
y f (x) x 2 <br />
<br />
<br />
(−2, 4) (−1, 1) (0, 0) (1, 1) (2, 4)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
s(t) 64 − 16t 2 [1, x]<br />
<br />
AV [1,x] <br />
s(x) − s(1)<br />
x − 1<br />
(64 − 16x2 ) − (64 − 16)<br />
x − 1<br />
<br />
16 − 16x2<br />
.<br />
x − 1<br />
x AV [1,x] x<br />
x 1 <br />
(x) 16−16x2<br />
x−1<br />
<br />
t 1 t x <br />
t 1 (x) x <br />
1 (1) 0/0<br />
x <br />
1 (x)
(−2) (−1)<br />
(0) (1) (2) <br />
<br />
<br />
<br />
a −1 a 0<br />
a 2 <br />
x <br />
a (x) <br />
<br />
x <br />
a 1<br />
(x) <br />
<br />
g<br />
3<br />
2<br />
1<br />
-2 -1 1 2 3<br />
-1<br />
y (x) <br />
<br />
<br />
<br />
<br />
<br />
<br />
f L x a <br />
<br />
x <br />
(x) <br />
(0) 1 4 <br />
<br />
<br />
<br />
L a
f x a L <br />
f L x a <br />
lim f (x) L<br />
x→a<br />
f (x) L x <br />
a f (x) x<br />
a f x a<br />
<br />
lim (x) 3, lim (x) 4, lim (x) 1,<br />
x→−1 x→0 x→2<br />
x → 1 <br />
<br />
<br />
<br />
x 1 x 1 <br />
x 1 <br />
<br />
x 1<br />
f f <br />
x a <br />
f (x) <br />
<br />
a <br />
<br />
a <br />
<br />
<br />
<br />
<br />
a <br />
<br />
<br />
<br />
f (x) 4−x2<br />
x+2 a −1 a −2 (x) sin <br />
π<br />
x a 3 a 0<br />
<br />
a −2<br />
f a −1
x f(x)<br />
−0.9 2.9<br />
−0.99 2.99<br />
−0.999 2.999<br />
−0.9999 2.9999<br />
−1.1 3.1<br />
−1.01 3.01<br />
−1.001 3.001<br />
−1.0001 3.0001<br />
x f(x)<br />
−1.9 3.9<br />
−1.99 3.99<br />
−1.999 3.999<br />
−1.9999 3.9999<br />
−2.1 4.1<br />
−<strong>2.0</strong>1 4.01<br />
−<strong>2.0</strong>01 4.001<br />
−<strong>2.0</strong>001 4.0001<br />
f<br />
5<br />
3<br />
1<br />
-3 -1 1<br />
<br />
f x −1<br />
<br />
f x −2<br />
f (x) [−4, 2]<br />
f x <br />
−1 lim x→−1 f (x) 3 <br />
f <br />
f f (x) 4−x2<br />
x+2<br />
<br />
<br />
<br />
x → −1 (4 − x 2 ) → (4 − (−1) 2 ) 3 <br />
(x + 2) → (−1 + 2) 1 x →−1 f <br />
lim x→−1 f (x) 3 1 3<br />
x →−2 f (−2) <br />
<br />
x →−2 (4−x 2 ) → (4−(−2) 2 ) 0 (x+2) → (−2+2) 0<br />
x →−2 f 0/0 <br />
<br />
<br />
f 4 x −2 <br />
f (x) <br />
4 − x<br />
lim f (x) 2<br />
lim<br />
x→−2 x→−2 x + 2<br />
(2 − x)(2 + x)<br />
lim<br />
.<br />
x→−2 x + 2<br />
x →−2 <br />
x −2 x <br />
−2 2+x<br />
x+2<br />
x
lim f (x) lim 2 − x.<br />
x→−2 x→−2<br />
2 − x <br />
4 lim x→−2 f (x) 4.<br />
<br />
x (x)<br />
2.9 0.84864<br />
2.99 0.86428<br />
2.999 0.86585<br />
2.9999 0.86601<br />
3.1 0.88351<br />
3.01 0.86777<br />
3.001 0.86620<br />
3.0001 0.86604<br />
x (x)<br />
−0.1 0<br />
−0.01 0<br />
−0.001 0<br />
−0.0001 0<br />
0.1 0<br />
0.01 0<br />
0.001 0<br />
0.0001 0<br />
2<br />
g<br />
-3 -1 1 3<br />
-2<br />
<br />
x 3<br />
<br />
x 0<br />
(x) [−4, 4]<br />
x → 3 <br />
0.866025 π x → π 3<br />
<br />
(x) sin( π x ) → sin( π 3 ) x → 3 sin( π √<br />
3 ) 3<br />
2<br />
<br />
<br />
√<br />
3<br />
lim (x) <br />
x→3 2 .<br />
x → 0 π x<br />
x <br />
π x <br />
x π x<br />
<br />
x 0 <br />
<br />
(x) sin <br />
π<br />
x x 0 <br />
<br />
x 0<br />
<br />
x 0 0 <br />
{0.1, 0.01, 0.001,...} <br />
(10 −k )(10 −k ) ( )<br />
π<br />
sin<br />
10 sin(10 k π) 0 <br />
−k<br />
k {0.3, 0.03, 0.003,...}<br />
<br />
(<br />
(3 · 10 −k ) sin<br />
)<br />
π<br />
3 · 10 −k<br />
( ) √<br />
10 k π 3<br />
sin <br />
3<br />
2 ≈ 0.866025.
2<br />
<br />
x → 0<br />
<br />
<br />
<br />
<br />
√<br />
3<br />
<br />
<br />
<br />
<br />
lim x→1<br />
x 2 −1<br />
x−1<br />
lim x→0<br />
(2+x) 3 −8<br />
x<br />
lim x→0<br />
√<br />
x+1−1<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
t s <br />
[a, b] AV [a,b] s(b)−s(a)<br />
b−a<br />
.<br />
a b a<br />
b → a <br />
a <br />
IV ta <br />
t a <br />
s(b) − s(a)<br />
IV ta lim AV [a,b] lim .<br />
b→a b→a b − a<br />
b b a + h h <br />
<br />
s(a + h) − s(a)<br />
IV ta lim AV [a,a+h] lim<br />
.<br />
h→0 h→0 h
s(t) t 2 <br />
s t <br />
<br />
[3, 3 + h] h > 0<br />
[3, 3.2] <br />
<br />
t 3 <br />
<br />
<br />
<br />
s t <br />
s t<br />
<br />
<br />
<br />
[0.5, 1]<br />
[1.5, 2.5] [0, 5] <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t 2 <br />
t 2 <br />
<br />
[1.5, 2.5] <br />
<br />
5<br />
3<br />
1<br />
1 3 5<br />
s<br />
t<br />
<br />
y s(t)
lim x→a f (x) L f x <br />
a L f (x) L <br />
x a<br />
lim x→a f (x) a f <br />
f <br />
x a <br />
<br />
<br />
f x → a<br />
<br />
<br />
<br />
<br />
f (x) <br />
<br />
lim f (x) <br />
x→−2<br />
lim f (x) <br />
x→0<br />
lim<br />
x→2<br />
f (x) <br />
lim<br />
x→4<br />
f (x)
lim<br />
θ→0<br />
sin(5θ)<br />
.<br />
θ<br />
θ <br />
<br />
sin(5θ)<br />
lim<br />
θ→0<br />
θ<br />
<br />
<br />
x ⎧⎪<br />
2 − 4, 0 ≤ x < 1<br />
f (x) ⎨2, x 1<br />
⎪<br />
⎩2x − 5, 1 < x<br />
<br />
lim<br />
x→1 +<br />
lim<br />
x→1 −<br />
f (x) <br />
f (x) <br />
lim f (x) <br />
x→1<br />
<br />
f (x) <br />
<br />
<br />
<br />
x 2 − 36<br />
lim<br />
x→−6 x + 6<br />
f (x) 16−x4<br />
x 2 −4 <br />
f <br />
x a 2 lim x→2 f (x), <br />
<br />
16−x4<br />
x 2 −4<br />
lim x→2 f (x)<br />
<br />
<br />
f (2) −8 <br />
16−x4<br />
x 2 −4<br />
−4− x2 . <br />
f <br />
y f (x)<br />
[1, 3] <br />
16−x<br />
lim 4<br />
x→2 x 2 −4
(x) − |x+3|<br />
x+3 <br />
<br />
a −3 lim x→−3 (x), <br />
<br />
|x+3|<br />
x+3 lim x→−3 (x)<br />
<br />
|a| a a ≥ 0 <br />
|a| −a a < 0<br />
(−3) −1 <br />
− |x+3|<br />
x+3<br />
−1. <br />
<br />
y (x) <br />
[−4, −2] <br />
lim x→−3 (x)<br />
<br />
<br />
3<br />
3<br />
-3 3<br />
-3 3<br />
-3<br />
-3<br />
y f (x) y (x) <br />
y f (x) <br />
f (−2) 2 lim x→−2 f (x) 1<br />
f (−1) 3 lim x→−1 f (x) 3<br />
f (1) lim x→1 f (x) 0
f (2) 1 lim x→2 f (x) <br />
y (x) <br />
(−2) 3 (−1) −1 (1) −2 (2) 3<br />
x −2, −1, 1 2 <br />
<br />
(0) lim x→0 (x) <br />
t 0 s t <br />
s(t) 100 cos(0.75t) · e −0.2t + 100<br />
[1, 1+<br />
h]<br />
h → 0
s t a <br />
t a + h <br />
AV [a,a+h] <br />
s(a + h) − s(a)<br />
.<br />
h<br />
y f (x).<br />
f f [a, a + h]<br />
<br />
f (a + h) − f (a)<br />
AV [a,a+h] .<br />
h
f [a, b] <br />
AV [a,b] <br />
f (b) − f (a)<br />
.<br />
b − a<br />
f <br />
<br />
f <br />
a a + h x <br />
<br />
(a, f (a))<br />
(a + h, f (a + h)) <br />
<br />
<br />
(a, f (a)) (a + h, f (a + h))<br />
<br />
<br />
y<br />
f<br />
<br />
(a, f (a)) <br />
(a + h, f (a + h))<br />
<br />
<br />
<br />
<br />
<br />
a<br />
a + h<br />
x<br />
y f (x) <br />
<br />
<br />
<br />
<br />
f <br />
f a f a <br />
f ′ (a) <br />
<br />
f x a
f x x a f ′ (a) <br />
<br />
f ′ f (a + h) − f (a)<br />
(a) lim<br />
,<br />
h→0 h<br />
f ′ (a) f a f <br />
x a <br />
<br />
<br />
f x a <br />
f [a, a + h] h → 0 <br />
<br />
x a x a<br />
<br />
y s(t) s ′ (a) <br />
t a<br />
f (a+h)− f (a)<br />
<br />
h<br />
f x <br />
s t <br />
s ′ (a) <br />
f (a+h)− f (a)<br />
<br />
h<br />
(a, f (a)) <br />
(a + h, f (a + h)) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(a, f (a)) (a + h, f (a + h)) (a, f (a)) h → 0
(a, f (a)) (a + h, f (a + h)) h <br />
y f (x)<br />
<br />
f (a+h)− f (a)<br />
m <br />
h<br />
h <br />
h → 0 <br />
(a, f (a)) <br />
<br />
<br />
y f (x) (a, f (a)) (a, f (a)) <br />
m f ′ (a)<br />
y<br />
y<br />
y<br />
y<br />
f<br />
f<br />
f<br />
f<br />
x<br />
x<br />
x<br />
x<br />
a<br />
a<br />
a<br />
a<br />
f (a, f (a))<br />
x a <br />
f (a, f (a))<br />
<br />
(a, f (a)) <br />
<br />
y f (x) (a, f (a)) <br />
x a<br />
y<br />
f<br />
a<br />
x<br />
f (a, f (a)) <br />
(a, f (a)) f <br />
(a, f (a)) f ′ (a)
f x x a f ′ (a) <br />
y f (x) (a, f (a))<br />
<br />
f (x) <br />
x − x 2 f ′ (2) <br />
<br />
<br />
<br />
f ′ f (2 + h) − f (2)<br />
(2) lim<br />
.<br />
h→0 h<br />
f f (2) 2−2 2 −2 f (2+h) (2+h)−(2+h) 2 .<br />
<br />
f ′ (2 + h) − (2 + h) 2 − (−2)<br />
(2) lim<br />
.<br />
h→0 h<br />
h h → 0 <br />
h <br />
<br />
f ′ 2 + h − 4 − 4h − h<br />
(2) 2 + 2<br />
lim<br />
.<br />
h→0 h<br />
<br />
f ′ −3h − h<br />
(2) 2<br />
lim .<br />
h→0 h<br />
h <br />
<br />
f ′ (2) lim<br />
h→0<br />
(−3 − h).<br />
h → 0 f ′ (2) −3
f ′ (2) <br />
y x − x 2 (2, −2)<br />
f ′ (2) f <br />
(2, −2) <br />
(2, −2) m f ′ (2) −3 <br />
<br />
<br />
<br />
-2<br />
m = f ′ (2)<br />
1 2<br />
-4<br />
y = x − x 2<br />
<br />
y x − x 2 (2, −2)<br />
<br />
f f (x) 3 − 2x<br />
f f <br />
x<br />
f [1, 4] [3, 7] [5, 5+ h]<br />
<br />
<br />
<br />
f x a 1 f ′ (1) <br />
<br />
f ′ (2) f ′ (π)<br />
f ′ (− √ 2) <br />
<br />
t s(t) −16t 2 +<br />
16t + 32
s <br />
<br />
s [1, 2] <br />
<br />
<br />
s <br />
t a 1 <br />
<br />
<br />
<br />
<br />
s <br />
[1, 2] <br />
<br />
s a 1 <br />
<br />
32<br />
16<br />
y<br />
a <br />
s ′ (a) <br />
<br />
<br />
<br />
1 2<br />
t<br />
y s(t)<br />
<br />
P t t<br />
P(t) 25000e t/5 <br />
<br />
P t 0 t 5 <br />
<br />
P <br />
<br />
<br />
<br />
P t a 2
P a 2<br />
h <br />
P ′ (2) <br />
<br />
<br />
<br />
P <br />
[2, 4] <br />
<br />
P a 2<br />
y<br />
<br />
P ′ (a) a <br />
<br />
<br />
P<br />
t<br />
y P(t)<br />
<br />
<br />
f (b)− f (a)<br />
f [a, b] <br />
b−a<br />
<br />
f x <br />
<br />
(a, f (a)) (b, f (b)) y f (x) [a, a+h]<br />
[a, b] <br />
<br />
f (a+h)− f (a)<br />
h<br />
<br />
x f x a<br />
f ′ (a) f a f a <br />
<br />
f ′ f (a + h) − f (a)<br />
(a) lim<br />
,<br />
h→0 h<br />
<br />
x a [a, a + h] h → 0<br />
f ′ (a)
f x x a <br />
y f (x) (a, f (a)) f ′ (a) <br />
(a, f (a))<br />
<br />
<br />
<br />
<br />
y f (x) <br />
x <br />
x <br />
x <br />
x <br />
x <br />
x 1.75 <br />
x 1.75 x <br />
(x) B (2.6, 8) <br />
A (2.55, 8.06) <br />
B
( ) <br />
′ (<br />
) <br />
f (x) <br />
<br />
<br />
f (9) f (12)<br />
f (9) − f (6) f (6) − f (3)<br />
f (6)− f (3)<br />
<br />
6−3<br />
<br />
f ′ (3) f ′ (12)<br />
f (9)− f (3)<br />
9−3<br />
(t) t 2 + 6t t 3 <br />
′ (3) <br />
f ′ (3) f (x) 3 x <br />
<br />
f ′ (3) ≈
y f (x) <br />
y f (x) <br />
<br />
y f (x) <br />
[−3, −1] <br />
y f (x) <br />
[0, 2]<br />
y f (x) <br />
x −3 <br />
y f (x) x 0<br />
<br />
f <br />
[−3, −1] [0, 2] <br />
<br />
<br />
f <br />
x −3 x 0 <br />
<br />
4<br />
-4 4<br />
-4<br />
y<br />
f<br />
x<br />
y f (x)<br />
<br />
<br />
3<br />
3<br />
-3 3<br />
-3 3<br />
-3<br />
-3<br />
y f (x) y (x)
y f (x) <br />
f [−3, 0] −2 f<br />
[1, 3] <br />
f x −1 −1 <br />
f x 2 <br />
y (x) <br />
(3)−(−2)<br />
5<br />
0 (1)−(−1)<br />
2<br />
−1 <br />
′ (2) 1 ′ (−1) 0<br />
P <br />
P(t) 1.15(1.014) t t <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y P(t) <br />
t <br />
<br />
<br />
<br />
<br />
f ′ (a) <br />
<br />
h y f (x) (a, f (a)) <br />
f ′ (a) <br />
<br />
<br />
f (x) x 2 − 3x a 2<br />
f (x) 1 x a 1<br />
f (x) √ x a 1<br />
f (x) 2 − |x − 1| a 1<br />
f (x) sin(x) a π 2
f <br />
f ′ <br />
f ′ (a) f ′ (x)<br />
f ′ (x) f (x)<br />
f f ′ <br />
<br />
y f (x) <br />
x a y f (x)<br />
x a f ′ (a) <br />
a a 1 a 3 <br />
<br />
a <br />
f ′ (1) f ′ (3) f ′ (a) f ′ (x)<br />
<br />
y f (x)<br />
<br />
f (x) 4x − x 2 <br />
f ′ (0) f ′ (1) f ′ (2) <br />
f ′ (3)<br />
f ′ (a) a <br />
f ′ (4) f ′ (5)<br />
<br />
f ′ (a) a <br />
f (x) f (x) 4x − x 2 <br />
f ′ (a) a<br />
<br />
f (x) 4x − x 2 <br />
f ′ (0) 4 f ′ (1) 2 f ′ (2) 0 f ′ (3) −2<br />
f ′ (4) −4 f ′ (5) −6. <br />
0 1 2 3
f ′ (a) a <br />
<br />
f ′ f (a + h) − f (a)<br />
(a) lim<br />
h→0 h<br />
4a + 4h − a 2 − 2ha − h 2 − 4a + a 2<br />
lim<br />
h→0 h<br />
lim<br />
h→0<br />
h(4 − 2a − h)<br />
h<br />
lim<br />
h→0<br />
4(a + h) − (a + h) 2 − (4a − a 2 )<br />
h<br />
lim<br />
h→0<br />
(4 − 2a − h).<br />
lim<br />
h→0<br />
4h − 2ha − h 2<br />
h<br />
4 2a hh → 0 (4 − 2a − h) →<br />
(4 − 2a) f ′ (a) 4 − 2a<br />
f ′ (3) 4−2(3) <br />
−2 a a <br />
<br />
<br />
a x <br />
x <br />
y f (x) <br />
f (x) 4x − x 2 f ′ (x) 4 − 2x.<br />
<br />
<br />
4<br />
3<br />
2<br />
m = 0<br />
m = 2<br />
m = −2<br />
4<br />
3<br />
2<br />
(0,4)<br />
(1,2)<br />
1<br />
m = 4<br />
m = −4<br />
1<br />
(2,0)<br />
-1<br />
-2<br />
-3<br />
1 2 3 4<br />
y = f (x)<br />
-1<br />
-2<br />
-3<br />
1 2 3 4<br />
(3,−2)<br />
y = f ′ (x)<br />
-4<br />
-4<br />
(4,−4)<br />
f (x) 4x − x 2 f ′ (x) 4 − 2x <br />
f f ′
f (x) 4x−x 2 <br />
f ′ (x) 4−2x <br />
<br />
<br />
<br />
<br />
x <br />
x <br />
x <br />
f y f ′ (x)<br />
f x<br />
y f (x) <br />
f (x) <br />
f ′ (x) <br />
<br />
<br />
a f ′ (a) a <br />
x <br />
<br />
f x <br />
f x x f ′ (x) f ′ (x) <br />
f (x+h)− f (x)<br />
lim h→0 h<br />
, <br />
<br />
y f (x) <br />
y f ′ (x) y f (x) <br />
y f ′ (x) <br />
<br />
y f (x) <br />
y f ′ (x) <br />
f 1 × 1 f ′<br />
f <br />
f ′
f<br />
g<br />
x<br />
x<br />
f ′<br />
g ′<br />
x<br />
x<br />
p<br />
q<br />
x<br />
x<br />
p ′<br />
q ′<br />
x<br />
x
s<br />
x<br />
x<br />
r ′<br />
s ′<br />
x<br />
x<br />
w<br />
z<br />
x<br />
x<br />
w ′<br />
z ′<br />
x<br />
x
f ′ <br />
f <br />
y f (x) 4x −<br />
x 2 f ′ (a) 4 − 2a <br />
f ′ (x) 4−2x f f ′ <br />
<br />
f ′ (x) f f ′ <br />
<br />
f ′ (x) <br />
y f (x) <br />
f ′ (x)<br />
<br />
<br />
<br />
<br />
<br />
f x p(z) p ′ (z)<br />
q(s) s 3<br />
f (x) 1<br />
p(z) z 2 G(y) √ y<br />
(t) t<br />
F(t) 1 t<br />
<br />
f ′ f (x+h)− f (x)<br />
(x) lim h→0 h<br />
<br />
x <br />
y f ′ (x) f f ′ <br />
<br />
f f ′ <br />
f ′ (a) <br />
f ′ (x) <br />
<br />
y f (x)
y f ′ (x) <br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
f (x) <br />
<br />
x <br />
<br />
x <br />
<br />
(x) 3x 2 −4 <br />
′ (x) lim [(<br />
)/h]<br />
h→0<br />
<br />
<br />
<br />
<br />
f (x)
f (x) <br />
<br />
<br />
<br />
f <br />
x <br />
f (x) x <br />
f (x) x <br />
f ′ (x) x <br />
f ′ (x) x <br />
<br />
f (x) 1<br />
x − 3
f ′ (2)<br />
f ′ (4)<br />
f ′ (5)<br />
f ′ (7)<br />
f f x<br />
f f (−2) 1 f ′ (−2) −2 f ′ (−1) −1 f ′ (0) 0<br />
f ′ (1) 1 f ′ (2) 2<br />
y f (x) <br />
<br />
y f ′ (x) <br />
y f ′ (x)<br />
y f (x) <br />
y f ′ (x) <br />
<br />
3<br />
3<br />
-3 3<br />
-3 3<br />
-3<br />
-3<br />
y f (x) y f ′ (x) <br />
(x) x 2 − x + 3<br />
′ (x)<br />
y (x) y ′ (x) <br />
′ (x) <br />
p ′ (x) p(x) 5x 2 −<br />
4x + 12.
′ (x) p ′ (x) <br />
<br />
<br />
y ′ (x) <br />
2<br />
2<br />
-2 2<br />
-2 2<br />
-2<br />
-2<br />
y (x) y ′ (x)<br />
x 0 < x < 2 ′ (x) <br />
y (x) <br />
0 < x < 2 y (x) <br />
<br />
x ′ (x) <br />
y (x)<br />
(0) 1 <br />
y (x)<br />
y f (x) <br />
y f ′ (x) <br />
f 1×1 <br />
f ′ f <br />
f ′
f<br />
f<br />
x<br />
x<br />
f ′<br />
f ′<br />
x<br />
x<br />
f<br />
f<br />
x<br />
x<br />
f ′<br />
f ′<br />
x<br />
x<br />
y f (x) y f ′ (x)
f ′ <br />
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y s(t) <br />
t v(t) s ′ (t) <br />
s(t) t v(t) s ′ (t) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t 0 <br />
t <br />
<br />
<br />
<br />
[57, 68]<br />
[68, 104]<br />
(57, 63.8) (104, 106.8)
[68, 104] <br />
<br />
<br />
<br />
t 80 <br />
<br />
<br />
100<br />
80<br />
60<br />
40<br />
20<br />
s<br />
(104,106.8)<br />
(57,63.8)<br />
(68,63.8)<br />
t<br />
20 40 60 80 100<br />
y s(t)<br />
<br />
<br />
t <br />
<br />
f x <br />
f ′ f (x + h) − f (x)<br />
(x) lim<br />
,<br />
h→0 h<br />
x a f ′ (a) <br />
(a, f (a))<br />
df dy<br />
dx<br />
<br />
dx f ′ (x) <br />
<br />
f x <br />
f (x+h)− f (x)<br />
<br />
h<br />
f x <br />
f ′ (x) f ′ (x) f <br />
x f <br />
<br />
<br />
y P(t) P <br />
t t 0 <br />
P ′ (2) 21.37 P <br />
t
f ′ f (x+h)− f (x)<br />
(x) x <br />
h<br />
<br />
h h <br />
<br />
<br />
<br />
y f (x) f (1) <br />
2.5 f (2) 3.25 f (3) 3.625 f ′ (2)<br />
f ′ f (2+h)− f (2)<br />
(2) lim h→0 h<br />
<br />
y f (x) <br />
h <br />
h −1 h 1 <br />
(2, 3.25)<br />
<br />
f ′ (2) ≈<br />
f (1) − f (2)<br />
1 − 2<br />
<br />
2.5 − 3.25<br />
−1<br />
0.75.<br />
<br />
f ′ (2) ≈<br />
f (3) − f (2)<br />
3 − 2<br />
<br />
3.625 − 3.25<br />
1<br />
0.375.<br />
(2, 3.25) <br />
(2, 3.25) <br />
<br />
f ′ (2) ≈<br />
0.75 + 0.375<br />
2<br />
0.5625.<br />
<br />
f ′ (2) f
3<br />
3<br />
2<br />
2<br />
1<br />
1<br />
1 2 3<br />
1 2 3<br />
y f (x) (1, 2.5) <br />
(2, 3.25) (2, 3.25) (3, 3.625) <br />
(1, 2.5) (3, 3.625) <br />
<br />
<br />
y f (x) <br />
f (1)− f (2)<br />
<br />
1−2<br />
0.75 <br />
f (3)− f (2)<br />
<br />
3−2<br />
0.375. <br />
(2, f (2)) f ′ (2) <br />
<br />
f (3) − f (1)<br />
3 − 1<br />
<br />
3.625 − 2.5<br />
2<br />
1.125<br />
2<br />
0.5625.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f ′ f (a + h) − f (a − h)<br />
(a) ≈ ,<br />
2h<br />
y f (x) (a −<br />
h, f (a − h)) (a + h, f (a + h))
F <br />
<br />
t <br />
t 0 15 30 45 60 75 90<br />
F(t) 70 180.5 251 296 324.5 342.8 354.5<br />
<br />
<br />
t 30 <br />
<br />
t 60 <br />
F ′ (75) F ′ (90)<br />
<br />
F(64) 330.28 F ′ (64) 1.341 <br />
<br />
t 65 t 66 <br />
<br />
[0, 90] <br />
<br />
r <br />
C(r) <br />
C(2000) 800<br />
C ′ (r)<br />
C(2000) 800 C ′ (2000) 0.35 C(2100) <br />
<br />
C ′ (2000) C ′ (3000) <br />
C ′ (5000) −0.1
C s <br />
C f (s) s <br />
f (80) 0.015 f (90) 0.02 <br />
f (100) 0.027 <br />
s 90 <br />
<br />
<br />
f (80) 0.015<br />
<br />
f ′ (90) <br />
f <br />
f ′ <br />
f ′ <br />
P(t) 400 − 330e −0.03t <br />
t <br />
P P ′ <br />
400<br />
◦ F<br />
16<br />
◦ F/min<br />
300<br />
y = P(t)<br />
12<br />
200<br />
8<br />
y = P ′ (t)<br />
100<br />
min<br />
4<br />
min<br />
20 40 60 80<br />
20 40 60 80<br />
P(t) 400 − 330e −0.03t P ′ (t) <br />
<br />
P ◦ P ′ ◦
y f (x) <br />
<br />
<br />
y f ′ (x) f x <br />
f <br />
<br />
<br />
f ′ f (a + h) − f (a − h)<br />
(a) ≈ ,<br />
2h<br />
y f (x) <br />
(a − h, f (a − h)) (a + h, f (a + h)) <br />
<br />
<br />
<br />
f ′ (7) 2 <br />
x 7 f <br />
f (8) <br />
f (7) <br />
x <br />
<br />
H <br />
H f (t) t <br />
f ′ (t) <br />
<br />
f ′ (35)<br />
<br />
| f ′ (35)| 1.2 f (35) 52 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
C C f ()<br />
f (300) 350
f ′ (300) 1.2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
W W f (c) <br />
c <br />
f (1500) 150<br />
<br />
<br />
<br />
<br />
<br />
<br />
f ′ (2000) 0<br />
<br />
<br />
<br />
<br />
<br />
<br />
f −1 (157) 2300<br />
<br />
<br />
<br />
<br />
<br />
<br />
f ′ (c) dW/dc <br />
<br />
f ′ <br />
<br />
<br />
<br />
f ′(<br />
) <br />
<br />
s t 2 − 5t + 12,<br />
t
t 4<br />
<br />
F t <br />
F(t) 75 + 110e −0.05t <br />
h 0.01 F ′ (10)<br />
F ′ (10) <br />
F ′ (10)<br />
F ′ (10) F ′ (20) <br />
y F ′ (t) <br />
0 ≤ t ≤ 30 <br />
T <br />
q T f (q)<br />
f (50) 0.75 <br />
<br />
s s(q) f ′ (q) <br />
<br />
f ′ (50) −0.02 <br />
<br />
v(t) 16 − 32t<br />
v t <br />
t 0 t 2<br />
<br />
v ′ (1) <br />
v ′ (1) <br />
t 1<br />
v ′ (t)<br />
V m<br />
V h(m)<br />
h(40000) 15500 h(55000) 13200 <br />
h [40000, 55000]
h(70000) 11100 <br />
h ′ (55000) <br />
<br />
h ′ (30000) h ′ (80000) <br />
V h(m) <br />
h ′ (m)
f <br />
<br />
<br />
<br />
<br />
y f (x) y f ′ (x) <br />
x a y f (x) <br />
(a, f (a)) <br />
<br />
f <br />
f ′ (x) <br />
<br />
f f ′ (x) <br />
f <br />
f ′ (x) <br />
<br />
f <br />
A<br />
B<br />
<br />
f <br />
f <br />
A <br />
f ′ (x) <br />
B
f B <br />
<br />
<br />
y f ′ (x) <br />
<br />
<br />
<br />
<br />
<br />
<br />
t <br />
y s(t) <br />
<br />
<br />
(2, 4) <br />
<br />
<br />
14<br />
10<br />
y<br />
s<br />
6<br />
2<br />
2 6 10<br />
t<br />
y s(t) <br />
<br />
<br />
t <br />
<br />
<br />
[0, 1] [1, 2] [2, 3] [3, 4] [4, 5] <br />
[0, 12]<br />
<br />
y s ′ (t)
y s ′ (t) <br />
s ′ (t) s ′ (t) <br />
s ′ (t) <br />
y v(t) <br />
v v ... v <br />
...<br />
y v ′ (t) <br />
v ′ (t) <br />
y v(t)<br />
y<br />
y<br />
2 6 10<br />
t<br />
2 6 10<br />
t<br />
y v(t) s ′ (t) y v ′ (t)<br />
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<br />
<br />
<br />
<br />
f (x) (a, b) f <br />
(a, b) x y (a, b) x < y f (x) < f (y)<br />
f (a, b) x y (a, b)<br />
x < y f (x) > f (y)
f (a, b) f <br />
(a, b) f ′ (x) > 0 x a < x < b f <br />
(a, b) f ′ (x) < 0 f ′ (a) 0 f <br />
x a<br />
<br />
<br />
−2 < x < 0 x ±2 <br />
x 0 f <br />
f ′ (x) 0 <br />
<br />
2<br />
A<br />
-2 2<br />
y = f (x)<br />
-2<br />
B<br />
<br />
−3 < x < −2 0 < x < 2<br />
−2 < x < 0 2 < x <<br />
3<br />
<br />
<br />
f <br />
f ′ f (x + h) − f (x)<br />
(x) lim<br />
.<br />
h→0 h<br />
f ′ <br />
y [ f ′ (x)] ′ <br />
y f (x) y f ′′ (x) <br />
f ′ f <br />
f
f ′′ f ′ (x + h) − f ′ (x)<br />
(x) lim<br />
.<br />
h→0 h<br />
<br />
y f ′′ (x) <br />
y f ′ (x) y f ′ (x) <br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x
−100 <br />
−2 <br />
x < y x y <br />
−100 −2 −100 <br />
−2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y x 2 y e x <br />
<br />
y −x 2 y −e x
f <br />
f ′ <br />
<br />
<br />
f <br />
f ′ <br />
<br />
<br />
<br />
f (a, b) f <br />
(a, b) f ′ (a, b) f (a, b) <br />
f ′ (a, b)<br />
<br />
<br />
<br />
t <br />
y s(t) <br />
<br />
y v(t) s ′ (t) y v ′ (t)
y s(t) <br />
<br />
14<br />
y<br />
s<br />
<br />
y v(t) s ′ (t) <br />
<br />
10<br />
<br />
<br />
<br />
<br />
<br />
<br />
a(t) a(t) <br />
v(t) a(t) <br />
s(t) <br />
6<br />
2<br />
2 6 10<br />
t<br />
s ′′ s ′ <br />
y s(t) <br />
<br />
<br />
t <br />
<br />
s <br />
v a<br />
v s <br />
v s <br />
v s <br />
a v <br />
a v <br />
a v <br />
a s <br />
a s <br />
a s <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
s v a f f ′
f ′′ f ′ <br />
f f ′ f ′′ <br />
f ′ f f ′ <br />
f ′′ <br />
F <br />
<br />
t F ′ (30) <br />
F ′ (60) <br />
<br />
t F(t)<br />
0 70<br />
15 180.5<br />
30 251<br />
45 296<br />
60 324.5<br />
75 342.8<br />
90 354.5<br />
t F ′ (t)<br />
0 <br />
15 6.03<br />
30 3.85<br />
45 2.45<br />
60 1.56<br />
75 1.00<br />
90 <br />
F(t)<br />
F ′ (t)<br />
F ′ (t)<br />
F ′′ (30)<br />
F ′′ (30) <br />
<br />
F(30) F ′ (30) F ′′ (30) <br />
<br />
<br />
<br />
<br />
f ′ f <br />
f <br />
f ′ f ′′ <br />
<br />
f f ′ f ′′
f ′ <br />
f f ′′ <br />
<br />
<br />
f 1 × 1 f ′<br />
f <br />
f ′ f ′′ <br />
f<br />
f<br />
x<br />
x<br />
f ′<br />
f ′<br />
x<br />
x<br />
f ′′<br />
f ′′<br />
x<br />
x<br />
f f ′ f ′′
f <br />
<br />
f <br />
f ′′ <br />
<br />
f <br />
<br />
<br />
<br />
y x 2 y e x <br />
y −x 2 y −e x <br />
<br />
<br />
<br />
<br />
<br />
f (x) <br />
<br />
f (4) <br />
f ′ (4) <br />
f ′′ (4)
f <br />
f ′ f ′′ <br />
f f ′ f ′′ <br />
<br />
f <br />
<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
<br />
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<br />
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<br />
<br />
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<br />
<br />
<br />
f ′<br />
f ′′<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
<br />
<br />
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<br />
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<br />
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<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t <br />
v(t) <br />
<br />
<br />
<br />
<br />
P(t) t <br />
P(t)
P(t) <br />
P(t) <br />
<br />
P(t) <br />
P(t) <br />
f ′ f <br />
f ′ f <br />
x <br />
f (x) x <br />
f (x) x <br />
f ′ (x) x <br />
f ′ (x) x <br />
f ′′ (x) x <br />
f ′′ (x) x <br />
y f (x) <br />
f (2) −3 f ′ (2) 1.5 f ′′ (2) −0.25<br />
f x 2 f x 2<br />
f (2.1) −3 −3 −3 <br />
f ′ (2.1) 1.5 1.5 1.5 <br />
y f (x) (2, f (2)) <br />
y (x)
y (x) <br />
(2, (2))<br />
<br />
(x) 0<br />
<br />
<br />
<br />
′ <br />
−3 < x < 3 <br />
<br />
<br />
<br />
′′ (2)<br />
y = g ′ (x)<br />
4<br />
2<br />
-3 -1 1 3<br />
y ′ (x)<br />
h t <br />
t 0.0 0.5 1.0 1.5 <strong>2.0</strong> 2.5 3.0 3.5 4.0 4.5 5.0<br />
h(t) 200 184.2 159.9 131.9 104.7 81.8 65.5 56.8 55.5 60.4 69.8<br />
t 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0<br />
h(t) 81.6 93.7 104.4 112.6 117.7 119.4 118.2 114.8 110.0 104.7<br />
h ′ (4.5) h ′ (5) h ′ (5.5) <br />
<br />
h ′′ (5)<br />
h ′′ (5) <br />
<br />
y h(t) <br />
<br />
−3 <<br />
x < 3 <br />
y f (x) f −3 < x < 3 −3 < x < 0 <br />
0 < x < 3<br />
y (x) −3 < x < 3 −3 < x < 0 <br />
0 < x < 3<br />
y h(x) h −3 < x < 3 −3 < x < −1 <br />
−1 < x < 1 1 < x < 3
y p(x) p −3 < x < 0 <br />
0 < x < 3
f L x → a <br />
x a x a<br />
f x a <br />
<br />
f x a <br />
<br />
<br />
<br />
<br />
<br />
x a <br />
<br />
<br />
L x a <br />
f (a) lim x→a f (x) <br />
f ′ (a) <br />
<br />
<br />
<br />
f −4 < x < 4 <br />
<br />
x 2 f <br />
sin( π x<br />
) <br />
a −3, −2, −1, 0, 1, 2, 3 lim x→a f (x)<br />
L <br />
lim x→a f (x) L <br />
<br />
a f <br />
f (a) a f (a) <br />
lim x→a f (x)<br />
a −3, −2, −1, 0, 1, 2, 3 f ′ (a)
f (a, f (a)) a <br />
f ′ (a)<br />
f<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
-2<br />
-3<br />
y f (x)<br />
<br />
f L x<br />
a lim x→a f (x) L f (x) <br />
L x a <br />
<br />
<br />
<br />
f a 1<br />
x f <br />
x 1 f f <br />
x f<br />
a 1<br />
f a 1 <br />
f L 1 x a <br />
<br />
lim<br />
x→a − f (x) L 1<br />
f (x) L 1 x <br />
a x < a L 1 f x<br />
a L 2 f x a <br />
lim<br />
x→a + f (x) L 2
f (x) L 2 x <br />
a x > a f <br />
<br />
lim f (x) 2 lim f (x) 3<br />
x→1− x→1 +<br />
f x → 1<br />
<br />
3<br />
f<br />
3<br />
g<br />
2<br />
2<br />
1<br />
1<br />
1<br />
f a 1<br />
a 1<br />
a 1 <br />
x <br />
a 1 <br />
x → 1 + lim x→1 (x) .<br />
<br />
<br />
f L x → a <br />
lim f (x) L lim f (x).<br />
x→a− x→a +<br />
x a <br />
x a <br />
f <br />
a −2 −1 <br />
x 2 lim x→2 + f (x) a −1 <br />
a 0
3(x + 2) + 2 −3 < x < −2<br />
2<br />
3<br />
(x + 2) + 1 −2 ≤ x < −1<br />
⎧⎪<br />
f (x) ⎨ 2(x + 2) + 1 −1 < x < 1<br />
2 x 1<br />
⎪<br />
⎩4 − x x > 1<br />
<br />
3<br />
3<br />
2<br />
1<br />
-2 -1 1 2<br />
-1<br />
y f (x) <br />
a −2, −1, 0, 1, 2 f (a)<br />
a −2, −1, 0, 1, 2 lim x→a − f (x) lim x→a + f (x)<br />
a −2, −1, 0, 1, 2 lim x→a f (x) <br />
<br />
a<br />
a <br />
lim f (x) f (a)<br />
x→a<br />
y f (x)<br />
◦ •
a 1 <br />
3<br />
f<br />
3<br />
g<br />
3 h<br />
2<br />
2<br />
2<br />
1<br />
1<br />
1<br />
f h a 1<br />
f (1) f a 1 <br />
f a 1 <br />
lim x→1 (x) 3 (1) 2 <br />
<br />
a 1 h <br />
a 1 <br />
lim h(x) 3 h(1).<br />
x→1<br />
h a 1 h <br />
<br />
f x a <br />
f x → a<br />
f x a <br />
lim x→a f (x) f (a).<br />
<br />
<br />
x a x → a <br />
x a [a, b] <br />
[a, b]
p(x) x 2 − 2x + 3 <br />
lim x→2 p(x)<br />
<br />
lim<br />
x→2 (x2 − 2x + 3) 2 2 − 2 · 2 + 3 3.<br />
<br />
<br />
<br />
<br />
<br />
f <br />
a lim x→a f (x) <br />
a f (a) <br />
a f lim x→a f (x) f (a)<br />
a f <br />
x a<br />
<br />
f <br />
x a f <br />
x a <br />
<br />
f x a f<br />
x a <br />
<br />
<br />
<br />
f<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
-2<br />
-3<br />
y f (x) <br />
<br />
<br />
f x a f ′ (a) <br />
f ′ (a) y f (x) <br />
(a, f (a)) f ′ (a) f <br />
(a, f (a)) f x a f
x a f (a) f (a) lim x→a f (x) <br />
<br />
f x a <br />
x a f x a <br />
f <br />
f x 1 <br />
x 1 <br />
<br />
<br />
f <br />
f a 1 lim x→1 f (x) 1 f (1).<br />
f<br />
1<br />
(1,1)<br />
1<br />
f a 1 a 1 <br />
(1, 1) <br />
f a 1 f ′ (1) <br />
f ′ (x) −1 x <br />
f ′ (x) +1 x +1 <br />
−1 x 1 <br />
f ′ (1) <br />
<br />
f <br />
(1, 1) f <br />
<br />
<br />
<br />
<br />
<br />
<br />
x a<br />
(a, f (a))
x a <br />
<br />
<br />
f x a f x a f <br />
x a f x a<br />
<br />
f x a <br />
(a, f (a))<br />
f x a f x a <br />
<br />
<br />
<br />
(x) |x| <br />
f <br />
<br />
<br />
x<br />
x 0<br />
<br />
′ (0) <br />
|h|<br />
lim h→0 h .<br />
′ (0) <br />
<br />
h<br />
a f <br />
x a <br />
<br />
<br />
f<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
-2<br />
-3<br />
y f (x) <br />
<br />
p x b lim x→b p(x)
f L x → a f x a <br />
x a <br />
x a <br />
x a<br />
f x a f (a) f x → a<br />
<br />
f x a<br />
f x a f ′ (a) f <br />
(a, f (a)) f x a <br />
(a, f (a)) <br />
(a, f (a))<br />
x a <br />
x a x a <br />
f <br />
x a f x a f x a<br />
f x a<br />
<br />
f (x) <br />
<br />
lim f (x) <br />
x→−2<br />
lim f (x) <br />
x→0<br />
lim<br />
x→2<br />
f (x) <br />
lim<br />
x→3<br />
f (x)
2x − 5, 0 ≤ x < 3<br />
⎧⎪<br />
f (x) ⎨4, x 3<br />
⎪<br />
⎩x 2 − 6x + 10, 3 < x<br />
<br />
lim<br />
x→3 +<br />
lim<br />
x→3 −<br />
f (x) <br />
f (x) <br />
lim f (x) <br />
x→3<br />
<br />
f (x) <br />
<br />
x x <br />
x x <br />
x x <br />
k <br />
k <br />
⎧⎪<br />
f (x) ⎨<br />
kx 0 ≤ x < 2<br />
⎪<br />
⎩9x 2 2 ≤ x.<br />
y p(x) <br />
p
p<br />
3<br />
3<br />
-3 3<br />
-3 3<br />
-3<br />
-3<br />
y p(x) y <br />
p ′ (x)<br />
a lim x→a p(x) <br />
a p a<br />
a p x a<br />
y p ′ (x)<br />
<br />
<br />
<br />
f a 2 a 2<br />
a 3 a 3<br />
h a −2 a −2 <br />
a −2<br />
p <br />
p(−1) 3 lim x→−1 p(x) 2<br />
p(0) 1 p ′ (0) 0<br />
lim x→1 p(x) p(1) p ′ (1) <br />
h(x) y h ′ (x) <br />
<br />
y h ′ (x) <br />
y h(x)
x y h ′ (x) <br />
y h(x)<br />
y h(x) h <br />
<br />
<br />
y h(x) y h ′ (x) <br />
y h(x)<br />
3<br />
3<br />
2<br />
1<br />
y = h ′ (x)<br />
-3 3<br />
-3 -2 -1 1 2 3<br />
-1<br />
-2<br />
-3<br />
-3<br />
y h(x) y h ′ (x)<br />
(x) √ |x|<br />
x 0<br />
<br />
′ (0) lim<br />
h→0<br />
√<br />
|h|<br />
h .<br />
′ (0) <br />
√<br />
|−0.01|<br />
h <br />
0.01<br />
<br />
h <br />
′ (0)<br />
y ′ (x)
y f (x) (a, f (a))<br />
<br />
f (a, f (a))<br />
<br />
<br />
<br />
<br />
<br />
y f (x) (a, f (a)) <br />
y f (x) <br />
f L <br />
f f<br />
<br />
<br />
f x a f ′ (a) <br />
y f (x) (a, f (a)) <br />
<br />
<br />
<br />
y (x) −x 2 + 3x + 2<br />
y ′ (x)<br />
y (x) x 2<br />
(2)<br />
y (x) (2, (2)) <br />
<br />
y (x)<br />
(2, (2))
y (x) (2, (2))<br />
m (x 0 , y 0 ) y − y 0 m(x − x 0 ) <br />
<br />
<br />
f x a <br />
y f (x) (a, f (a)) f ′ (a) <br />
(a, f (a)) m f ′ (a) <br />
y − f (a) f ′ (a)(x − a),<br />
y f ′ (a)(x − a) + f (a) <br />
f (a) f (x) <br />
a <br />
f ′ (a) f ′ (x) <br />
<br />
a<br />
f <br />
(a, f (a)) f
y<br />
y = f (x)<br />
y = f (x)<br />
(a, f (a))<br />
(a, f (a))<br />
y = f ′ (a)(x − a)+ f (a)<br />
y = L(x)<br />
a<br />
x<br />
y f (x) (a, f (a)) <br />
y L(x) <br />
x a f (x) ≈ L(x)<br />
<br />
<br />
y f (x) (a, f (a)) f x a <br />
y <br />
y f ′ (a)(x − a) + f (a)<br />
x y <br />
L(x) <br />
L(x) f ′ (a)(x − a) + f (a)<br />
f (a, f (a)) <br />
L(x) x <br />
a f (x) ≈ L(x)<br />
y f (x) <br />
L(x) 3 − 2(x − 1) (1, 3) <br />
f f (x) x f (1.2) <br />
f (1.2) ≈ L(1.2) <br />
f (1.2) ≈ L(1.2) 3 − 2(1.2 − 1) 3 − 2(0.2) 2.6.<br />
y L(x)
L(x) f (a) + f ′ (a)(x − a) L(x) ≈ f (x) x a <br />
<br />
f (x) ≈ f (a) + f ′ (a)(x − a) x a.<br />
<br />
y L(x) f a<br />
y (x) <br />
a −1 L(x) −2 + 3(x + 1)<br />
L(−1) L ′ (−1)<br />
(−1) ′ (−1) <br />
(−1.03) <br />
(−1) <br />
(−1.03)<br />
′′ (−1) 2. <br />
y (x) a −1<br />
x −1 y L(x) <br />
y (x) <br />
y L(x) y (x)<br />
y L(x) <br />
y f (x) <br />
L(x) f (a) + f ′ (a)(x − a) L(a) f (a)<br />
L L ′ (x) f ′ (a) <br />
x L ′ (a) f ′ (a) L <br />
f (a, f (a))
y L(x) <br />
<br />
f <br />
<br />
<br />
<br />
<br />
f ′′ (a) < 0 f ′′ (a) 0 f ′′ (a) > 0 f ′′ (a) > 0 f <br />
<br />
f ′′ (a) < 0 <br />
f <br />
f ′′ (a) 0 f ′′ x a <br />
<br />
f f (x) L(x) x<br />
<br />
<br />
<br />
f <br />
f ′′ (a) > 0 <br />
L(x) ≤ f (x) x a<br />
<br />
f (x) <br />
<br />
f x<br />
f (2) −1<br />
y f ′ (x) <br />
f ′′ (a) 0 f ′′ x a
2<br />
2<br />
y = f ′ (x)<br />
2<br />
2<br />
x<br />
2<br />
x<br />
2<br />
x<br />
y f ′ (x) y f (x) <br />
y f ′′ (x)<br />
f <br />
a 2 <br />
L(x) f (2, −1)<br />
f (<strong>2.0</strong>7) <br />
<br />
y f ′′ (x) <br />
<br />
y f (x) <br />
x 2 <br />
y f (x) x 2 <br />
y L(x) <br />
y f (x) <br />
f (<strong>2.0</strong>7) <br />
<br />
<br />
<br />
<br />
<br />
sin(x)<br />
lim<br />
x→0 x<br />
<br />
sin(x)<br />
x<br />
<br />
f (x) sin(x) (0, 0) L(x) x x sin(x) ≈ x<br />
x <br />
sin(x)<br />
≈ x x x 1,
sin(x)<br />
lim<br />
x→0 x<br />
1.<br />
<br />
<br />
y f (x) (a, f (a)) <br />
<br />
y − f (a) f ′ (a)(x − a).<br />
<br />
y f (x) <br />
<br />
y L(x) L(x) f (a) + f ′ (a)(x − a)<br />
f (x) ≈ L(x) x x a<br />
L(x) f (a) + f ′ (a)(x − a) <br />
L(a) f (a) L ′ (a) f ′ (a) <br />
y f (x) x a <br />
<br />
f ′′ (a) <br />
y f (x) f <br />
<br />
√ 25.3 <br />
f (x) √ x f (x) x 25 <br />
y mx + b m b<br />
m <br />
b <br />
√ 25.3<br />
<br />
f (x) x a y 2x − 3
a<br />
a <br />
f (a)<br />
f (a) <br />
f (3.1)<br />
f (3.1) <br />
<br />
<br />
f (x) f (125) 76 f ′ (125) 8 f (128.5)<br />
f (128.5) <br />
H <br />
H f (t) t <br />
f ′ (t) <br />
<br />
f ′ (25)<br />
<br />
| f ′ (25)| 0.6 f (25) 65 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y p(x) a 3 L(x) −2x + 5<br />
p(3) p ′ (3) <br />
p(2.79)<br />
p ′′ (3) 0 p ′′ (x) < 0 x < 3 <br />
<br />
p ′′ (x) > 0 x > 3 <br />
y p(x) x 3 y L(x)
F <br />
t <br />
<br />
t F(t)<br />
0 70<br />
15 180.5<br />
30 251<br />
45 296<br />
60 324.5<br />
75 342.8<br />
90 354.5<br />
<br />
F ′ (60) <br />
<br />
y L(t) y F(t) a 60<br />
F(63) <br />
<br />
y s(t)<br />
s(t) t <br />
t 9 s(9) 4 <br />
t 9 −1.2 <br />
0.08 <br />
t 9.34<br />
<br />
t 9 <br />
<br />
f f ′ (x) (x − 1)e −x2 <br />
y f (x)<br />
x f ′ (x) 0 <br />
f ′ <br />
x f ′′ (x) > 0 <br />
f <br />
f (2) −3 f (1.88) <br />
f x 2 <br />
f (1.88)
f (x) <br />
f ′ (x)<br />
f (x) x n <br />
f (x) a x <br />
y f (x) y kf(x) <br />
k <br />
y f (x) y (x) y <br />
f (x) + (x) <br />
<br />
f ′ f f <br />
x y f (x) x <br />
<br />
<br />
<br />
<br />
<br />
f ′ f (x + h) − f (x)<br />
(x) lim<br />
,<br />
h→0 h<br />
f ′ (x) <br />
f (x) <br />
f (x) x f ′ (x) 1 <br />
f (x)
x <br />
(x) 4x 7 − sin(x) + 3e x <br />
′ (x)<br />
f (x) x n n 1, 2, 3,... <br />
<br />
x<br />
f ′ (x) f (x) x 2 <br />
f ′ (x) f (x) x 3 <br />
f ′ (x) f (x) x 4 (a+b) 4 <br />
a 4 +4a 3 b +6a 2 b 2 +4ab 3 + b 4 (x + h) 4 <br />
<br />
f (x) x 5 f (x) x 13 <br />
f (x) x n <br />
n f (x) x n n <br />
f ′ (x)<br />
<br />
f ′ <br />
<br />
f (x) x 2 f ′ (x) <br />
f ′ (x) 2x y<br />
x y x <br />
dy<br />
dx<br />
<br />
Δy<br />
Δy<br />
Δx<br />
<br />
Δx<br />
<br />
y x dy<br />
dx <br />
y x 2 dy<br />
dx 2x.<br />
dy<br />
dx<br />
<br />
<br />
d<br />
dx [□]<br />
□ x <br />
d<br />
dx [x2 ] 2x.
x <br />
f (z) z 2 f ′ (z) 2z y t 2 dy<br />
dt<br />
2t <br />
d<br />
dq [q2 ] 2q <br />
[<br />
f ′′ (z) d df<br />
]<br />
d<br />
dz dz 2 f<br />
<br />
dz 2<br />
<br />
<br />
<br />
<br />
f (x) c <br />
d<br />
<br />
dx<br />
[c] 0<br />
<br />
<br />
c f (x) c f ′ (x) 0<br />
f (x) 7 f ′ d<br />
(x) 0 <br />
dx [√ 3] 0.<br />
<br />
n f (x) x n f ′ (x) nx n−1 <br />
n <br />
<br />
<br />
f (x) x n f ′ (x) nx n−1 <br />
(z) <br />
z −3 ′ (z) −3z −4 h(t) t 7/5 dh<br />
dt<br />
7 5 t2/5 d<br />
<br />
dq [qπ ] πq π−1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
a f (x) a x f ′ (x) a x ln(a)<br />
f (x) 2 x f ′ (x) 2 x ln(2) p(t) 10 t <br />
p ′ (t) 10 t ln(10) a e e <br />
<br />
d<br />
dx [e x ] e x ln(e) e x
ln(e) 1 e x <br />
<br />
<br />
x 2 <br />
2 x <br />
<br />
<br />
<br />
<br />
<br />
h(z) <br />
h ′ (z) dh<br />
dz<br />
<br />
f (t) π<br />
p(x) 3 1/2<br />
d<br />
<br />
(z) 7 z<br />
r(t) ( √ dq [q−1 ]<br />
2) t m(t) 1<br />
h(w) w 3/4 t 3<br />
<br />
<br />
<br />
<br />
<br />
p(t) 3t 5 −7t 4 +t 2 −9<br />
t <br />
<br />
y f (x) <br />
y kf(x) <br />
k |k| <br />
y 0 k < 0 <br />
y kf(x) k y f (x) <br />
k <br />
k <br />
<br />
<br />
k f (x) f ′ (x) <br />
d<br />
dx [kf(x)] kf′ (x)
(t) 3 · 5 t ′ (t) 3 · 5 t ln(5)<br />
d<br />
<br />
dz [5z−2 ] 5(−2z −3 )<br />
y f (x)<br />
y (x) y ( f + )(x) <br />
( f + )(x) f (x) + (x) <br />
<br />
<br />
<br />
<br />
f (x) (x) f ′ (x) ′ (x) <br />
d<br />
dx [ f (x) + (x)] f ′ (x) + ′ (x)<br />
<br />
<br />
y ( f − )(x) <br />
f (x) − (x) y f (x) + (−1 · (x)) <br />
<br />
d<br />
dx [ f (x) + (−1 · (x))] f ′ (x) − ′ (x) <br />
<br />
<br />
d<br />
dw (2w + w 2 ) 2 w ln(2) + 2w h(q) <br />
3q 6 − 4q −3 h ′ (q) 3(6q 5 ) − 4(−3q −4 ) 18q 5 + 12q −4 <br />
<br />
<br />
<br />
<br />
<br />
<br />
f ′ (x)<br />
h ′ (z) dr/dt <br />
f (x) x 5/3 − x 4 + 2 x<br />
s(y) (y 2 + 1)(y 2 − 1)<br />
(x) 14e x + 3x 5 − x<br />
h(z) √ q(x) x3 −x+2<br />
x<br />
z + 1 + 5 z<br />
z 4 p(a) 3a 4 − 2a 3 + 7a 2 − a + 12<br />
r(t) √ 53 t 7 − πe t + e 4<br />
<br />
f
f <br />
f <br />
f <br />
<br />
<br />
<br />
<br />
p(t) 7t 5 − 4t 3 + 8t p <br />
p ′ (t) 35t 4 − 12t 2 + 8 <br />
<br />
(z) 7 · 2 z ′ (z) 7 · 2 z ln(2) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
h(z) √ z + 1 z<br />
z 4<br />
<br />
P(t) 2(1.37) t + 32 t <br />
<br />
<br />
<br />
<br />
p(a) 3a 4 − 2a 3 + 7a 2 − a + 12 <br />
a −1<br />
<br />
<br />
<br />
y f (x) f <br />
f ′ (x) df<br />
dx dy<br />
dx d<br />
dx [ f (x)]
f f (x) x n <br />
f ′ (x) nx n−1 n <br />
<br />
<br />
a f (x) a x f ′ (x) <br />
a x ln(a)<br />
<br />
<br />
<br />
f (x) (x) f ′ (x) ′ (x) a b <br />
<br />
d af(x) + b(x) af ′ (x) + b ′ (x).<br />
dx<br />
<br />
y x 15/16 <br />
dy<br />
dx <br />
f (x) 1<br />
x 17 <br />
f ′ (x) <br />
<br />
y 9 √ x<br />
dy<br />
dx <br />
f (t) 2t 2 − 2t + 15<br />
f ′ (t) <br />
y 4t 12 − 7 √ t + 4 t <br />
dy<br />
dt<br />
<br />
y √ x(x 2 + 4)<br />
dy<br />
dx <br />
y x5 + 2<br />
<br />
x<br />
dy<br />
dx
f (2, 27) f <br />
f (x) 4x 3 − 4x 2 + 11<br />
y <br />
f (x) x 3 + 3x 2 − 189x + 1 x f ′ (x) 0 <br />
<br />
x <br />
f <br />
f (2) 5 (2) −3 f ′ (2) −1/2 ′ (2) 2<br />
h h(x) 3 f (x) − 4(x) h(2) <br />
h ′ (2)<br />
y h(x) (2, h(2))<br />
p p(x) −2 f (x) + 1 2<br />
(x) p <br />
a 2 <br />
p(<strong>2.0</strong>3) p (2, p(2))<br />
p q <br />
<br />
p<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
q<br />
-2<br />
-3<br />
p q <br />
x p x q <br />
<br />
r(x) p(x) + 2q(x) x r <br />
r ′ (−2) r ′ (0)
y r(x) (2, r(2))<br />
r(t) t t s(t) arccos(t) <br />
r ′ (t) t t (ln(t)+1) s ′ (t) − √ 1 <br />
1−t 2<br />
0 < t < 1<br />
w(t) 3t t − 2 arccos(t) w ′ (t)<br />
y w(t) ( 1 2 , w( 1 2 ))<br />
v(t) t t + arccos(t) v t 1 2 <br />
f (x) a x a <br />
f (x) d<br />
dx [ax ] <br />
<br />
<br />
<br />
f ′ a<br />
(x) x · a h − a x<br />
lim<br />
.<br />
h→0 h<br />
f ′ (x) a x a h − 1 · lim .<br />
h→0 h<br />
h <br />
L lim<br />
h→0<br />
a h − 1<br />
h<br />
a 2 a 3<br />
L 1 f <br />
a x <br />
a 2 <br />
3 a <br />
a h − 1<br />
L lim 1.<br />
h→0 h<br />
ln(2) ln(3) <br />
d<br />
dx [2x ] d<br />
dx [3x ]<br />
<br />
f (x) e x
d<br />
dx [ax ] a x ln(a)<br />
y sin(x) y cos(x) <br />
<br />
sin(x) cos(x) <br />
<br />
<br />
<br />
f ′ (x) f (x) <br />
f (x) x n f ′ (x) nx n−1 <br />
n <br />
a f (x) a x <br />
f ′ (x) a x ln(a) <br />
<br />
<br />
<br />
<br />
(x) 2 x <br />
<br />
x −2, −1, 0, 1, 2 <br />
y (x)<br />
<br />
<br />
′ (0) <br />
h <br />
y (x) x 0 <br />
y ′ (x) <br />
y (x)<br />
′ (x) <br />
c(x) c ln(2) <br />
′ (x) 2 x ln(2)
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
-2 -1 1 2<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
-2 -1 1 2<br />
y (x) 2 x y ′ (x)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
d<br />
d<br />
dx<br />
[sin(x)] <br />
dx<br />
[cos(x)] <br />
<br />
f (x) sin(x) <br />
1 × 1 <br />
1.57 × 1 π/2 <br />
−2π −π π<br />
-1<br />
1<br />
2π<br />
1<br />
−2π −π π 2π<br />
-1<br />
y f (x) sin(x)<br />
x −2π, − 3π 2 , −π, − π 2 , 0, π 2 ,π,3π 2<br />
, 2π,
y f (x)<br />
<br />
<br />
f ′ (0) <br />
h <br />
y f (x) x 0 <br />
f ′ (2π) f ′ (−2π)<br />
y f ′ (x) <br />
y f (x)<br />
f (x) sin(x)<br />
(x) cos(x) <br />
1 × 1 <br />
1.57 × 1 π/2 <br />
1<br />
1<br />
−2π −π π<br />
-1<br />
2π −2π −π π<br />
-1<br />
2π<br />
y (x) cos(x)<br />
x −2π, − 3π 2 , −π, − π 2 , 0, π 2 ,π, 3π 2<br />
, 2π, <br />
y (x)<br />
<br />
<br />
′ ( π 2<br />
) <br />
h <br />
y (x) x π 2<br />
<br />
′ (− 3π 2<br />
) <br />
y ′ (x) <br />
y (x)<br />
(x) cos(x)<br />
<br />
<br />
sin 2 (x) + cos 2 (x) 1 cos(x − π 2<br />
) sin(x)
x<br />
d<br />
[sin(x)] cos(x) <br />
d<br />
dx<br />
[cos(x)] − sin(x).<br />
dx<br />
<br />
<br />
<br />
<br />
f (x) sin(x) <br />
cos(x) <br />
<br />
<br />
h(t) 3 cos(t) − 4 sin(t)<br />
y f (x) 2x + sin(x)<br />
2<br />
<br />
x π 6 <br />
y (x) x 2 + 2 cos(x) <br />
x π 2 <br />
p(z) z 4 + 4 z + 4 cos(z) − sin( π 2 )<br />
P(t) 24 + 8 sin(t) <br />
P t <br />
<br />
P <br />
<br />
<br />
<br />
f (x) a x (a > 1) <br />
f ′ (x) <br />
a x f (x) 2 x <br />
d<br />
dx [2x ] 2 x ln(2) <br />
y sin(x) y cos(x) <br />
d<br />
<br />
dx<br />
[sin(x)] cos(x) <br />
<br />
sin(x + h) cos(x + h)
d<br />
dx<br />
[cos(x)] − sin(x)<br />
<br />
<br />
<br />
<br />
V(t) 24 · 1.07 t + 6 sin(t) <br />
t t 0 <br />
<br />
<br />
V ′′ (2) <br />
<br />
0 ≤ t ≤ 20 V(t) 24 · 1.07 t + 6 sin(t) <br />
<br />
A(t) 24 · 1.07 t V(t) 24 · 1.07 t + 6 sin(t) <br />
A ′ (t) V ′ (t) 6 sin(t) <br />
V(t)<br />
f (x) 3 cos(x) − 2 sin(x) + 6<br />
y f (x) a π 4 <br />
y f (x) a π<br />
a π 2<br />
f <br />
a 3π 2<br />
y f (x) <br />
<br />
<br />
<br />
d<br />
dx<br />
[sin(x)] cos(x) f (x) sin(x) <br />
<br />
f ′ sin(x + h) − sin(x)<br />
(x) lim<br />
.<br />
h→0 h<br />
α β sin(α + β) <br />
sin(α) cos(β) + cos(α) sin(β) <br />
f ′ sin(x)(cos(h) − 1) + cos(x) sin(h)<br />
(x) lim<br />
.<br />
h→0 h<br />
h x <br />
<br />
f ′ cos(h) − 1<br />
sin(h)<br />
(x) sin(x) · lim + cos(x) · lim .<br />
h→0 h<br />
h→0 h
h <br />
cos(h) − 1<br />
lim<br />
h→0 h<br />
sin(h)<br />
lim .<br />
h→0 h<br />
f ′ (x)<br />
<br />
d<br />
dx<br />
[cos(x)] − sin(x).
x n <br />
a x sin(x) cos(x)<br />
<br />
<br />
f (x) 7x 11 − 4 · 9 x + π sin(x) − √ 3 cos(x),<br />
f <br />
f ′ (x) 77x 10 − 4 · 9 x ln(9) + π cos(x) + √ 3 sin(x)<br />
<br />
<br />
<br />
p(z) z 3 cos(z),<br />
<br />
q(t) sin(t)<br />
2 t .<br />
<br />
<br />
<br />
<br />
<br />
f f (t) 2t 2 (t) <br />
t 3 + 4t<br />
f ′ (t) ′ (t)
p(t) 2t 2 (t 3 + 4t) p(t) f (t) · (t) <br />
p 2t 2 p ′ (t) <br />
<br />
p ′ (t) f ′ (t) · ′ (t)<br />
q(t) t3 +4t<br />
2t 2<br />
q(t) (t)<br />
f (t)<br />
q <br />
q <br />
t q ′ (t) <br />
<br />
q ′ (t) ′ (t)<br />
f ′ (t) <br />
<br />
<br />
<br />
p(x) f (x) · (x) <br />
f f <br />
<br />
<br />
N(t) <br />
t t 0 <br />
S(t) <br />
t S(t) <br />
t <br />
V(t) N(t) · S(t) <br />
V(t) N(t) · S(t) ,<br />
<br />
N ′ (t) <br />
S ′ (t) <br />
<br />
<br />
N S V <br />
<br />
N(100) 520 S(100) 27.50 <br />
<br />
<br />
<br />
N ′ (100) 12 S ′ (100) 0.75
S(100) · N ′ (100) 27.50 <br />
<br />
· 12<br />
<br />
<br />
<br />
330<br />
.<br />
<br />
V <br />
<br />
<br />
<br />
N(100) · S ′ <br />
(100) 520 · 0.75<br />
<br />
390 <br />
.<br />
<br />
<br />
<br />
V ′ (100) S(100) · N ′ (100) + N(100) · S ′ (100) 330 + 390 720 <br />
.<br />
<br />
<br />
<br />
p f P(x) <br />
f (x) · (x) P ′ (x) f (x) ′ (x) + (x) f ′ (x). <br />
<br />
<br />
<br />
f P(x) f (x) · (x) <br />
<br />
P ′ (x) f (x) ′ (x) + (x) f ′ (x).<br />
<br />
P f <br />
f <br />
<br />
<br />
V(101) N(101) · S(101) 532 · 28.25 15029 <br />
V(100) N(100) · S(100) 520 · 27.50 14300 <br />
V(101) V(100) + 720 15020. <br />
N ′ (100) 12
P f <br />
P <br />
<br />
<br />
P(z) z 3 · cos(z) P <br />
z 3 cos(z) P ′ <br />
z 3 − sin(z) cos(z) <br />
3z 2 <br />
P ′ (z) z 3 (− sin(z)) + cos(z)3z 2 −z 3 sin(z) + 3z 2 cos(z).<br />
<br />
<br />
<br />
<br />
m(w) 3w 17 4 w m ′ (w)<br />
h(t) (sin(t) + cos(t))t 4 h ′ (t)<br />
y f (x) <br />
a 1 f f (x) e x sin(x)<br />
L(x) y (x) <br />
a −1 (x) (x 2 + x)2 x <br />
<br />
<br />
Q(x) Q(x) <br />
f (x)/(x) f Q ′ <br />
f f ′ ′ Q (x) 0<br />
Q f / <br />
f (x) Q(x) · (x).<br />
f <br />
f ′ (x) Q(x) ′ (x) + (x)Q ′ (x).<br />
Q ′ <br />
Q ′ (x) <br />
Q ′ (x)(x) f ′ (x) − Q(x) ′ (x).
(x)<br />
Q ′ (x) f ′ (x) − Q(x) ′ (x)<br />
.<br />
(x)<br />
Q(x) f (x)<br />
(x)<br />
. <br />
<br />
Q ′ (x) <br />
<br />
f ′ (x) − f (x)<br />
(x) ′ (x)<br />
(x)<br />
f ′ (x) − f (x)<br />
(x) ′ (x)<br />
(x)<br />
· (x)<br />
(x)<br />
(x) f ′ (x) − f (x) ′ (x)<br />
(x) 2 .<br />
<br />
<br />
f Q(x) f (x)<br />
(x)<br />
x ′ (x) 0 <br />
Q ′ (x) (x) f ′ (x) − f (x) ′ (x)<br />
(x) 2 .<br />
<br />
Q<br />
f Q ′ <br />
<br />
Q(t) sin(t)/2 t <br />
sin(t) 2 t Q ′ <br />
2 t cos(t) sin(t) <br />
2 t ln(2) (2 t ) 2 <br />
Q ′ (t) 2t cos(t) − sin(t)2 t ln(2)<br />
(2 t ) 2 .<br />
Q ′ (t) 2 t <br />
<br />
Q ′ (t) <br />
cos(t) − sin(t) ln(2)<br />
2 t .
f (x) f ′ (x) <br />
<br />
r(z) <br />
v(t) <br />
3z<br />
z 4 +1 r′ (z)<br />
sin(t)<br />
cos(t)+t 2 v ′ (t)<br />
R(x) x2 − 2x − 8<br />
x 2 <br />
− 9<br />
x 0<br />
I <br />
<br />
I(t) 100t<br />
e t ,<br />
I t I ′ (0.5)<br />
I ′ (2) I ′ (5) <br />
<br />
<br />
<br />
<br />
<br />
x<br />
f (x) x sin(x) 2<br />
+<br />
cos(x) + 2 .<br />
<br />
f <br />
<br />
f ′ (x) d<br />
dx<br />
[<br />
x 2<br />
x sin(x) +<br />
cos(x) + 2<br />
[<br />
d<br />
d<br />
[x sin(x)] +<br />
dx dx<br />
]<br />
x 2<br />
cos(x) + 2<br />
<br />
<br />
<br />
f ′ (x) (x cos(x) + sin(x)) + (cos(x) + 2)2x − x2 (− sin(x))<br />
(cos(x) + 2) 2<br />
<br />
d<br />
dx [] <br />
]
x cos(x) + sin(x) + 2x cos(x) + 4x2 + x 2 sin(x)<br />
(cos(x) + 2) 2 .<br />
<br />
s(y) y · 7y<br />
y 2 + 1 .<br />
s <br />
d<br />
dy<br />
<br />
<br />
<br />
(y 2 + 1) · <br />
d<br />
s ′ dy y · 7<br />
y<br />
− y · 7 y d ·<br />
dy y 2 + 1 <br />
(y) <br />
(y 2 + 1) 2 .<br />
<br />
d<br />
dy y · 7<br />
y<br />
<br />
<br />
d<br />
<br />
dy y 2 + 1 <br />
<br />
s ′ (y) (y2 + 1)[y · 7 y ln(7) + 7 y · 1] − y · 7 y [2y]<br />
(y 2 + 1) 2 .<br />
s ′ (y) <br />
s s <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f (r) (5r 3 + sin(r))(4 r − 2 cos(r)) f ′ (r)<br />
p(t) cos(t)<br />
t 6 · 6 t p′ (t)<br />
(z) 3z 7 e z − 2z 2 sin(z) +<br />
z<br />
z 2 +1 . ′ (z)<br />
t <br />
s(t) 3 cos(t)−sin(t) t 1<br />
e t<br />
f (x) (x) f (3)
−2 f ′ (3) 7 (3) 4 ′ (3) −1 p(x) f (x) · (x) q(x) f (x)<br />
(x) <br />
p ′ (3) q ′ (3)<br />
<br />
<br />
f f ′ (a) <br />
f x x a y f (x)<br />
(a, f (a))<br />
<br />
<br />
<br />
<br />
P f <br />
P(x) f (x)(x) <br />
P ′ (x) f (x) ′ (x) + (x) f ′ (x).<br />
Q f <br />
Q(x) f (x)<br />
(x) <br />
Q ′ (x) (x) f ′ (x) − f (x) ′ (x)<br />
(x) 2 .<br />
<br />
<br />
<br />
F <br />
2a(x) − 5b(x)<br />
F(x) ,<br />
c(x) · d(x)<br />
F <br />
F <br />
<br />
<br />
<br />
f (x) <br />
<br />
f (x) x · 12 x<br />
f ′ (x)
f (x) <br />
<br />
f (x) (x 8 − 7 √<br />
x)8<br />
x<br />
f ′ (x) <br />
z <br />
dz<br />
dt <br />
z 9t + 4<br />
4t + 5<br />
h(r) <br />
<br />
h ′ (r) <br />
h(r) <br />
r 2<br />
15r + 11<br />
s(q) 4 cos q sin q<br />
s ′ (q) <br />
f (x) x 4 cos x<br />
f ′ (x) <br />
h(t) t sin t + cos t<br />
h ′ (t) <br />
h(x) f (x) · (x) k(x) f (x)/(x) <br />
<br />
f (x)<br />
(x)
h ′ (0) <br />
k ′ (−2) <br />
<br />
F(4) 2, F ′ (4) 4, H(4) 3, H ′ (4) 1<br />
G(z) F(z) · H(z) G ′ (4) <br />
G(w) F(w)/H(w) G ′ (4) <br />
f <br />
f (2) 5 (2) −3 f ′ (2) −1/2 ′ (2) 2<br />
h h(x) (x) · f (x) h(2) <br />
h ′ (2)<br />
y h(x) (2, h(2)) h <br />
<br />
r r(x) (x)<br />
f (x)<br />
r <br />
a 2 <br />
r(<strong>2.0</strong>6) r <br />
r (2, r(2))<br />
r(t) t t s(t) arccos(t) <br />
r ′ (t) t t (ln(t)+1) s ′ (t) − √ 1 <br />
1−t 2<br />
0 < t < 1<br />
w(t) t t arccos(t) w ′ (t)<br />
y w(t) ( 1 2 , w( 1 2 ))<br />
v(t) <br />
t t<br />
arccos(t) v t 1 2 <br />
p q
(x) p(x)·q(x) r ′ (−2)<br />
r ′ (0)<br />
x r ′ (x)<br />
<br />
<br />
<br />
y r(x) (2, r(2))<br />
z(x) q(x)<br />
p(x) z′ (0) <br />
z ′ (2)<br />
x z ′ (x)<br />
<br />
<br />
p<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
q<br />
-2<br />
-3<br />
p <br />
q <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t 0 A(t) <br />
t Y(t) t <br />
C(t) <br />
C(t) A(t) Y(t) <br />
C(0) <br />
C ′ (t) A(t) A ′ (t) Y(t) Y ′ (t) <br />
<br />
C ′ (0) <br />
C(1)<br />
f (v) v <br />
f (v) <br />
v f (80) 0.05 f ′ (80) <br />
0.0004<br />
(v) v <br />
f (v) (v) (80) ′ (80)
h(v) v <br />
h(v) <br />
v h(v) f (v) h(80)<br />
h ′ (80)
tan(x) cot(x) sec(x) csc(x) <br />
<br />
<br />
<br />
<br />
(x,y)<br />
1<br />
sin(θ)<br />
θ<br />
cos(θ)<br />
<br />
θ (x, y) x<br />
y θ <br />
cos(θ) sin(θ) cos(θ) x <br />
θ sin(θ) y <br />
<br />
sin 2 (θ) + cos 2 (θ) 1,
tan(θ) sin(θ)<br />
cos(θ)<br />
cos(θ)<br />
cot(θ) <br />
sin(θ)<br />
<br />
sec(θ) <br />
cos(θ) 1 <br />
csc(θ) <br />
sin(θ) 1 <br />
<br />
<br />
<br />
y tan(x)<br />
f (x) tan(x) tan(x) <br />
sin(x)<br />
cos(x) <br />
f <br />
f ′ (x) <br />
f ′ (x) <br />
cos(x) cos(x) + sin(x) sin(x)<br />
cos 2 .<br />
(x)<br />
<br />
f ′ (x)<br />
sec(x) <br />
cos(x) 1 f ′ (x) <br />
<br />
x f ′ (x) <br />
f <br />
<br />
<br />
<br />
<br />
(x) cot(x) ′ (x) (x) cos(x)<br />
sin(x)<br />
<br />
<br />
′ sin(x)(− sin(x)) − cos(x) cos(x)<br />
(x) <br />
sin 2 (x)<br />
− sin2 (x) + cos 2 (x)<br />
sin 2 (x)<br />
′ (x) − 1 <br />
sin 2 (x)
csc(x) 1<br />
sin(x) ′ <br />
′ (x) − csc 2 (x).<br />
′ sin(x) 0 <br />
π <br />
<br />
x x kπ k 0, ±1, ±2,...<br />
d<br />
dx [cot(x)] − csc2 (x).<br />
<br />
<br />
<br />
x x (2k+1)π<br />
2<br />
k ±1, ±2,...<br />
d<br />
dx [tan(x)] sec2 (x).<br />
<br />
<br />
h(x) sec(x) sec(x) <br />
cos(x) 1 <br />
h<br />
h ′ (x) <br />
sin(x) cos(x)<br />
h ′ (x)<br />
tan(x) sec(x)<br />
h ′ h<br />
p(x) csc(x) csc(x) <br />
sin(x) 1 <br />
p<br />
p ′ (x) <br />
sin(x) cos(x)<br />
p ′ (x)<br />
cot(x) csc(x)
p ′ p<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
tan(x) cot(x) sec(x) csc(x) <br />
<br />
<br />
<br />
<br />
f (x) 5 sec(x) − 2 csc(x) f <br />
x π 3 <br />
p(z) z 2 sec(z) − z cot(z) p <br />
z π 4 <br />
h(t) tan(t)<br />
t 2 + 1 − 2e t cos(t) h ′ (t)<br />
(r) r sec(r)<br />
5 r ′ (r)<br />
<br />
<br />
<br />
<br />
s(t) <br />
15 sin(t)<br />
e t .<br />
s t <br />
t ≥ 0 <br />
ds/dt <br />
s ′ (2)
d<br />
dx [tan(x)] sec2 (x),<br />
d<br />
dx [cot(x)] − csc2 (x),<br />
d<br />
d<br />
[sec(x)] sec(x) tan(x), [csc(x)] − csc(x) cot(x).<br />
dx dx<br />
<br />
<br />
x x kπ 2<br />
k ±1, ±2,...<br />
<br />
<br />
<br />
<br />
<br />
h(t) t tan t + sin t<br />
h ′ (t) <br />
f (x) 5 tan(x) <br />
x<br />
f ′ (x) <br />
f ′ (4) <br />
f (x) tan(x) − 5 <br />
sec(x)<br />
f ′ (x) <br />
f ′ (1) <br />
f (x) 2x2 tan(x)<br />
<br />
sec(x)<br />
f ′ (x) <br />
f ′ (3) <br />
y 2 tan x (π/4, 2) <br />
y mx+b m <br />
b
t <br />
h(t) 3 + 2 cos(t) <br />
1.2 t<br />
t 2<br />
t 2<br />
t 2<br />
f (x) sin(x) cot(x)<br />
f ′ (x)<br />
x f (x) cos(x)<br />
<br />
<br />
f <br />
f ′ <br />
p(z) <br />
p(z) <br />
z tan(z)<br />
z 2 sec(z) + 1 + 3e z + 1.<br />
p ′ (z)<br />
p z 0<br />
z 0p
C(x) f ((x)) <br />
f C ′ (x) f f ′ ′ <br />
<br />
<br />
<br />
f (x) sin(x)<br />
(x) x 2 <br />
f <br />
s(x) 3x 2 − 5 sin(x),<br />
p(x) x 2 sin(x), <br />
q(x) sin(x)<br />
x 2 .<br />
s ′ p ′ <br />
q ′ <br />
<br />
s(x) 3(x) − 5 f (x) p(x) (x) · f (x) q(x) f (x)<br />
(x) .<br />
<br />
<br />
C(x) sin(x 2 ),<br />
x <br />
C(x) x <br />
<br />
x −→ x 2 −→ sin(x 2 ).<br />
f x <br />
f <br />
C(x) f ((x)) sin(x 2 )<br />
C f <br />
x f <br />
<br />
<br />
C(x) f ((x)) f C ′ (x)
f f ′ ′ <br />
p(x) f (x) · (x) f <br />
C(x) f ((x)) <br />
f <br />
C ′ f <br />
<br />
<br />
C(x) sin(x 2 ).<br />
<br />
<br />
<br />
f <br />
f ((x)) <br />
<br />
h(x) tan(2 x )<br />
p(x) 2 x tan(x)<br />
r(x) (tan(x)) 2<br />
m(x) e tan(x)<br />
w(x) √ x + tan(x)<br />
z(x) √ tan(x)<br />
<br />
<br />
C(x) sin(x 2 ) <br />
<br />
<br />
<br />
<br />
f (x) −4x +7 (x) 3x −5 C(x) f ((x))<br />
C ′ (x) C ′ f <br />
<br />
f <br />
C(x) f ((x))<br />
f (3x − 5)<br />
− 4(3x − 5) + 7<br />
− 12x + 20 + 7<br />
− 12x + 27.<br />
C ′ (x) −12 f ′ (x) −4 ′ (x) 3 C ′ <br />
f ′ ′
f <br />
<br />
<br />
<br />
<br />
C(x) sin(2x). C <br />
C ′ C ′ <br />
<br />
<br />
<br />
C(x) sin(2x) 2 sin(x) cos(x).<br />
C ′ (x) 2 sin(x)(− sin(x)) + cos(x)(2 cos(x)) 2(cos 2 (x) − sin 2 (x)).<br />
<br />
cos(2x) cos 2 (x) − sin 2 (x).<br />
C ′ (x) <br />
C ′ (x) 2 cos(2x).<br />
C(x) sin(2x) C ′ (x) 2 cos(2x) (x) <br />
2x f (x) sin(x) C(x) f ((x)) ′ (x) 2 f ′ (x) <br />
cos(x) C ′ (x) <br />
C ′ (x) 2 cos(2x) ′ (x) f ′ ((x)).<br />
C(x) f ((x)) C ′ <br />
<br />
f f ′ (x) x<br />
<br />
C(x) f ((x)) C<br />
x x x<br />
f (x)
x f (x) C<br />
C(x) f ((x)) x <br />
C ′ (x) f ′ ((x)) ′ (x).<br />
<br />
C f <br />
C ′ <br />
<br />
<br />
f <br />
<br />
<br />
r(x) (tan(x)) 2 .<br />
r (x) tan(x) f (x) x 2 <br />
f <br />
f (x) x 2<br />
f ′ (x) 2x<br />
f ′ ((x)) 2 tan(x)<br />
(x) tan(x)<br />
′ (x) sec 2 (x)<br />
r ′ (x) f ′ ((x)) ′ (x) r(x) <br />
(tan(x)) 2 <br />
r ′ (x) 2 tan(x) sec 2 (x).<br />
r(x) r(x) tan 2 (x) <br />
<br />
cos 4 (x) sin 5 (x) sec 2 (x) <br />
<br />
<br />
<br />
<br />
f f ((x)) f ′ (x) ′ (x) f ′ ((x))<br />
<br />
h(x) cos(x 4 )<br />
p(x) √ tan(x)<br />
s(x) 2 sin(x)<br />
z(x) cot 5 (x)<br />
m(x) (sec(x) + e x ) 9
h(t) 3 t2 +2t sec 4 (t)<br />
h <br />
h(t) a(t) · b(t) a(t) 3 t2 +2t b(t) sec 4 (t) <br />
h a<br />
b <br />
a ′ (t)<br />
b ′ (t)<br />
a(t) f ((t)) 3 t2 +2t f <br />
f (t) 3 t<br />
(t) t 2 + 2t<br />
f ′ (t) 3 t ln(3) ′ (t) 2t + 2<br />
f ′ ((t)) 3 t2 +2t ln(3)<br />
a ′ (t) f ′ ((t)) ′ (t) 3 t2 +2t ln(3)(2t + 2).<br />
b b(t) r(s(t)) sec 4 (t) r <br />
<br />
r(t) t 4<br />
r ′ (t) 4t 3<br />
r ′ (s(t)) 4 sec 3 (t)<br />
s(t) sec(t)<br />
s ′ (t) sec(t) tan(t)<br />
<br />
b ′ (t) r ′ (s(t))s ′ (t) 4 sec 3 (t) sec(t) tan(t) 4 sec 4 (t) tan(t).<br />
h <br />
h(t) 3 t2 +2t sec 4 (t) <br />
h ′ (t) 3 t2 +2t d dt [sec4 (t)] + sec 4 (t) d dt [3t2 +2t ].<br />
a b 3 t2 +2t sec 4 (t) <br />
<br />
h ′ (t) 3 t2 +2t 4 sec 4 (t) tan(t) + sec 4 (t)3 t2 +2t ln(3)(2t + 2).
p(r) 4 √ r 6 + 2e r<br />
m(v) sin(v 2 ) cos(v 3 )<br />
h(y) cos(10y)<br />
e 4y +1<br />
s(z) 2 z2 sec(z)<br />
c(x) sin(e x2 )<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y √ e x + 3 <br />
x 0<br />
1<br />
s(t) <br />
(t 2 <br />
+ 1) 3<br />
t s t <br />
t 1 <br />
<br />
h <br />
P P <br />
30e −0.0000323h dP/dh <br />
dP/dh <br />
<br />
1000 <br />
f (x) (x) <br />
<br />
x f(x) f ′ (x) (x) ′ (x)<br />
−1 2 −5 −3 4<br />
2 −3 4 −1 2<br />
f <br />
C(x) f ((x)) C ′ (2) <br />
D(x) f ( f (x)) D ′ (−1)
d<br />
dx<br />
[sin(x)] cos(x) <br />
d<br />
dx [sin(u(x))],<br />
u x <br />
<br />
d<br />
dx [sin(u(x))] cos(u(x)) · u′ (x).<br />
d<br />
dx [ax ] a x ln(a) <br />
d<br />
dx [au(x) ] a u(x) ln(a) · u ′ (x).<br />
<br />
<br />
<br />
u(x) <br />
<br />
<br />
d (5x + 7)<br />
10 10(5x + 7) 9 · 5,<br />
dx<br />
d<br />
dx [tan(17x)] 17 sec2 (17x), <br />
d e<br />
−3x −3e −3x .<br />
dx<br />
<br />
x <br />
<br />
h(x) 2 sin(x) x −→ sin(x) −→ 2 sin(x) .<br />
C(x) f ((x)) <br />
f C ′ (x) f f ′ ′<br />
<br />
C ′ (x) f ′ ((x)) ′ (x).
f (x) e 4x (x 2 + 3 x )<br />
f ′ (x) <br />
<br />
v(t) t 5 e −ct<br />
c <br />
v ′ (t) <br />
<br />
y √ e −5t2 + 5<br />
dy<br />
dt<br />
<br />
<br />
f (x) axe −bx+10<br />
a b <br />
f ′ (x) <br />
<br />
f (x)<br />
x 2 (x) <br />
h(x) f ((x))
h ′ (1) <br />
h ′ (2) <br />
h ′ (3) <br />
<br />
F(2) 3, F ′ (2) 4, F(4) 1, F ′ (4) 5 G(1) 3, G ′ (1) 4, G(4) 2, G ′ (4) 7<br />
<br />
<br />
H(4) H(x) F(G(x))<br />
H ′ (4) H(x) F(G(x))<br />
H(4) H(x) G(F(x))<br />
H ′ (4) H(x) G(F(x))<br />
H ′ (4) H(x) F(x)/G(x)<br />
f (x) 9x sin(2x)<br />
f ′ (x) <br />
f (x) x 3 (x) sin(x)<br />
h(x) f ((x)) h <br />
x π 4 .<br />
x 0.25 h(x) f ((x)) r(x) ( f (x))<br />
<br />
h(x) f ((x)) r(x) ( f (x)) <br />
<br />
u(x) <br />
u u ′ <br />
p(x) e u(x)<br />
q(x) u(e x )<br />
r(x) cot(u(x))<br />
s(x) u(cot(x))<br />
a(x) u(x 4 )<br />
b(x) u 4 (x)
p q <br />
<br />
p<br />
3<br />
2<br />
1<br />
-3 -2 -1 1 2 3<br />
-1<br />
q<br />
-2<br />
-3<br />
p q <br />
C(x) p(q(x)) C ′ (0) C ′ (3)<br />
x C ′ (x) <br />
Y(x) q(q(x)) Z(x) q(p(x)) Y ′ (−2) Z ′ (0)<br />
h <br />
<br />
V π 3 h2 (12 − h).<br />
<br />
h 1 <br />
<br />
t <br />
h(t) sin(πt) + 1 t h <br />
<br />
t 2<br />
<br />
t 2
arcsin(x) arctan(x)<br />
f ′ f f ′ <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f (x) e x (x) ln(x) <br />
<br />
<br />
<br />
<br />
<br />
y 5 9<br />
(x − 32) x<br />
y <br />
y 5 9<br />
(x − 32) x x <br />
y <br />
C(x) 5 9<br />
(x − 32) <br />
F(y) <br />
y <br />
F(y) <br />
F(y)<br />
p(x) F(C(x)) <br />
F C p(x) <br />
<br />
r(y) C(F(y)) F C <br />
r(y)
C ′ (x) F ′ (y) <br />
<br />
f : A → B A <br />
B A f B f <br />
: B → A ( f (a)) a a A <br />
f ((b)) b b B f <br />
f −1 f f <br />
f y f (x)<br />
<br />
f −1 (y) f −1 ( f (x)) x,<br />
y f (x) x f −1 (y) <br />
<br />
x y<br />
<br />
f : A → B<br />
f f <br />
f −1 f −1 f f −1<br />
f <br />
f −1 ( f (x)) x x f f ( f −1 (y)) y y <br />
f <br />
y f (x) x f −1 (y)<br />
f f −1 <br />
y f (x) (x, y) f <br />
x f −1 (y) (y, x) f −1 <br />
f f −1 y x <br />
y x <br />
y f (x) 2 x <br />
(−1, 1 2 ) ( 1 2<br />
, −1) <br />
y x<br />
f <br />
f
y = f (x)<br />
2<br />
(−1, 1 2 )<br />
-2 2<br />
y = x<br />
-2<br />
( 1 2 ,−1)<br />
y = f −1 (x)<br />
y f (x) y f −1 (x)<br />
<br />
y f (x) e x <br />
x f −1 (y) ln(y) y e x x ln(y) ln(e x ) x<br />
x e ln(y) y y.<br />
<br />
(x) ln(x) <br />
<br />
(x) ln(x) <br />
e (x) x <br />
d<br />
dx<br />
[<br />
e<br />
(x) ] d<br />
dx [x].<br />
1 <br />
e (x) ′ (x) 1.<br />
′ (x) ′ (x) <br />
′ (x) 1<br />
e (x) .<br />
(x) ln(x) e (x) e ln(x) x <br />
′ (x) 1 x .
x<br />
d<br />
dx [ln(x)] 1 x <br />
<br />
<br />
d<br />
dx [ln(x)] 1 x<br />
<br />
x <br />
<br />
<br />
<br />
<br />
f (t) ln(t 2 + 1) f ′ (t) 1<br />
t 2 +1 · 2t<br />
<br />
f (x) e x<br />
f −1 (x) ln(x)<br />
<br />
<br />
y = e x<br />
<br />
8<br />
y e x <br />
<br />
B<br />
A (ln(0.5), 0.5)<br />
m A 0.5<br />
4<br />
B (ln(5), 5) <br />
m B 5 <br />
B ′<br />
A ′ B ′ <br />
A<br />
<br />
-4 4 8<br />
y x <br />
A ′<br />
<br />
x <br />
y = ln(x)<br />
<br />
d<br />
dx [ln(x)] 1 x A′ <br />
-4<br />
(0.5, ln(0.5)) m A ′ 1<br />
0.5<br />
2 <br />
B ′ (5, ln(5)) m B ′ 1 5 <br />
y <br />
e x y ln(x) <br />
<br />
x<br />
m A ′ 1<br />
m A<br />
m B ′ 1<br />
m B<br />
<br />
y f (x) <br />
y x <br />
x y
h(x) x 2 ln(x)<br />
p(t) ln(t)<br />
e t +1<br />
s(y) ln(cos(y) + 2)<br />
z(x) tan(ln(x))<br />
m(z) ln(ln(z))<br />
<br />
<br />
<br />
[− π 2 , π 2 ]<br />
<br />
f (x) sin(x) [− π 2 , π 2 ]<br />
[−1, 1] f −1 <br />
f −1 : [−1, 1] → [− π 2 , π 2 ].<br />
f −1 <br />
f −1 (y) arcsin(y). <br />
<br />
<br />
y sin(x) x arcsin(y)<br />
<br />
y <br />
y π 6<br />
− π<br />
1 2<br />
2 <br />
arcsin( 1 2 ) π 6<br />
<br />
sin( π 6 ) 1 2 .<br />
π<br />
2<br />
− π 2<br />
(1, π 2 )<br />
f −1 f<br />
( π 2 ,1)<br />
π<br />
2<br />
f (x) sin(x)<br />
[− π 2 , π 2 ]<br />
f −1 (x) arcsin(x)
h(x) arcsin(x) <br />
h ′ (x) h(x) x <br />
sin(h(x)) x.<br />
<br />
d<br />
dx [sin(h(x))] d<br />
dx [x],<br />
1 <br />
<br />
cos(h(x))h ′ (x) 1.<br />
h ′ (x) <br />
h ′ (x) <br />
1<br />
cos(h(x)) .<br />
h(x) arcsin(x) h ′ (x) cos(arcsin(x))<br />
x <br />
<br />
θ arcsin(x)θ <br />
x <br />
θ <br />
1 <br />
x <br />
√ 1 − x 2 <br />
<br />
θ arcsin(x) sin(θ) x<br />
<br />
cos(arcsin(x)) <br />
cos(arcsin(x)) cos(θ) √ 1 − x 2 .<br />
1<br />
θ<br />
√<br />
1 − x 2<br />
x<br />
<br />
θ arcsin(x)<br />
h(x) arcsin(x) <br />
h ′ 1<br />
(x) <br />
cos(arcsin(x))<br />
<br />
h ′ (x) <br />
1<br />
√<br />
1 − x 2 .
x −1 < x < 1<br />
d<br />
dx [arcsin(x)] 1<br />
√ .<br />
1 − x 2<br />
<br />
<br />
r(x) arctan(x) <br />
<br />
<br />
r ′ (x) r ′ (x) <br />
r(x)<br />
r(x) arctan(x) r ′ (x) <br />
x<br />
θ θ arctan(x) <br />
<br />
x 1 cos(arctan(x))<br />
1 x<br />
r ′ (x)<br />
<br />
<br />
<br />
<br />
<br />
f (x) x 3 arctan(x) + e x ln(x)<br />
p(t) 2 t arcsin(t)<br />
h(z) (arcsin(5z) + arctan(4 − z)) 27<br />
s(y) cot(arctan(y))<br />
m(v) ln(sin 2 (v) + 1)<br />
( ) ln(w)<br />
(w) arctan<br />
1 + w 2
y x (ln(2), 2) <br />
f (x) e x f ′ (ln(2)) 2 <br />
(2, ln(2)) f −1 (x) ln(x) <br />
( f −1 ) ′ (2) 1 2 f ′ (ln(2))<br />
<br />
f <br />
y f (x) x (y) f ((x)) x<br />
x f −1 <br />
x<br />
d<br />
dx [ f ((x))] d<br />
dx [x],<br />
<br />
f ′ ((x)) ′ (x) 1.<br />
′ (x) ′ 1<br />
(x) <br />
f ′ ((x))<br />
. <br />
(x, (x)) <br />
f ((x), f ((x))) ((x), x)<br />
m = f ′ (a)<br />
y = f (x)<br />
(a,b)<br />
(b,a)<br />
m = g ′ (b)<br />
y = g(x)<br />
y f (x) y (x) f −1 (x) <br />
<br />
y f (x) <br />
y (x) (a, b) f <br />
(b, a) f (a) b (b) a <br />
′ (x) 1/ f ′ ((x)) x b<br />
′ (b) <br />
1<br />
f ′ ((b)) 1<br />
f ′ (a) ,
(b, a) <br />
f (a, b) <br />
y x<br />
<br />
f (a, b) <br />
f f ′ (a) 0 <br />
′ (b) 1<br />
f ′ (a) .<br />
x ′ ′ (x) 1/ f ′ ((x)).<br />
ln(x) arcsin(x) arctan(x) <br />
(x) ln(x)<br />
f (x) e x <br />
′ (x) <br />
1<br />
f ′ ((x)) 1<br />
e ln(x) 1 x .<br />
<br />
d<br />
x<br />
dx [ln(x)] 1 x <br />
d<br />
x −1 < x < 1<br />
dx [arcsin(x)] √ 1 <br />
1−x 2<br />
d<br />
x<br />
dx [arctan(x)] 1 <br />
1+x 2<br />
f x <br />
′ ′ 1<br />
(x) <br />
f ′ ((x)) .<br />
<br />
f (t) <br />
f (t) ln(t 2 + 7)<br />
f ′ (t) <br />
(t) <br />
<br />
(t) cos(ln(t))<br />
′ (t) <br />
h(w) <br />
<br />
h(w) 5w arcsin w
h ′ (w) <br />
x > 0 f (x) arctan x + arctan(1/x)<br />
f ′ (x) <br />
f <br />
(x 0 , y 0 ) (2, 4) (x 1 , y 1 ) (2.5, 4.6) f <br />
<br />
h(x) ( f (x)) 5 <br />
h ′ (2) <br />
(x) f −1 (x) <br />
′ (4) <br />
<br />
f (x) 6 sin −1 x 4<br />
f ′ (x) <br />
arcsin(x) sin −1 (x) sin(x)<br />
f (x) 7x 4 arctan(7x 2 ) f ′ (x).<br />
f ′ (x) <br />
f (x) 8 sin(x) sin −1 (x) f ′ (x)<br />
f ′ (x)
f (x) ln(2 arctan(x) + 3 arcsin(x) + 5)<br />
r(z) arctan(ln(arcsin(z)))<br />
q(t) arctan 2 (3t) arcsin 4 (7t)<br />
(v) ( )<br />
arctan(v)<br />
ln<br />
arcsin(v)+v 2<br />
y f (x) <br />
<br />
<br />
f ′ (1)<br />
y <br />
f −1 (x) <br />
<br />
f <br />
<br />
( f −1 ) ′ (−1) <br />
y = f (x)<br />
y f (x)<br />
f (x) 1 4 x3 + 4.<br />
y f (x) f <br />
f <br />
f ′ (x) ′ (x) f ′ (2) ′ (6) f ′ (2)<br />
′ (6) <br />
h(x) x + sin(x)<br />
y h(x) h <br />
<br />
h −1 <br />
( π 2 , π 2<br />
+ 1) y h(x) <br />
(h −1 ) ′ ( π 2 + 1)
x <br />
x<br />
dy<br />
dx<br />
y <br />
x<br />
dy<br />
dx<br />
<br />
<br />
<br />
x <br />
x y <br />
y x<br />
x 2 + y 2 = 16<br />
4 x 3 − y 3 = 6xy<br />
A<br />
-4 4<br />
x<br />
B<br />
-4<br />
x 2 + y 2 16 <br />
x 2 + y 2 16 <br />
x 3 − y 3 6xy<br />
<br />
x <br />
y A (−3, √ 7)<br />
B (−3, − √ 7) <br />
y f (x)<br />
x <br />
<br />
<br />
dy<br />
dx<br />
<br />
y x
dy<br />
dx<br />
(x, y) <br />
y y x <br />
y x <br />
<br />
d<br />
dx [e u(x) ] e u(x) u ′ (x)<br />
f x <br />
d<br />
<br />
dx [ f (x)] f ′ (x) <br />
x <br />
f c<br />
<br />
<br />
<br />
<br />
d<br />
dx x 2 + f (x) <br />
d<br />
<br />
<br />
d<br />
dx x 2 f (x) <br />
<br />
c + x + f (x)<br />
2<br />
d<br />
dx<br />
dx<br />
d<br />
dx<br />
f (x 2 ) <br />
xf(x) + f (cx) + cf(x)<br />
<br />
<br />
<br />
x <br />
x 2 + y 2 16. <br />
<br />
<br />
dy<br />
dx<br />
<br />
y x y <br />
f (x) y <br />
x dy<br />
dx<br />
<br />
x 2 + y 2 16 y x<br />
x<br />
d x 2 + y 2 d<br />
dx<br />
dx [16] .<br />
<br />
<br />
d x<br />
2 + d y<br />
2 0.<br />
dx dx<br />
<br />
x <br />
x
x y x <br />
<br />
d<br />
dx x<br />
2<br />
2x. y y x <br />
d<br />
dx [y2 ] <br />
d<br />
dx [ f (x)2 ] <br />
<br />
d<br />
dx [y2 ] 2y 1 dy<br />
dx<br />
. <br />
x 2 + y 2 16 <br />
2x + 2y dy<br />
dx 0.<br />
dy<br />
dy<br />
dx<br />
<br />
dx <br />
2x 2y<br />
dy<br />
dx − 2x<br />
2y − x y .<br />
<br />
dy<br />
dx − x y <br />
<br />
x y <br />
<br />
x −4 4 <br />
<br />
<br />
<br />
<br />
<br />
(a, b) <br />
m r b a<br />
<br />
(a, b) <br />
m t − a b <br />
<br />
(0, 4) (0, −4)<br />
(−4, 0) (4, 0) <br />
<br />
dy<br />
dx − x y <br />
m t = − a b<br />
(a,b)<br />
m r = b a<br />
x 2 +y 2 16<br />
(a, b) <br />
<br />
m r m t
x 3 + y 2 − 2xy 2 <br />
(−1, 1)<br />
3<br />
y<br />
-3 3<br />
x<br />
-3<br />
x 3 + y 2 − 2xy 2<br />
<br />
x<br />
d x 3 + y 2 − 2xy d<br />
dx<br />
dx [2] ,<br />
<br />
d<br />
dx [x3 ] + d<br />
dx [y2 ] − d [2xy] 0.<br />
dx<br />
<br />
y x <br />
y x <br />
3x 2 + 2y dy<br />
dx<br />
− [2x<br />
dy<br />
dx<br />
+ 2y] 0.<br />
dy<br />
dx<br />
<br />
dy<br />
dx <br />
dy<br />
dx<br />
<br />
3x 2 − 2y <br />
2y dy<br />
dx<br />
− 2x<br />
dy<br />
dx 2y − 3x2 .
dy<br />
dx <br />
dy<br />
dx (2y − 2x) 2y − 3x2 .<br />
(2y − 2x) <br />
dy<br />
dx<br />
<br />
2y − 3x2<br />
2y − 2x .<br />
<br />
dy<br />
dx<br />
x y <br />
(−1, 1) dy<br />
dx<br />
<br />
dy<br />
dx<br />
(−1,1)<br />
<br />
2(1) − 3(−1)2<br />
2(1) − 2(−1) −1 4 .<br />
<br />
<br />
dy<br />
dx <br />
<br />
dy<br />
dx<br />
<br />
dy<br />
dx<br />
<br />
dy<br />
dy<br />
dx <br />
dx <br />
dy<br />
dx<br />
x y <br />
dy<br />
dx<br />
(a,b)<br />
dy<br />
dx (a, b) f ′ (a) f ′<br />
<br />
d<br />
dx<br />
d<br />
dx [x2 + y 2 ]<br />
dy<br />
<br />
dx<br />
<br />
x x 2 + y 2 <br />
y x <br />
dy<br />
dx (x2 + y 2 )<br />
y x x 2 + y 2
x y 5 − 5y 3 + 4y <br />
<br />
<br />
y x<br />
<br />
dy/dx<br />
3<br />
y<br />
<br />
<br />
x y 5 − 5y 3 + 4y <br />
(0, 1)<br />
<br />
<br />
x y 5 − 5y 3 + 4y <br />
<br />
-3 3<br />
-3<br />
x<br />
x y 5 − 5y 3 + 4y<br />
<br />
<br />
<br />
dy<br />
dx<br />
x y<br />
dy<br />
dx<br />
<br />
p(x, y)<br />
q(x, y) .<br />
<br />
<br />
q(x, y) 0 p(x, y) 0 <br />
x y p(x, y) 0 <br />
<br />
<br />
<br />
<br />
<br />
y(y 2 −1)(y −2) x(x −1)(x −<br />
2)
dy<br />
dx (x − 1)(x − 2) + x(x − 2) + x(x − 1)<br />
(y 2 − 1)(y − 2) + 2y 2 (y − 2) + y(y 2 − 1) .<br />
<br />
(x, y) <br />
<br />
<br />
<br />
<br />
(x, y) <br />
<br />
<br />
<br />
<br />
<br />
<br />
x 1<br />
2<br />
1<br />
-1<br />
y<br />
1 2 3<br />
x<br />
y(y 2 −1)(y−2) x(x−1)(x−<br />
2)<br />
dy/dx<br />
<br />
x 3 − y 3 6xy (−3, 3)<br />
sin(y) + y x 3 + x (0, 0)<br />
3xe −xy y 2 (0.619061, 1)<br />
<br />
x y y <br />
x <br />
x y x <br />
<br />
<br />
x y <br />
x dy<br />
dx<br />
<br />
dy<br />
dx
dy<br />
dy<br />
dx<br />
x y<br />
dx<br />
<br />
<br />
dy<br />
dy<br />
dx<br />
0 <br />
dx<br />
<br />
<br />
<br />
dy/dx x y x 5 y − x − 9y − 8 0<br />
dy<br />
dx <br />
dy<br />
dx x y x ln y + y2 6lnx<br />
dy<br />
dx <br />
dy/dx x y arcsin(x 3 y) xy 3 <br />
dy<br />
dx <br />
x 3 + 2xy + y 2 64 (1, 7)<br />
<br />
<br />
<br />
xy 3 +4xy <br />
40 (8, 1)<br />
(8, 1)<br />
2y 3 + y 2 − y 5 x 4 − 2x 3 + x 2 <br />
<br />
<br />
<br />
sin(x + y) + cos(x − y) 1 <br />
( π 2 , π 2 )<br />
<br />
d<br />
dx [ax ] a x ln(a) d<br />
dx [ln(x)] 1 x<br />
<br />
<br />
y a x x <br />
y a<br />
x <br />
y x<br />
dy<br />
dy<br />
dx y <br />
dx<br />
x
0 0 <br />
lim x→∞ f (x) L lim x→a f (x) ∞<br />
∞ ∞ <br />
<br />
<br />
<br />
0 0 f f ′ (x) <br />
<br />
f ′ f (x + h) − f (x)<br />
(x) lim<br />
,<br />
h→0 h<br />
h → 0 ( f (x + h) − f (x)) → 0 <br />
f <br />
f ′ (x) 0 0 <br />
0 0 <br />
f ′ (x) x <br />
f ′ (x) x 2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
h h(x) x5 +x−2<br />
x 2 −1<br />
<br />
h<br />
x 5 + x − 2<br />
lim<br />
x→1 x 2 <br />
− 1<br />
<br />
h x 1 f (x) x 5 + x − 2 (x) x 2 − 1 <br />
f a 1 L f (x) L (x)<br />
<br />
h(x) ≈ L f (x)<br />
L (x)<br />
x a 1
L f (x)<br />
lim<br />
x→1 L (x) .<br />
lim x→1 h(x)<br />
h(x) x 1 <br />
lim x→1 h(x)<br />
0 0 <br />
f<br />
L f<br />
L f ≈ f<br />
a<br />
L g<br />
a<br />
L g ≈ g<br />
g<br />
f a <br />
L f L x a a <br />
<br />
0 0<br />
<br />
<br />
h(x) <br />
h(x) f (x)<br />
(x)<br />
f x a <br />
f (a) (a) 0 <br />
lim x→a h(x). <br />
f f x <br />
x a <br />
L f L x a <br />
f (x)<br />
(x)<br />
x → a
f (x) (x) x → a <br />
<br />
x → a<br />
L f (x) f ′ (a)(x − a) + f (a) L (x) ′ (a)(x − a) + (a) <br />
x <br />
a f L f L <br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
L f (x)<br />
x→a L (x)<br />
f ′ (a)(x − a) + f (a)<br />
lim<br />
x→a ′ (a)(x − a) + (a) .<br />
f (a) 0 (a) 0<br />
f (a)<br />
(a) <br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
x→a<br />
f ′ (a)(x − a)<br />
′ (a)(x − a)<br />
lim<br />
x→a<br />
f ′ (a)<br />
′ (a) ,<br />
x a x−a<br />
x−a<br />
f ′ (a)<br />
′ (a)<br />
x <br />
lim<br />
x→a<br />
f (x)<br />
(x) f ′ (a)<br />
′ (a) .<br />
1 <br />
′ (a) 0 <br />
f ′ (a)<br />
′ (a)<br />
<br />
<br />
<br />
f x a f (a) (a) 0 <br />
′ f (x)<br />
(a) 0 lim x→a (x) f ′ (a)<br />
′ (a) .<br />
<br />
<br />
′ x a<br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
x→a<br />
f ′ (x)<br />
′ (x) ,<br />
<br />
f (x)<br />
(x) 0 0<br />
x → a
f ′ (x)<br />
′ (x)<br />
. <br />
<br />
x 5 + x − 2<br />
lim<br />
x→1 x 2 − 1 ,<br />
x 5 + x − 2 5x 4 + 1<br />
lim<br />
x→1 x 2 lim 6 − 1 x→1 2x 2 3.<br />
<br />
<br />
<br />
<br />
lim x→0<br />
ln(1+x)<br />
x<br />
lim x→π<br />
cos(x)<br />
x<br />
lim x→1<br />
2ln(x)<br />
1−e x−1<br />
lim x→0<br />
sin(x)−x<br />
cos(2x)−1<br />
<br />
f<br />
m = f ′ (a)<br />
m = f ′ (a)<br />
a<br />
m = g ′ (a)<br />
a<br />
m = g ′ (a)<br />
g<br />
f <br />
<br />
f x a f (x)<br />
(x)
x → a <br />
f ′ (a) 2 ′ (a) −1 <br />
lim<br />
x→a<br />
f (x)<br />
(x) f ′ (a)<br />
′ (a) 2<br />
−1 −2.<br />
f <br />
<br />
f (a) (a) 0 <br />
f (x)<br />
(x)<br />
x → a f x a<br />
<br />
<br />
2<br />
g<br />
f<br />
2<br />
1<br />
2 p 1<br />
q<br />
1<br />
r<br />
1 2 3 4<br />
1 2 3 4<br />
1 2 3 4<br />
-1<br />
-1<br />
-1<br />
-2<br />
-2<br />
-2<br />
s<br />
<br />
f (2) f ′ (2) (2) ′ (2)<br />
f (x)<br />
lim<br />
x→2<br />
(x) <br />
p(2) p ′ (2) q(2) q ′ (2) <br />
p(x)<br />
lim<br />
x→2<br />
q(x) <br />
r(2) r ′ (2) s(2) s ′ (2) <br />
r(x)<br />
lim<br />
x→2<br />
s(x)<br />
<br />
r(x)<br />
lim<br />
x→2<br />
s(x)
∞<br />
∞ <br />
∞ <br />
∞ <br />
f (x) 1 x<br />
<br />
x 0 <br />
f <br />
x → 0 x → 0 + <br />
f (x) <br />
f (x) <br />
x <br />
x > 0 <br />
<br />
<br />
<br />
<br />
1<br />
1<br />
f (x)= 1 x<br />
f (x) 1 x <br />
lim x→a f (x) L f (x) L <br />
x a <br />
L a ∞ f (x) 1 x <br />
<br />
1<br />
lim<br />
x→0 + x ∞,<br />
1 x<br />
x <br />
<br />
1<br />
lim<br />
x→∞ x 0,<br />
1 x<br />
x x<br />
<br />
lim x→a f (x) ∞ f (x)<br />
x a <br />
lim x→∞ f (x) L f (x) L x<br />
<br />
−∞ f (x) 1 x<br />
<br />
1<br />
lim<br />
x→0 − x<br />
−∞ lim<br />
1<br />
x→−∞ x 0.
lim f (x) ∞<br />
x→∞<br />
f (x) x <br />
<br />
lim<br />
x→∞ x2 ∞.<br />
<br />
lim x→a f (x) ∞ x a f lim x→∞ f (x) <br />
L y L f −∞ <br />
x → a − x → a + <br />
<br />
x →∞ x →−∞ <br />
<br />
∞<br />
8<br />
y = e x<br />
64<br />
y = f (x)<br />
1<br />
y = sin(x)<br />
4<br />
-2 2<br />
10<br />
-4 4 8<br />
-4<br />
y = ln(x)<br />
-64<br />
y = g(x)<br />
x →±∞ <br />
f (x) x 3 − 16x (x) x 4 − 16x 2 − 8<br />
e x lim x→∞ e x ∞ lim x→−∞ e x 0,<br />
e −x <br />
lim x→∞ e −x 0 lim x→−∞ e −x ∞. <br />
lim x→0 + ln(x) −∞ lim x→∞ ln(x) ∞. e x ln(x) <br />
x →∞ <br />
<br />
p(x) a n x n + a n−1 x n−1 + ···a 1 x + a 0 <br />
a n n n <br />
a n lim x→∞ p(x) ∞ lim x→−∞ p(x) ∞ <br />
a n lim x→∞ p(x) −∞ lim x→−∞ p(x) −∞ <br />
n lim x→∞ p(x) ∞ lim x→−∞ p(x) −∞ <br />
a n f lim x→∞ p(x) −∞ <br />
lim x→−∞ p(x) ∞ a n
x →∞ <br />
−1 <br />
1 x →∞lim x→∞ sin(x) <br />
x → ∞<br />
<br />
q(x) 3x2 − 4x + 5<br />
7x 2 + 9x − 10 .<br />
(3x 2 − 4x + 5) →∞ x →∞ (7x 2 + 9x − 10) →∞ x →∞ <br />
lim x→∞ q(x) ∞ ∞ <br />
0 0 <br />
1 <br />
x 2<br />
lim q(x) lim<br />
x→∞ x→∞<br />
3x 2 − 4x + 5<br />
7x 2 + 9x − 10 ·<br />
1<br />
x 2<br />
1<br />
x 2<br />
lim<br />
x→∞<br />
3 − 4 1 x + 5 1 x 2<br />
7 + 9 1 x − 10 1 x 2 3 7<br />
1 → 0 1 x 2<br />
x<br />
→ 0 x →∞ q <br />
y 3 7 <br />
x →∞<br />
<br />
x 2<br />
lim<br />
x→∞ e x ?<br />
x 2 →∞ e x →∞ <br />
<br />
<br />
∞<br />
f ±∞ x → a<br />
a ∞<br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
x→a<br />
f ′ (x)<br />
′ (x) .<br />
′ (x) 0 <br />
a a ∞ <br />
<br />
x<br />
lim 2<br />
x→∞ e<br />
x 2 →∞<br />
x<br />
e x →∞ <br />
x 2<br />
lim<br />
x→∞ e x lim 2x<br />
x→∞ e x .
∞ ∞ 2x <br />
x 2 <br />
x 2<br />
lim<br />
x→∞ e x lim 2x<br />
x→∞ e x lim 2<br />
x→∞ e x .<br />
2 e x →∞ x →∞ 2<br />
e x<br />
<br />
x 2<br />
lim<br />
x→∞ e x 0.<br />
→ 0 x →∞ <br />
<br />
<br />
lim x→∞<br />
x<br />
ln(x)<br />
lim x→∞<br />
e x +x<br />
2e x +x 2<br />
lim x→0 + ln(x)<br />
1<br />
x<br />
lim x→<br />
π<br />
2<br />
− tan(x)<br />
x− π 2<br />
lim x→∞ xe −x<br />
f (x)<br />
(x)<br />
<br />
∞ ∞ <br />
f x →∞ <br />
lim<br />
x→∞<br />
f (x)<br />
(x) 0,<br />
f (x)<br />
f lim x→∞ (x) ∞ lim x→∞ f (x)<br />
(x)<br />
f <br />
x<br />
lim 2<br />
x→∞ e<br />
0, e x x 2 3x<br />
lim 2 −4x+5<br />
x<br />
x→∞ 7x 2 +9x−10<br />
3 7 <br />
f (x) 3x 2 − 4x + 5 (x) 7x 2 + 9x − 10 <br />
<br />
0 0 <br />
<br />
f (a) (a) 0<br />
f a <br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
x→a<br />
f ′ (x)<br />
′ (x) .<br />
x →∞ x ∞<br />
lim x→∞ f (x) L f (x)
L x lim x→a f (x) <br />
∞ f (x) x <br />
a<br />
<br />
∞ ∞ f <br />
±∞ x → a a ∞ <br />
lim<br />
x→a<br />
f (x)<br />
(x) lim<br />
x→a<br />
f ′ (x)<br />
′ (x) .<br />
<br />
lim<br />
x→a<br />
(x) <br />
f (x)<br />
f (x) <br />
(x)<br />
f (x)<br />
lim <br />
x→a (x)<br />
<br />
<br />
f (x)<br />
lim <br />
x→a (x)<br />
<br />
<br />
ln(x/4)<br />
lim<br />
x→4 x 2 − 16 <br />
<br />
∞<br />
−∞<br />
1 − cos(7x)<br />
lim<br />
x→0 1 − cos(6x) <br />
7 x − 6 x − 1<br />
lim<br />
x→1 x 2 <br />
− 1<br />
<br />
13x 2<br />
lim<br />
x→∞ e 8x <br />
f <br />
f (3) (3) 0 f ′ (3) ′ (3) 0 f ′′ (3) −2 ′′ (3) 1 h <br />
h(x) f (x)<br />
(x)<br />
f x 3<br />
<br />
lim h(x).<br />
x→3
R(x) <br />
3(x − a)(x − b)<br />
5(x − a)(x − c) ,<br />
a b c x <br />
R R a b c<br />
(x) x 2x x > 0 lim x→0 + (x)<br />
0 0 0 k 0 k > 0<br />
b 0 1 b 0 0 0 <br />
h(x) ln((x)) h(x) 2x ln(x).<br />
h(x) 2ln(x)<br />
1 <br />
x<br />
lim x→0 + h(x)<br />
lim x→0 + h(x) lim x→0 + (x).<br />
f lim x→∞ f (x) ∞<br />
f (x)<br />
lim x→∞ (x) ∞ lim x→∞ (x) 0<br />
ln(x) √ x<br />
ln(x) n√ xn <br />
e x <br />
x n ln(x) n
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
s(t) −16t 2 + 32t + 48 <br />
<br />
<br />
<br />
<br />
<br />
<br />
f f (c) f (c) ≥ f (x) x<br />
f f (c) f (c) ≤ f (x)<br />
x f <br />
(c) <br />
(c, (c)) <br />
<br />
(a, (a)) (b, (b))
40<br />
V<br />
y = s(t)<br />
(c,g(c))<br />
30<br />
20<br />
10<br />
(a,g(a))<br />
(b,g(b))<br />
1 2<br />
y = g(x)<br />
s(t) −16t 2 + 24t + 32 ( 3 4<br />
, 41) <br />
<br />
f f (c) <br />
f (c) ≥ f (x) x c f (c) f (c) ≤<br />
f (x) x c f <br />
(b) (b, (b)) <br />
(a) (a, (a)) (c)<br />
<br />
<br />
<br />
<br />
<br />
h <br />
<br />
c h(c) h<br />
c h(c) h<br />
h [−3, 3] <br />
<br />
h [−3, 3] <br />
c h ′ (c) 0
c <br />
h ′ (c) <br />
<br />
h <br />
h ′ (c) <br />
<br />
<br />
h ′ (c) h <br />
<br />
y = h(x)<br />
2<br />
1<br />
-2 -1 1 2<br />
-1<br />
-2<br />
h <br />
[−3, 3]<br />
<br />
(c, f (c)) <br />
c <br />
<br />
<br />
<br />
<br />
<br />
c <br />
<br />
c c <br />
c <br />
c <br />
<br />
f ′ (c) 0 f ′ (c)
c <br />
f x c c <br />
f f ′ (c) 0 f ′ (c) <br />
f <br />
<br />
<br />
<br />
<br />
c (c, f (c)) <br />
f (c) <br />
<br />
<br />
<br />
p f p <br />
x p f p f ′ <br />
p f p f ′ <br />
p<br />
<br />
<br />
f f ′ (x) e −2x (3−<br />
x)(x + 1) 2 f <br />
<br />
f ′ (x) <br />
f f ′ (x) x <br />
f ′ (x) 0 <br />
e −2x (3 − x)(x + 1) 2 0<br />
x 3 x −1 f <br />
x e −2x 0<br />
f ′ (x) <br />
f ′ <br />
<br />
f ′ x < −1 <br />
f ′ <br />
f
f ′ (x)=e −2x (3 − x)(x + 1) 2<br />
sign( f ′ )<br />
behav( f )<br />
+++ +++<br />
+<br />
+<br />
INC −1<br />
INC 3<br />
+ − +<br />
−<br />
DEC<br />
f <br />
f ′ (x) e −2x (3 − x)(x + 1) 2 <br />
<br />
f ′ (x) <br />
x < −1 x −2 e −2x <br />
(3 − x) (x + 1) 2 x −2 e −2x (x + 1) 2 <br />
x (3 − x) x −2 <br />
f ′ + ++ <br />
+ <br />
x −1 f ′ f ′ <br />
f <br />
f f ′ <br />
f −1 < x < 3 f ′ f x > 3<br />
f <br />
f ′ f ′ x 3 f ′ <br />
f x 3 f <br />
x −1 f x −1 f <br />
x −1<br />
(x) x 2 <br />
′ (x) (x+4)(x−1)2<br />
x−2<br />
<br />
x 2<br />
<br />
<br />
<br />
<br />
lim x→∞ ′ (x)
y (x)<br />
<br />
<br />
f ′′ <br />
f ′ f <br />
y f (x) <br />
f ′ (p) 0 <br />
f <br />
p<br />
f <br />
<br />
f p <br />
f ′ (p) 0 f ′′ (p) p p<br />
f ′′ f <br />
f <br />
p f <br />
f ′ (p) 0 f ′′ (p) < 0 f p <br />
f <br />
f ′′ (p) <br />
<br />
p f f ′ (p) 0 f ′′ (p) <br />
0 f p f ′′ (p) < 0 f <br />
p f ′′ (p) > 0<br />
f ′′ (p) 0 <br />
f ′′ (p) 0 f <br />
<br />
f (x) x 4 (x) −x 4 h(x) x 3 p 0
f (x) f ′ (x) 3x 4 − 9x 2 <br />
f f <br />
<br />
f <br />
f ′ (x) 3x 4 −9x 2 f <br />
3x 4 − 9x 2 0 <br />
0 3x 2 (x 2 − 3) 3x 2 (x + √ 3)(x − √ 3),<br />
x 0, ± √ 3 f <br />
f <br />
f ′ (x)=3x 2 (x + √ 3)(x − √ 3)<br />
sign( f ′ )<br />
behav( f )<br />
+ −− ++−<br />
INC<br />
√<br />
− 3<br />
+<br />
−<br />
DEC 0<br />
++−<br />
−<br />
DEC<br />
+++<br />
+<br />
√<br />
3<br />
INC<br />
f f ′ (x) 3x 4 − 9x 2 3x 2 (x 2 − 3)<br />
f (−∞, − √ 3) ( √ 3, ∞) f <br />
(− √ 3, 0) (0, √ 3) <br />
f x − √ 3 x √ 3 f <br />
x 0 f ′ <br />
x 0<br />
f ′ (x) 3x 4 − 9x 2 <br />
f ′′ (x) 12x 3 − 18x f ′′ <br />
f ′′ (x) 0 <br />
0 12x 3 − 18x 12x<br />
(<br />
x 2 − 3 )<br />
√<br />
12x 3<br />
+<br />
2<br />
2<br />
x <br />
x −<br />
<br />
√<br />
3 ,<br />
2<br />
<br />
√<br />
3<br />
x 0, ±<br />
2 f ′′ <br />
f ′
( √<br />
f ′′ (x)=12x x +<br />
3<br />
2<br />
)( √<br />
x −<br />
3<br />
2<br />
)<br />
sign( f ′′ )<br />
behav( f )<br />
−−−<br />
−<br />
CCD<br />
√<br />
3<br />
−<br />
2<br />
− + −<br />
+<br />
CCU 0<br />
++−<br />
+++<br />
−<br />
+<br />
CCD √ CCU<br />
3<br />
2<br />
f f ′′ (x) 12x 3 − 18x <br />
√ )<br />
12x<br />
(x 2 2 − <br />
3<br />
2<br />
√ √<br />
3<br />
f (−∞, −<br />
2 ) (0, 3<br />
2<br />
) <br />
√ √<br />
3<br />
(−<br />
2 , 0) ( 3<br />
2 , ∞)<br />
<br />
f <br />
A<br />
B<br />
C<br />
f<br />
D<br />
E<br />
− √ 3 − √ 1.5<br />
√<br />
1.5<br />
√<br />
3<br />
f <br />
A (− √ 3, f (− √ 3)) f A <br />
E ( √ 3, f ( √ 3) f <br />
A B <br />
B D
C f <br />
<br />
f C C<br />
f f <br />
f <br />
y <br />
f x − √ 3 <br />
f (− √ 3) f <br />
f <br />
B C D f <br />
p <br />
f f (p, f (p)) f <br />
f f ′ (p) 0<br />
f ′ (p) f ′′ (p) 0 f ′′ (p) <br />
<br />
<br />
f ′ f <br />
f ′′ f <br />
<br />
′′ <br />
<br />
x <br />
<br />
<br />
<br />
<br />
′ (−1.67857351) 0 <br />
<br />
<br />
<br />
<br />
g ′′<br />
2<br />
1<br />
1 2<br />
y ′′ (x)<br />
′′ (x) ′′
(x) <br />
<br />
<br />
f (x) a(x −<br />
h) 2 + k a h k <br />
<br />
<br />
h(x) x 2 + cos(kx) k <br />
<br />
<br />
h k<br />
k 1, 3, 5, 10<br />
h(x) x 2 + cos(3x) <br />
<br />
k <br />
<br />
<br />
h<br />
h <br />
k ≤ √ 2 <br />
<br />
k > √ 2<br />
12<br />
8<br />
4<br />
-2 2<br />
<br />
k h <br />
<br />
y h(x)<br />
<br />
f p f ′ (p) 0<br />
f ′ (p) <br />
<br />
<br />
f f ′ f <br />
f ′ f <br />
f f
f f <br />
f x p<br />
f ′′ (p) f ′′ f <br />
f x p f ′ (p) 0 <br />
f ′′ (p) < 0 f p f <br />
f ′ (p) 0 f ′′ (p) > 0 f p f ′ (p) 0 f ′′ (p) 0<br />
f p <br />
<br />
f (x) 3e −9x2<br />
f (x)<br />
x <br />
x <br />
x <br />
x <br />
x <br />
x <br />
f (x) 2x 4 +27x 3 −21x 2 +15 <br />
<br />
<br />
<br />
f f ′ f ′′ <br />
<br />
f f ′ f ′′ <br />
<br />
f x<br />
f (0) −1/2
y f ′ (x) <br />
2<br />
1<br />
x<br />
1<br />
f ′ x<br />
1<br />
x<br />
y f ′ (x) y f (x) <br />
y f ′′ (x)<br />
f f <br />
f f <br />
y f ′′ (x)<br />
f f <br />
<br />
y f (x)<br />
′ (2) 0 <br />
1 < x < 2 2 < x < 3 ′ (x) <br />
x 2 <br />
′′ (x) x 1 < x < 3 <br />
′′ (x) x 2<br />
x 2 <br />
<br />
h
h(x) 0 <br />
h(x) 0 <br />
<br />
<br />
lim x→∞ h ′ (x) 3 <br />
<br />
y h(x) <br />
x →∞<br />
y h(x) <br />
<br />
<br />
h ′<br />
y h ′ (x)<br />
p p ′′ (x) (x + 1)(x − 2)e −x <br />
p p<br />
√<br />
5−1<br />
x <br />
2<br />
p p <br />
x <br />
√<br />
5−1<br />
2<br />
<br />
(2, 12<br />
e 2 ) y p(x) p ′ (2) − 5 e 2 <br />
y p(x) x 2
d + a<br />
d<br />
f (t)=asin(b(t − c)) + d<br />
c<br />
c + 2π b<br />
f (t) a sin(b(t − c)) + d a b c d<br />
f (t) a sin(b(t −<br />
c)) + d a 2π b <br />
c d
a b c <br />
d <br />
<br />
<br />
y mx + b m y<br />
(0, b) y mx + b <br />
x a <br />
m <br />
b <br />
<br />
<br />
<br />
<br />
<br />
a h k a 0 f<br />
f (x) a(x − h) 2 + k<br />
f f <br />
a h k<br />
<br />
a h k f ′ (x)<br />
f a h k<br />
a < 0 f <br />
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(x) axe −bx ,<br />
a b <br />
a b
′ (x) <br />
′ (x) ax d<br />
dx<br />
[<br />
e<br />
−bx ] + e −bx d<br />
dx [ax],<br />
<br />
′ (x) axe −bx (−b) + e −bx (a).<br />
′ (x) 0 <br />
′ (x) <br />
0 ae −bx (−bx + 1).<br />
a 0 e −bx 0 x <br />
−bx + 1 0. x x 1 b<br />
<br />
<br />
′ (x) ae −bx (1 − bx) <br />
x 1 b<br />
<br />
<br />
g ′ (x)=ae −bx (1 − bx)<br />
sign(g ′ )<br />
behav(g)<br />
++ +−<br />
+<br />
−<br />
INC<br />
1<br />
b<br />
DEC<br />
(x) axe −bx <br />
′ (x) ae −bx (1 − bx) ae −bx <br />
(1 − bx) x 1 b<br />
<br />
−b (1 − bx) x < 1 b x > 1 b<br />
<br />
x < 1 b x > 1 b<br />
<br />
( 1 b , ( 1 b<br />
)) <br />
′ (x) −abxe −bx +ae −bx <br />
<br />
′′ (x) −abxe −bx (−b) + e −bx (−ab) + ae −bx (−b).<br />
<br />
′′ (x) ab 2 xe −bx − 2abe −bx abe −bx (bx − 2).
g ′′ (x)=abe −bx (bx − 2)<br />
sign(g ′′ )<br />
behav(g)<br />
+− ++<br />
−<br />
+<br />
CCD CCU<br />
2<br />
b<br />
(x) axe −bx <br />
abe −bx <br />
′′ (bx − 2) x 2 b<br />
(bx − 2)<br />
(b) (bx − 2) x < 2 b <br />
x > 2 b ′′ <br />
x < 2 b x > 2 b <br />
<br />
ax<br />
lim (x) lim<br />
x→∞ x→∞ axe−bx lim<br />
x→∞ e . bx<br />
∞ ∞ <br />
lim x→∞ (x) 0 <br />
lim (x) lim<br />
x→−∞ x→−∞ axe−bx −∞,<br />
ax →−∞ e −bx →∞ x →−∞ <br />
(x) → 0<br />
<br />
<br />
a b
a<br />
b e−1<br />
global max<br />
1<br />
b<br />
2<br />
b<br />
inflection pt<br />
g(x)=axe −bx<br />
(x) axe −bx <br />
b <br />
a a <br />
( 1 b , ( 1 b )) ( 1 b , a b e−1 )<br />
a <br />
<br />
<br />
<br />
x → ∞ <br />
<br />
<br />
<br />
<br />
<br />
p(x) x 3 − ax a 0<br />
<br />
p ′ (x) p <br />
p <br />
p <br />
p a a <br />
p<br />
p ′′ (x) p <br />
p a <br />
p<br />
p(x)
a > 0 a < 0 <br />
<br />
p(x) x 3 − ax a <br />
p <br />
a<br />
h(x) a(1 −<br />
e −bx ), a b <br />
h <br />
x h <br />
<br />
<br />
x <br />
lim x→∞ a(1 − e −bx ) lim x→−∞ a(1 − e −bx )<br />
b <br />
<br />
h<br />
a b<br />
A<br />
L(t) A c k <br />
1+ce −kt<br />
L(t) A(1+ce −kt ) −1 L ′ (t) <br />
L L <br />
<br />
L ′′ (t) Ack 2 e −kt ce −kt − 1<br />
(1 + ce −kt ) , 3<br />
t L ′′ (t) 0 <br />
t <br />
<br />
A<br />
A<br />
lim t→∞ lim<br />
1+ce −kt t→−∞ <br />
1+ce −kt<br />
L(x) <br />
<br />
L<br />
A c k <br />
L(t) <br />
<br />
A
C T D <br />
T C<br />
2 − D<br />
3 D 2 .<br />
<br />
D <br />
dT/dD <br />
<br />
D <br />
(x) −2 ≤ x ≤ 2<br />
′ (x) (x)<br />
(x) <br />
(x) <br />
x <br />
<br />
[−2, 2]<br />
x <br />
x <br />
(−2) −8 (0)<br />
(0) <br />
<br />
−1 < (0) ≤ 3 ∞ <br />
(2) (0)
(2)<br />
(0)<br />
< > <br />
p(x) x 3 − ax 2 a > 0<br />
<br />
x 0 x a<br />
p<br />
p ′′ p ′′ (x) 0 <br />
p<br />
p <br />
a a <br />
<br />
q(x) e−x<br />
x−c<br />
c > 0<br />
q x c<br />
lim x→∞ q(x) lim x→−∞ q(x)<br />
q ′ (x) q<br />
q <br />
q<br />
<br />
<br />
E(x) e − (x−m)2<br />
2s 2 m s <br />
E ′ (x) E<br />
E <br />
<br />
E ′′ (x) <br />
( )<br />
E ′′ (x) e − (x−m)2 (x − m) 2 − s 2<br />
2s 2 s 4 .<br />
x E ′′ (x) 0<br />
lim x→∞ E(x) lim x→−∞ E(x)<br />
E <br />
y E(x) m s
f x p f (p) ≥ f (x) x p <br />
p f (p) ≥ f (x) x f <br />
f <br />
x c <br />
x a f (c) f (x) <br />
x f (a) <br />
f (x) x a <br />
f <br />
f x b<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
global max<br />
relative max<br />
relative min<br />
f<br />
a b c<br />
f
3<br />
f (x) 2 + <br />
1+(x+1) 2<br />
f <br />
f <br />
f <br />
f <br />
<br />
lim x→∞ f (x) lim x→−∞ f (x)<br />
f (x) > 2 x<br />
f <br />
<br />
<br />
<br />
(−∞, ∞)<br />
<br />
3<br />
f (x) 2 + <br />
1+(x+1) 2<br />
x x 0 ≤ x ≤ 4 <br />
x [0, 4] <br />
f <br />
<br />
<br />
<br />
<br />
<br />
(x) 1 3 x3 − 2x + 2.<br />
−2 ≤ x ≤ 3<br />
−2 ≤ x ≤ 3<br />
x <br />
[−2, 3]<br />
−2 ≤ x ≤ 2<br />
−2 ≤ x ≤ 1<br />
<br />
[a, b] <br />
a b
f [a, b] f <br />
[a, b] x m a ≤<br />
x m ≤ b f (x m ) ≤ f (x) x [a, b] x M <br />
[a, b] f (x M ) ≥ f (x) x [a, b] m f (x m ) M f (x M )<br />
m ≤ f (x) ≤ M x [a, b]<br />
<br />
[a, b] <br />
<br />
<br />
<br />
<br />
<br />
<br />
f [a, b]<br />
f <br />
f <br />
<br />
f <br />
<br />
<br />
<br />
h(x) xe −x [0, 3]<br />
p(t) sin(t) + cos(t) [− π 2 , π 2 ]<br />
q(x) x2<br />
x−2 [3, 7]<br />
f (x) 4 − e −(x−2)2 (−∞, ∞)<br />
h(x) xe −ax [0, 2 a<br />
] a > 0<br />
f (x) b − e −(x−a)2 (−∞, ∞) a, b > 0
2<br />
g<br />
2<br />
g<br />
2<br />
g<br />
-2 3<br />
-2 2<br />
-2 1<br />
<br />
[−2, 3]<br />
[−2, 2] [−2, 1] <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x 20 − x
x <br />
<br />
<br />
x 3<br />
20−x<br />
<br />
20−x<br />
4<br />
<br />
x<br />
3<br />
x<br />
20 − x<br />
20−x<br />
4<br />
<br />
<br />
<br />
x 0 ≤ x ≤ 20 <br />
x 0<br />
5 × 5<br />
<br />
A Δ 1 2 bh 1 2 · x<br />
3 · x√ 3<br />
6<br />
<br />
√ 3 <br />
A □ s 2 <br />
20−x 2<br />
4 <br />
A(x) <br />
√<br />
3x<br />
2 ( ) 20 − x 2<br />
36 + .<br />
4<br />
[0, 20] <br />
<br />
A(x)<br />
A ′ (x) <br />
√ ( )(<br />
3x 20 − x<br />
18 + 2 − 1 ) √<br />
3<br />
<br />
4 4 18 x + 1 8 x − 5 2 .<br />
A ′ (x) 0 x 180<br />
4 √ ≈ 11.3007 A <br />
3+9<br />
[0, 20]<br />
A <br />
( )<br />
) 2<br />
A <br />
≈ 10.8741<br />
180<br />
4 √ 3+9<br />
A(0) 25<br />
A(20) <br />
√<br />
3<br />
36<br />
√<br />
3(<br />
180<br />
4 √ 3+9 )2<br />
4<br />
+<br />
(<br />
20−<br />
180<br />
4 √ 3+9<br />
4<br />
√<br />
100<br />
(400) <br />
9 3 ≈ 19.2450
x ≈ 11.3007 <br />
10.8741 <br />
<br />
<br />
<br />
<br />
y A(x) <br />
[0, 20] <br />
25<br />
20<br />
15<br />
10<br />
5<br />
<br />
y = A(x)<br />
5 10 15 20<br />
<br />
<br />
10 × 15 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
V <br />
<br />
V x <br />
<br />
V<br />
V
f f ′′ (x) < 0 x<br />
a b a < b ′ (x) < 0 x < a<br />
′ (x) < 0 a < x < b ′ (x) > 0 x > b<br />
h a b a < b h ′ (x) < 0 x < a<br />
h ′ (x) > 0 a < x < b h ′ (x) < 0 x > b lim x→∞ h(x) 0 <br />
lim x→−∞ h(x) 0<br />
p x a p ′′ (x) > 0 x < a <br />
p ′′ (x) < 0 x > a<br />
<br />
<br />
p(x) x 3 − a 2 x [0, a] a > 0<br />
r(x) axe −bx [ 1<br />
2b<br />
, b] a, b > 0
w(x) a(1 − e −bx ) [b, 3b] a, b > 0<br />
s(x) sin(kx) [ π 3k , 5π<br />
6k ]<br />
[a, b] <br />
[a, b] <br />
<br />
<br />
[a, b] <br />
f ′ (x) ≤ 0 x [a, b]<br />
c a < c < b ′ (x) > 0 x < c ′ (x) < 0<br />
x > c<br />
h(a) h(b) h ′′ (x) < 0 x [a, b]<br />
p(a) > 0 p(b) < 0 c a < c < b p ′ (x) < 0 x < c<br />
p ′ (x) > 0 x > c<br />
s(t) 3 sin(2(t − π 6<br />
)) + 5. s <br />
<br />
s<br />
[ π 6 , 7π 6 ]<br />
[0, π 2 ] [0, 2π]<br />
[ π 3 , 5π 6 ]
x y <br />
<br />
<br />
x y
x y<br />
x y <br />
<br />
<br />
x y <br />
<br />
x y
V x 2 y 4x + y 108 <br />
x y <br />
y 108 − 4x y V(x) x 2 (108 − 4x)<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
0.027
P <br />
Q P<br />
Q <br />
<br />
<br />
Z P <br />
Q <br />
<br />
<br />
<br />
P Z <br />
<br />
<br />
P<br />
3<br />
Z<br />
cabin<br />
Q<br />
2<br />
P Z <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
xy x <br />
f (x) 25 − x 2 x <br />
<br />
y 25 − x 2 <br />
<br />
a b <br />
f (x) b − ax 2 <br />
4 × 24
θ<br />
1 1<br />
2<br />
<br />
θ
printed<br />
material<br />
<br />
<br />
<br />
<br />
x <br />
y 12 − x 2
y x <br />
y x <br />
dy<br />
dx <br />
y x x<br />
<br />
<br />
t <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
d 12<br />
d 16 <br />
<br />
<br />
r V 4 3 πr3 <br />
V r t <br />
V r t dV<br />
dt<br />
dr<br />
dt V 4 3 πr3 t <br />
dV<br />
dt<br />
r dr<br />
dt <br />
<br />
V r
dr dV<br />
dt<br />
<br />
dt <br />
dV<br />
dt<br />
t <br />
<br />
dV<br />
dt<br />
4πr 2 dr<br />
dt<br />
<br />
dr<br />
dt<br />
<br />
d 12<br />
d 16 <br />
d 12 d 16<br />
<br />
<br />
t <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
h<br />
r
V 1 3 πr2 h.<br />
V r h t <br />
<br />
t<br />
d<br />
dt [V] d dt<br />
[ 1<br />
h]<br />
3 πr2 .<br />
d<br />
dV<br />
<br />
dt<br />
[V] <br />
dt<br />
r<br />
h t <br />
<br />
dV<br />
dt d [ 1<br />
h]<br />
dt 3 πr2<br />
1 3 πr2 d dt [h] + 1 3 πh d dt [r2 ]<br />
1 dh<br />
πr2<br />
3 dt + 1 dr<br />
πh2r<br />
3 dt<br />
<br />
d<br />
y x<br />
dx [y2 ] 2y dy<br />
dx <br />
d dt [r2 ] 2r dr<br />
dt <br />
<br />
dV<br />
dt 1 dh<br />
πr2<br />
3 dt + 2 3 πrhdr dt ,<br />
V h r <br />
<br />
<br />
<br />
<br />
<br />
<br />
t<br />
h 1 dh<br />
2<br />
r t <br />
dt<br />
1 dr<br />
2 dt<br />
<br />
dV<br />
dt<br />
r h r <br />
dr<br />
dh<br />
dt<br />
h <br />
dt<br />
<br />
dV<br />
dt 1 3 πr2 · 1 dr<br />
2 dt + 2 3 πr · 1<br />
2 r · dr<br />
dt .<br />
<br />
<br />
dV<br />
dt
dV<br />
dt<br />
10 3 <br />
r 4 r 4 dV<br />
dt<br />
10 <br />
<br />
10 1 3 π42 · 1<br />
2<br />
dr<br />
dt <br />
r4<br />
+ 2 3 π4 · 1<br />
2 4 · dr<br />
dt <br />
r4<br />
8 3 π dr<br />
dt <br />
r4<br />
+ 16<br />
3 π dr<br />
dt <br />
<br />
<br />
dr<br />
dr<br />
dt<br />
<br />
r4 dt<br />
10 <br />
<br />
r4<br />
8π dr<br />
dt<br />
<br />
r4<br />
dr<br />
dt <br />
10<br />
r4<br />
8π ≈ 0.39789<br />
<br />
dh<br />
dh<br />
dt<br />
r 4 <br />
dt<br />
1 dr<br />
2 dt<br />
t <br />
dh<br />
dt <br />
5 ≈ 0.19894 .<br />
8π<br />
r4<br />
<br />
dr dr<br />
dt<br />
<br />
dt<br />
<br />
r4<br />
r t t <br />
r t r 4<br />
t <br />
r <br />
h r h 1 2<br />
r t <br />
r h <br />
h 1 2<br />
r <br />
V 1 3 πr2 h V r <br />
V 1 ( ) 1<br />
3 πr2 2 h 1 6 πr3 .<br />
r4<br />
.<br />
t <br />
dV<br />
dt 1 dr<br />
πr2<br />
2 dt ,<br />
dr<br />
dt<br />
<br />
dh<br />
dt
t <br />
<br />
<br />
<br />
dr<br />
dt<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
r4 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
r h t <br />
<br />
t <br />
h
t<br />
<br />
<br />
h 3 h 4 h 5<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
r h <br />
h <br />
dV<br />
dt<br />
dh<br />
dt
h<br />
dh<br />
dt<br />
t <br />
h <br />
<br />
<br />
<br />
dV<br />
dt 1 dh<br />
πh2<br />
16 dt ,<br />
<br />
dh<br />
dt 16<br />
πh 2 dV<br />
dt .<br />
dV<br />
dh<br />
dt<br />
h <br />
dt<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x <br />
s <br />
<br />
<br />
<br />
<br />
x s<br />
dx<br />
dt<br />
<br />
ds<br />
dt
90 ′ <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x y z <br />
x 2 + y 2 z 2 <br />
t <br />
2x dx<br />
dt<br />
+ 2y<br />
dy<br />
dt<br />
2z<br />
dz<br />
dt ,<br />
x y z <br />
<br />
<br />
10 <br />
<br />
15 <br />
h r <br />
V 1 3 πr2 h<br />
15
8600.0 3 / <br />
1<strong>2.0</strong> <br />
4.0 24.0 /<br />
5.0 <br />
<br />
<br />
3 /<br />
<br />
<br />
<br />
<br />
<br />
<br />
25<br />
60<br />
3<br />
15<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ <br />
θ
t <br />
s t s(t) 64 − 16(t − 1) 2 <br />
[a, b] <br />
s(b)−s(a)<br />
b−a<br />
<br />
t <br />
s ′ s(t + h) − s(t)<br />
(t) lim<br />
.<br />
h→0 h<br />
<br />
<br />
<br />
f
f f ′ <br />
<br />
<br />
f ′ f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
v(t) 3<br />
8<br />
mph<br />
8<br />
miles<br />
4<br />
4<br />
hrs<br />
hrs<br />
1 2<br />
1 2<br />
y v(t) y s(t)<br />
<br />
<br />
<br />
<br />
y v(t)<br />
s(t) t <br />
s(0) 0
y s(t)<br />
t s <br />
v<br />
<br />
<br />
<br />
<br />
1<br />
A 2<br />
y = v(t)<br />
3 mph v(t)=2<br />
1<br />
hrs<br />
3 mph hrs<br />
A 1<br />
1 2 3<br />
1 2 3<br />
<br />
<br />
[1, 1.5] A 1 y v(t) [1, 1.5] <br />
A 1 2 <br />
· 1 1 .<br />
2<br />
<br />
v(t) [a, b] <br />
[a, b] A <br />
A v(a)(b − a) v(a)Δt,<br />
Δt t <br />
v(t) [a, b] v(a) <br />
<br />
<br />
<br />
v(t)
[1, 1.5] v(1) 2.5 v(1.5) ≈ 2.1 <br />
<br />
A 2 v(1)Δt 2.5 <br />
· 1 1.25 .<br />
2<br />
v [1, 1.5] A 2 1.25<br />
<br />
<br />
<br />
[0, 3] <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
3 mph y = v(t)<br />
1<br />
hrs<br />
1 2 3<br />
<br />
y v(t) [0, 3]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 <strong>2.0</strong>0<br />
v(t) 1.500 1.789 1.938 1.992 <strong>2.0</strong>00 <strong>2.0</strong>08 <strong>2.0</strong>63 2.211 2.500
t 0 t 2 <br />
Δt 0.5 <br />
<br />
<br />
<br />
[0, 2] <br />
<br />
v<br />
v(t) 0.5t 3 − 1.5t 2 +<br />
1.5t + 1.5 v <br />
<br />
s s <br />
s ′ v<br />
<br />
s(2) − s(0) <br />
<br />
3 mph y = v(t)<br />
2<br />
1<br />
hrs<br />
1 2<br />
y v(t)<br />
<br />
<br />
<br />
<br />
<br />
v <br />
v
y v(t) <br />
t v <br />
s s v <br />
t <br />
<br />
<br />
v(t) 3 <br />
8<br />
mph<br />
8<br />
miles<br />
s(t)=3t<br />
4<br />
v(t)=3<br />
4<br />
A = 3 · 1.25 = 3.75<br />
1 2<br />
hrs<br />
s(0.25)=0.75<br />
s(1.5)=4.5<br />
1 2<br />
hrs<br />
v(t) 3 s(t) 3t<br />
<br />
<br />
s(t) 3t s ′ (t) 3 s(t) 3t <br />
v(t) s(1.5) 4.5 s(0.25) 0.75 <br />
t 0.25 t 1.5 <br />
s(1.5) − s(0.25) 4.5 − 0.75 3.75 .<br />
[0.25, 1.5] <br />
[0.25, 1.5] <br />
<br />
<br />
<br />
v s s ′ v <br />
s(0) 0 <br />
s(0)
s v f ′ f <br />
f G G ′ <br />
G (x) 3x 2 + 2x <br />
G(x) x 3 + x 2 G ′ (x) (x) <br />
H(x) x 3 + x 2 + 5 <br />
H <br />
<br />
v(t) 32 − 32t t v <br />
0 ≤ t ≤ 2<br />
t <br />
<br />
sv s(0) 0<br />
s(1) − s( 1 2<br />
) <br />
y v(t) <br />
t 1 2<br />
t 1 <br />
<br />
<br />
<br />
[0, 1]<br />
s(2) − s(0)<br />
<br />
<br />
<br />
y v(t) <br />
24<br />
12<br />
-12<br />
-24<br />
ft/sec<br />
v(t)=32 − 32t<br />
sec<br />
1 2<br />
y v(t)
s(0) 0 <br />
s 1 <br />
s −1 <br />
<br />
t 0 3 <br />
4 <br />
<br />
3 <br />
t 0 t 3 <br />
<br />
<br />
t 0 t 1.5 <br />
D [0,1.5] 3 · 1.5 4.5 .<br />
t 1.5 t 2 <br />
D [1.5,2] 4 · 0.5 2 .<br />
<br />
D [2,3] 3 · 1 3 ,<br />
<br />
D D [0,1.5] + D [1.5,2] + D [2,3] 4.5 + 2 + 3 9.5 .<br />
1.5 < t < 2 v −4 <br />
<br />
<br />
4.5 − 2 + 3 5.5 .
4.5<br />
3.0<br />
mph<br />
y = v(t)<br />
(3,5.5)<br />
(1.5,4.5)<br />
1.5<br />
A 1 = 4.5<br />
A 3 = 3<br />
hrs<br />
4.5 miles y = s(t)<br />
3.0<br />
1.5<br />
(2,2.5)<br />
hrs<br />
-1.5<br />
-3.0<br />
1 3<br />
A 2 = 2<br />
-1.5<br />
-3.0<br />
1 3<br />
-4.5<br />
-4.5<br />
<br />
<br />
A 1 <br />
4.5 3 · 1.5 A 2 A 3 <br />
A 2 <br />
<br />
<br />
<br />
<br />
D A 1 + A 2 + A 3 4.5 + 2 + 3 9.5 .<br />
<br />
t t <br />
<br />
s(3) − s(0) (+4.5) + (−2) + (+3) 5.5 ,<br />
−2 [1.5, 2] <br />
<br />
<br />
<br />
y s(t) −4 <br />
1.5 < t < 2 −4 <br />
<br />
<br />
s
v t <br />
<br />
<br />
<br />
4 m/sec y = v(t)<br />
2<br />
sec<br />
8<br />
4<br />
2 4 6 8<br />
2 4 6 8<br />
-2<br />
-4<br />
-4<br />
-8<br />
<br />
t 0 s(0) 1<br />
<br />
0 ≤ t ≤ 2 t 2<br />
<br />
<br />
t 8 <br />
<br />
t 0 t 8<br />
t 1, 2, 3,...,8 <br />
y s(t)
y v(t) <br />
s <br />
<br />
<br />
<br />
<br />
<br />
t <br />
<br />
<br />
f F f F<br />
f F ′ f <br />
v v s <br />
s ′ v v [a, b] <br />
s(b) − s(a) [a, b]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f (t) 9t f (t) <br />
t 0 t 10
x f (t) 12 − 4t <br />
f (t) t 0 t 4<br />
<br />
<br />
<br />
<br />
v <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t <br />
v(t) <br />
<br />
<br />
<br />
<br />
v(t) t 2 − 6t + 8 <br />
[−2, 5]<br />
<br />
<br />
<br />
s(t)
0 ≤ t ≤ 4 t s(0) 1<br />
10<br />
mph<br />
y = v(t)<br />
10<br />
miles<br />
6<br />
6<br />
2<br />
hrs<br />
2<br />
hrs<br />
-2<br />
1 2 3 4 5<br />
-2<br />
1 2 3 4 5<br />
-6<br />
-6<br />
-10<br />
-10<br />
y v(t) <br />
y s(t)<br />
<br />
<br />
0 ≤ t ≤ 2 t 2<br />
<br />
0 ≤ t ≤ 4 <br />
<br />
y s(t) <br />
0 ≤ t ≤ 4 s<br />
<br />
t v(t) 500 − 32t <br />
<br />
a t a<br />
y v(t) t <br />
0 ≤ t ≤ a <br />
s v s s ′ (t) v(t)<br />
s(a) − s(0) <br />
<br />
s(5) − s(1) <br />
t <br />
v v
y v(t) <br />
<br />
1<br />
y = v(t)<br />
-1<br />
1 2 3 4 5 6 7<br />
y v(t) <br />
<br />
0 ≤ t ≤ 2<br />
s(5) − s(2) y s(t) <br />
<br />
[0, 2] [2, 4][5, 7]<br />
s <br />
s <br />
<br />
<br />
<br />
t <br />
<br />
t 0 6 12 18 24 30<br />
p(t) 7 8 10 13 18 35<br />
<br />
<br />
<br />
<br />
y p(t) t <br />
[0, 30]
v <br />
v(t) y v(t) t <br />
v(t) <br />
t <br />
<br />
<br />
<br />
<br />
A 1 A 2 A 3 <br />
D <br />
[a, b] <br />
y = v(t)<br />
D A 1 + A 2 + A 3 ,<br />
<br />
[a, b] <br />
A 1<br />
A 2<br />
A 3<br />
s(b) − s(a) A 1 − A 2 + A 3 .<br />
a<br />
b<br />
<br />
<br />
v(t) < 0 <br />
A 2 <br />
D s(b) − s(a) <br />
A 1 A 2 A 3
t v(t) 0.25t 3 − 1.5t 2 + 3t + 0.25 <br />
0 ≤ t ≤ 2 <br />
<br />
3 mph hrs<br />
3 mph hrs<br />
3 mph hrs<br />
y = v(t)<br />
y = v(t)<br />
y = v(t)<br />
2<br />
2<br />
2<br />
1<br />
A 4 B 4<br />
A 3 B 3<br />
A 2 B 2 2<br />
B 1 C 1<br />
1<br />
1<br />
C<br />
C 3<br />
C 4<br />
A 1<br />
1 2<br />
1 2<br />
1 2<br />
y v(t) <br />
[0, 2]<br />
y v(t) <br />
[0, 2] <br />
<br />
<br />
<br />
S A 1 + A 2 + A 3 + A 4<br />
y v(t) <br />
<br />
A 3 v(1) · 1<br />
2 2 · 1<br />
2 1.
T B 1 + B 2 + B 3 + B 4 .<br />
<br />
<br />
U C 1 + C 2 + C 3 + C 4 .<br />
S T U D<br />
[0, 2] <br />
<br />
<br />
<br />
<br />
<br />
<br />
Σ S <br />
<br />
1 + 2 + 3 + ···+ 100,<br />
<br />
<br />
∑100<br />
k 1 + 2 + 3 + ···+ 100.<br />
k1<br />
∑ 100<br />
k1<br />
k k k k <br />
<br />
<br />
10∑<br />
k1<br />
(k 2 + 2k) (1 2 + 2 · 1) + (2 2 + 2 · 2) + (3 2 + 2 · 3) + ···+ (10 2 + 2 · 10),<br />
<br />
n∑<br />
f (k) f (1) + f (2) + ···+ f (n).<br />
k1<br />
<br />
<br />
n
∑ 5<br />
k1 (k2 + 2)<br />
3 + 7 + 11 + 15 + ···+ 27<br />
∑ 6<br />
i3 (2i − 1) 4 + 8 + 16 + 32 + ···+ 256<br />
<br />
∑ 6<br />
i1 1 2 i<br />
<br />
y v(t) [a, b] <br />
<br />
[a, b] <br />
<br />
y f (x) [a, b] <br />
f <br />
f (x) ≥ 0<br />
[a, b] f <br />
[a, b]<br />
<br />
a<br />
b<br />
x 0<br />
x 1 x 2<br />
··· x i x i+1<br />
△x<br />
··· x n−1<br />
x n<br />
[a, b] n Δx<br />
n n <br />
[a, b] Δx b−a<br />
n<br />
<br />
x 1 x 0 + Δx x 2 x 0 + 2Δx x i a + iΔx, <br />
<br />
[x i , x i+1 ] <br />
y f (x)<br />
<br />
<br />
<br />
<br />
f [a, b]
[x i , x i+1 ] <br />
<br />
A i+1 f (x i ) · Δx,<br />
y = f (x)<br />
A 1 A 2 ··· A i+1 ···<br />
A n<br />
x 0 x 1 x 2 x i x i+1 x n−1<br />
x n<br />
[a, b] n Δx <br />
y f (x) [a, b] <br />
L n <br />
<br />
L n A 1 + A 2 + ···+ A i+1 + ···+ A n<br />
f (x 0 ) · Δx + f (x 1 ) · Δx + ···+ f (x i ) · Δx + ···+ f (x n−1 ) · Δx.<br />
<br />
L n <br />
∑n−1<br />
i0<br />
f (x i )Δx.<br />
0 n − 1 <br />
n L n f <br />
[a, b]
y = f (x)<br />
y = f (x)<br />
B 1 B 2 ··· B i+1 ···<br />
C 1 C 2 ··· C i+1 ···<br />
C n<br />
B n<br />
x 0 x 1 x 2 x i x i+1 x n−1<br />
x n<br />
x 0 x 1 x 2 x i x i+1 x n−1<br />
x n<br />
<br />
<br />
[x i , x i+1 ] B i+1 f (x i+1 ) · Δx, <br />
<br />
R n B 1 + B 2 + ···+ B i+1 + ···+ B n<br />
f (x 1 ) · Δx + f (x 2 ) · Δx + ···+ f (x i+1 ) · Δx + ···+ f (x n ) · Δx<br />
n∑<br />
f (x i )Δx.<br />
i1
R n f [a, b] <br />
<br />
x i+1 x i + x i+1<br />
2<br />
x i+1 [x i , x i+1 ] <br />
C 1 <br />
C 1 f (x 1 ) · Δx.<br />
<br />
M n C 1 + C 2 + ···+ C i+1 + ···+ C n<br />
f (x 1 ) · Δx + f (x 2 ) · Δx + ···+ f (x i+1 ) · Δx + ···+ f (x n ) · Δx<br />
n∑<br />
f (x i )Δx,<br />
i1<br />
M n f [a, b]<br />
f (x) ≥ 0 [a, b] L n R n M n <br />
y f (x) [a, b] <br />
f <br />
y v(t) <br />
[a, b] <br />
v<br />
<br />
<br />
<br />
x ∗ i+1 [x i, x i+1 ] x i ≤ x ∗ i+1 ≤ x i+1<br />
<br />
f (x ∗ 1 ) · Δx + f (x∗ 2 ) · Δx + ···+ f (x∗ i+1 ) · Δx + ···+ f (x∗ n) · Δx <br />
n∑<br />
f (x ∗ i )Δx.<br />
<br />
<br />
<br />
L n R n M n <br />
<br />
<br />
i1<br />
<br />
t v(t) 2 9 (t − 3)2 + 2<br />
<br />
2 ≤ t ≤ 5
[2, 5] L 4 R 4 M 4 <br />
L 4 R 4 M 4 <br />
L 4 R 4 <br />
f <br />
v f [a, b] L n <br />
f [a, b] R n <br />
f [a, b] <br />
<br />
<br />
L n <br />
∑n−1<br />
i0<br />
f (x i )Δx,<br />
f <br />
f [a, b] L n <br />
f <br />
y = f (x)<br />
y = f (x)<br />
y = f (x)<br />
A 1<br />
A 2<br />
A 3<br />
a b c d<br />
a b c d<br />
a b c d<br />
f <br />
f [a, d]<br />
<br />
[a, d] <br />
b ≤ x ≤ c [b, c] <br />
f (x i )Δx <br />
y f (x) x <br />
[b, c]
f [a, b] <br />
[a, b] f <br />
[a, b] f <br />
L 24 ≈ A 1 − A 2 + A 3 ,<br />
L 24 <br />
A 1 A 3 f <br />
A 2 f A 1 − A 2 + A 3<br />
f [a, d] <br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
<br />
v t <br />
v(t) 1 2 t2 − 3t + 7 2 .<br />
M 5 v [1, 5] <br />
Δt t 0 t 1 ··· t 5 <br />
<br />
<br />
<br />
[1, 5]<br />
<br />
[1, 5]<br />
M 10 M 20 <br />
n <br />
<br />
<br />
<br />
a b
f (x ∗ )Δx <br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
L n R n M n <br />
L n f (x 0 )Δx + f (x 1 )Δx + ···+ f (x n−1 )Δx <br />
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx <br />
M n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx <br />
∑n−1<br />
i0<br />
f (x i )Δx,<br />
n∑<br />
f (x i )Δx,<br />
i1<br />
n∑<br />
f (x i )Δx,<br />
i1<br />
x 0 a x i a + iΔx x n b Δx b−a<br />
n<br />
<br />
x i (x i−1 + x i )/2<br />
<br />
f (x) −x2<br />
4 +<br />
2x [2, 6]<br />
<br />
<br />
<br />
y f (x)
y −x2<br />
4<br />
+ 2x [2, 6]<br />
f (x) −x2<br />
4 +<br />
2x [2, 6]<br />
<br />
<br />
<br />
y f (x) <br />
<br />
y −x2<br />
4<br />
+ 2x [2, 6]<br />
0 ≤ t ≤ 8
y e x n 2 ∫ 3<br />
2 e x dx<br />
<br />
<br />
<br />
n 2 ∫ 3<br />
2 e x dx<br />
<br />
<br />
<br />
<br />
<br />
f (x) 3x + 4<br />
M 4 y f (x) [2, 5] <br />
Δx x 0 , x 1 ,...,x 4 <br />
<br />
<br />
y f (x) x [2, 5]<br />
<br />
n 4 M n L 4 R 4 <br />
<br />
M n <br />
n <br />
x [a, b]<br />
S <br />
S ((1.4) 2 + 1) · 0.4 + ((1.8) 2 + 1) · 0.4 + ((2.2) 2 + 1) · 0.4 + ((2.6) 2 + 1) · 0.4 + ((3.0) 2 + 1) · 0.4.<br />
S f [a, b] <br />
S f x [a, b] <br />
S <br />
<br />
S S <br />
<br />
R <br />
S
t 0 0.3 0.6 0.9 1.2 1.5 1.8<br />
v(t) 100 88 74 59 40 19 0<br />
<br />
<br />
<br />
<br />
<br />
[0, 1.8] L 6 R 6 1 2 (L 6 + R 6 )<br />
v(t) [0, 1.8] <br />
<br />
<br />
<br />
r <br />
<br />
M 4 [0, 4]<br />
M 4 <br />
r(t) 0.5e 0.5t r L 5 [0, 4]<br />
<br />
<br />
<br />
4<br />
3<br />
tons/week<br />
y = r(t)<br />
2<br />
1<br />
weeks<br />
1 2 3 4<br />
r(t)
f [a, b]<br />
<br />
<br />
<br />
<br />
<br />
<br />
y = f (x)<br />
y = f (x)<br />
y = f (x)<br />
A 1<br />
A 2<br />
A 3<br />
a b c d<br />
a b c d<br />
a b c d<br />
f <br />
f [a, d]<br />
<br />
<br />
<br />
n
f (x) sin(2x)−<br />
x 2<br />
10 + 3 [1, 7] R 10 4.90595<br />
<br />
<br />
<br />
<br />
x <br />
<br />
<br />
<br />
<br />
<br />
n 5<br />
f (x) 2x + 1 <br />
<br />
<br />
f (x) 2x + 1 [1, 4] <br />
L n M n R n n 5 n 25 n 100 <br />
f (x) 2x + 1 x [1, 4]<br />
f (x) 2x + 1
x [1, 4]<br />
<br />
<br />
f (x) x 2 + 1 [1, 4] <br />
<br />
L n M n R n n 5 n 25 n 100 <br />
f (x) x 2 + 1 x [1, 4]<br />
f (x) x 2 +1 <br />
x [1, 4] <br />
f (x) 2x + 1 <br />
[1, 4] L 5 16.2<br />
<br />
<br />
<br />
L n M n R n <br />
[a, b] <br />
x ∗ i+1 [x i, x i+1 ] <br />
n →∞
lim<br />
n→∞ L n lim<br />
n→∞<br />
R n lim<br />
n→∞<br />
M n lim<br />
n→∞<br />
n∑<br />
f (x ∗ i )Δx.<br />
f <br />
<br />
f [a, b] <br />
∫ b<br />
a<br />
f (x) dx <br />
∫ b<br />
n∑<br />
f (x) dx lim f (x ∗<br />
n→∞<br />
i )Δx,<br />
a<br />
Δx b−a<br />
n x i a + iΔx i 0,...,n x ∗ i<br />
x i−1 ≤ x ∗ i<br />
≤ x i i <br />
1,...,n<br />
∫ a b <br />
f ∫ b<br />
a<br />
f (x) dx <br />
<br />
<br />
∫ b<br />
a<br />
f (x) dx y f (x) x <br />
[a, b]<br />
i1<br />
i1<br />
<br />
f <br />
A 1 A 2 A 3 <br />
f x <br />
[a, b] [b, c] [c, d]<br />
<br />
∫ b<br />
∫ c<br />
f (x) dx A 1 , f (x) dx −A 2 ,<br />
a<br />
<br />
∫ d<br />
c<br />
∫ d<br />
a<br />
b<br />
f (x) dx A 3 ,<br />
f (x) dx A 1 − A 2 + A 3 .<br />
y = f (x)<br />
A 1<br />
A 2<br />
A 3<br />
a b c d<br />
f <br />
[a, d]
v [a, b] <br />
s(b) − s(a) <br />
s(b) − s(a) <br />
∫ b<br />
a<br />
v(t) dt.<br />
[a, b] ∫ b<br />
a<br />
v(t) dt <br />
[a, b], <br />
v <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x <br />
∫ 4<br />
1 (x2 + 1) dx <br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ 4<br />
1<br />
(2x + 1) dx <br />
<br />
x <br />
<br />
A 1 2 (3 +<br />
9) · 3 18<br />
∫ 4<br />
1<br />
(2x + 1) dx 18.<br />
9<br />
3<br />
∫ 4<br />
1<br />
f (x)=2x + 1<br />
(2x + 1)dx<br />
1 4<br />
f (x) <br />
2x + 1 x [1, 4]
∫ 1<br />
0 3xdx<br />
∫ 4<br />
−1<br />
(2 − 2x) dx<br />
∫ 1<br />
√<br />
−1 1 − x 2 dx<br />
y = g(x)<br />
1<br />
-3 -2 -1 1 2 3 4<br />
-1<br />
<br />
<br />
<br />
∫ 4<br />
−3<br />
(x) dx <br />
<br />
<br />
f [a, b] <br />
f x <br />
<br />
<br />
<br />
∫ a<br />
a<br />
f (x) dx a <br />
<br />
f a ∫ a<br />
a<br />
f (x) dx 0.
∫ b<br />
∫ c<br />
f (x) dx A 1 , f (x) dx A 2 ,<br />
a<br />
<br />
∫ c<br />
a<br />
b<br />
f (x) dx A 1 + A 2 ,<br />
<br />
<br />
y = f (x)<br />
A 1 A 2<br />
a b c<br />
f a b c <br />
y f (x)<br />
[a, c]<br />
∫ c<br />
a<br />
f (x) dx <br />
∫ b<br />
a<br />
f (x) dx +<br />
∫ c<br />
b<br />
f (x) dx.<br />
a < b < c <br />
a b c <br />
<br />
<br />
f a b <br />
∫ a<br />
b<br />
∫ b<br />
f (x) dx −<br />
a<br />
f (x) dx.<br />
a b <br />
Δx b−a<br />
a−b<br />
n<br />
b a Δx <br />
n<br />
− b−a<br />
n<br />
<br />
<br />
<br />
<br />
f <br />
k <br />
d<br />
dx [kf(x)] kf′ (x),
f <br />
d<br />
dx [ f (x) + (x)] f ′ (x) + ′ (x).<br />
<br />
<br />
<br />
<br />
<br />
<br />
B = 2 f (x i )△x<br />
y = 2 f (x)<br />
a<br />
A = f (x i )△x<br />
A<br />
x i x i+1<br />
y = f (x)<br />
b<br />
a<br />
x i<br />
B<br />
x i+1<br />
b<br />
y f (x) y 2 f (x) [a, b]<br />
<br />
x x <br />
<br />
f (x i ) 2 f (x i ) <br />
<br />
A B B 2A <br />
n <br />
y 2 f (x) <br />
y f (x) 2 <br />
k
f k <br />
∫ b<br />
a<br />
k · f (x) dx k<br />
∫ b<br />
a<br />
f (x) dx.<br />
f <br />
C =(f (x i )+g(x i ))△x<br />
A = f (x i )△x<br />
A<br />
f<br />
B = g(x i )△x<br />
B<br />
g<br />
C<br />
f + g<br />
a<br />
x i<br />
x i+1<br />
b<br />
a<br />
x i<br />
x i+1<br />
b<br />
a<br />
x i<br />
x i+1<br />
b<br />
y f (x) y (x) [a, b] <br />
y f (x) + (x)<br />
f <br />
f + ( f + )(x i ) f (x i ) + (x i )<br />
f <br />
A B C C A + B <br />
<br />
<br />
<br />
<br />
f <br />
∫ b<br />
a<br />
[ f (x) + (x)] dx <br />
∫ b<br />
a<br />
f (x) dx +<br />
∫ b<br />
a<br />
(x) dx.<br />
<br />
f c k<br />
∫ b<br />
∫ b<br />
∫ b<br />
[cf(x) ± k(x)] dx c f (x) dx ± k (x) dx.<br />
a<br />
a<br />
a
f <br />
x 2 x 3 <br />
∫ 2<br />
0 f (x) dx −3 ∫ 5<br />
2<br />
f (x) dx 2<br />
∫ 2<br />
0 (x) dx 4 ∫ 5<br />
2<br />
(x) dx −1<br />
∫ 2<br />
0 x2 dx 8 3 ∫ 5<br />
2 x2 dx 117<br />
3<br />
∫ 2<br />
0 x3 dx 4 ∫ 5<br />
2 x3 dx 609<br />
4<br />
<br />
<br />
∫ 2<br />
5<br />
∫ 5<br />
0<br />
f (x) dx<br />
<br />
(x) dx<br />
∫ 5<br />
0 ( f (x) + (x)) dx<br />
∫ 5<br />
2 (3x2 − 4x 3 ) dx<br />
∫ 0<br />
5 (2x3 − 7(x)) dx<br />
<br />
<br />
<br />
n y 1 y 2 ... y n <br />
<br />
AVG y 1 + y 2 + ···+ y n<br />
.<br />
n<br />
n <br />
<br />
R n f <br />
<br />
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx ( f (x 1 ) + f (x 2 ) + ···+ f (x n ))Δx.<br />
Δx b−a<br />
n<br />
<br />
R n ( f (x 1 ) + f (x 2 ) + ···+ f (x n )) · b − a<br />
n<br />
(b − a) f (x 1) + f (x 2 ) + ···+ f (x n )<br />
. <br />
n<br />
n <br />
(b − a) n <br />
n
f <br />
f [a, b] <br />
f (x 1 ) + f (x 2 ) + ···+ f (x n )<br />
f AVG[a,b] lim<br />
.<br />
n→∞ n<br />
f [a, b] <br />
lim n→∞ R n ∫ b<br />
a<br />
f (x) dx <br />
n →∞ <br />
∫ b<br />
a<br />
f (x) dx (b − a) · f AVG[a,b] .<br />
<br />
f AVG[a,b] <br />
<br />
f [a, b] [a, b] <br />
<br />
f AVG[a,b] 1 ∫ b<br />
b − a · f (x) dx.<br />
a<br />
<br />
f a b (b − a) <br />
<br />
f [a, b]<br />
∫ b<br />
a f (x)dx y = f (x)<br />
f AVG[a,b]<br />
(b − a) · f AVG[a,b]<br />
y = f (x)<br />
A 1<br />
A 2<br />
y = f (x)<br />
a<br />
b<br />
a<br />
b<br />
a<br />
b<br />
y f (x) [a, b]<br />
∫ b<br />
a<br />
f (x) dx <br />
(b − a) f AVG[a,b] <br />
y <br />
f AVG[a,b] (b − a) · f AVG[a,b] <br />
∫ b<br />
a<br />
f (x) dx
A 1 A 2 <br />
<br />
<br />
<br />
<br />
v(t) √ 4 − (t − 2) 2 <br />
0 ≤ t ≤ 4 t v <br />
<br />
y v(t) y √ 4 − (t − 2) 2 <br />
∫ 4<br />
0<br />
v(t) dt <br />
v(t) <br />
∫ 4<br />
0<br />
v(t) dt <br />
v(t) [0, 4] <br />
t 0 t 4 t<br />
∫ 4<br />
0<br />
v(t) dt <br />
<br />
<br />
[0, 4]<br />
<br />
f [a, b] <br />
<br />
<br />
<br />
<br />
<br />
<br />
f [a, b] ∫ b<br />
a<br />
f (x) dx <br />
f [a, b] <br />
∫ b<br />
a<br />
f (x) dx lim<br />
n→∞<br />
n∑<br />
f (x ∗ i )Δx,<br />
i1<br />
Δx b−a<br />
n x i a + iΔx i 0,...,n x ∗ i<br />
x i−1 ≤ x ∗ i<br />
≤ x i <br />
i 1,...,n
∫ b<br />
a<br />
f (x) dx f <br />
[a, b] <br />
[a, b] f [a,b] 1<br />
b−a · ∫ b<br />
a<br />
f (x) dx. <br />
v ∫ b<br />
a<br />
v(t) dt <br />
[a, b] v <br />
v(t) dt [a, b]<br />
∫ b<br />
a<br />
<br />
<br />
<br />
<br />
∫ b<br />
∫ b<br />
∫ b<br />
[cf(x) ± k(x)] dx c f (x) dx ± k (x) dx<br />
a<br />
a<br />
a<br />
f [a, b] c k <br />
<br />
f (x) <br />
<br />
<br />
<br />
∫ b<br />
a<br />
f (x)dx <br />
∫ c<br />
b<br />
f (x)dx <br />
∫ c<br />
a<br />
f (x)dx <br />
∫ c<br />
a | f (x)|dx <br />
f (x) <br />
∫ 0<br />
−4 f (x)dx <br />
A ∫ 6<br />
−4 f (x)dx <br />
A
f (x) 6x + 5 [2, 6]<br />
<br />
f (x) (x)<br />
f (x)<br />
(x)<br />
<br />
f (x) 0 ≤ x ≤ 2<br />
<br />
<br />
(x) 0 ≤ x ≤ 2<br />
<br />
<br />
f (x) · (x) 0 ≤ x ≤ 2<br />
<br />
<br />
<br />
( f ) · ( f · )<br />
y f (x) <br />
<br />
∫ 3<br />
−3<br />
f (x) dx ≈<br />
<br />
f <br />
<br />
∫ 13<br />
11.5<br />
f (x)dx <br />
∫ 14.5<br />
10<br />
f (x)dx 5,<br />
∫ 11.5<br />
10<br />
f (x)dx 6,<br />
∫ 14.5<br />
13<br />
f (x)dx 8
∫ 11.5<br />
13<br />
(5 f (x) − 6)dx <br />
<br />
v <br />
<br />
t 0 t 4<br />
<br />
<br />
<br />
[0, 4]<br />
v <br />
<br />
[0, 4]<br />
<br />
<br />
[0, 4]<br />
<br />
[0, 4]<br />
<br />
[0, 4]<br />
<br />
[0, 1.5] <br />
s(0) 0<br />
2 ft/sec 1<br />
sec<br />
-1<br />
-2<br />
1 2 3 4<br />
y = v(t)<br />
<br />
<br />
v(t) t(t − 1)(t − 3) <br />
0 ≤ t ≤ 4<br />
<br />
[0, 4]<br />
<br />
[0, 4] <br />
<br />
<br />
[0, 4]<br />
<br />
[0, 4]
[0, 4] <br />
f <br />
f <br />
2<br />
y = f (x)<br />
2<br />
y = g(x)<br />
1<br />
1<br />
1 2 3 4<br />
1 2 3 4<br />
-1<br />
-1<br />
-2<br />
-2<br />
f <br />
∫ 1<br />
0<br />
[ f (x) + (x)] dx<br />
∫ 4<br />
1<br />
[2 f (x) − 3(x)] dx<br />
h(x) (x) − f (x) [0, 4]<br />
c <br />
∫ 4<br />
0<br />
cdx<br />
∫ 4<br />
0<br />
[ f (x) + (x)] dx<br />
f (x) 3 − x 2 (x) 2x 2 <br />
[−1, 1] y f (x) <br />
y f (x) [−1, 1]<br />
[−1, 1] y (x) <br />
y (x) [−1, 1]<br />
<br />
y f (x) y (x) [−1, 1]<br />
∫ 1<br />
−1 [ f (x) −<br />
(x)] dx
p(x) ≥ q(x) x [a, b] y p(x)<br />
y q(x) <br />
∫ b<br />
a<br />
[p(x) − q(x)] dx.
v(t) <br />
[a, b] [a, b] <br />
<br />
y v(t) t [a, b] <br />
y v(t) [a, b] <br />
t <br />
∫ b<br />
a<br />
v(t) dt <br />
<br />
<br />
<br />
<br />
<br />
D <br />
[a, b] <br />
D A 1 + A 2 + A 3 ,<br />
<br />
[a, b] <br />
s(b) − s(a) A 1 − A 2 + A 3 .<br />
A 1 A 2 A 3 <br />
<br />
<br />
<br />
<br />
y = v(t)<br />
a<br />
A 1<br />
A 2<br />
A 3<br />
b
v(t) −32t + 16 v t <br />
s(t) t <br />
s v v s s ′ (t) v(t) <br />
s(t) <br />
s s(0) 32<br />
<br />
s( 1 2 ) − s(0) s(2) − s( 1 2<br />
) s(2) − s(0) <br />
<br />
<br />
y v(t) [0, 2] <br />
y v(t) t [0, 2] <br />
<br />
<br />
<br />
a<br />
D = ∫ b<br />
a v(t)dt<br />
= s(b) − s(a)<br />
y = v(t)<br />
b<br />
<br />
s(t) <br />
<br />
v(t) <br />
v(t) <br />
[a, b] <br />
<br />
D D <br />
s(b) − s(a) <br />
<br />
<br />
<br />
D ∫ b<br />
a<br />
v(t) dt<br />
v
s(b) − s(a) <br />
∫ b<br />
a<br />
v(t) dt.<br />
<br />
<br />
s(b) − s(a) [a, b] <br />
[a, b] ∫ b<br />
a<br />
v(t) dt<br />
<br />
s s v <br />
v s s v <br />
v(t) 3t 2 + 40 <br />
[1, 5]<br />
D <br />
∫ 5<br />
1<br />
v(t) dt <br />
∫ 5<br />
1<br />
(3t 2 + 40) dt s(5) − s(1),<br />
s v t 3 3t 2 <br />
40t 40 s(t) t 3 +40t s <br />
v(t) s ′ (t) 3t 2 + 40 v <br />
D <br />
∫ 5<br />
1<br />
3t 2 + 40 dt s(5) − s(1)<br />
(5 3 + 40 · 5) − (1 3 + 40 · 1) 284 .<br />
<br />
<br />
<br />
<br />
<br />
s <br />
s <br />
<br />
<br />
<br />
<br />
<br />
<br />
140<br />
120<br />
100<br />
80<br />
60<br />
40<br />
20<br />
y = v(t)<br />
D = ∫ 5<br />
1 v(t)dt<br />
= 284<br />
1 3 5<br />
<br />
v(t) 3t 2 + 40 [1, 5]
f f F <br />
F f F ′ (x) f (x) x V <br />
s V v <br />
<br />
∫ b<br />
V(b) − V(a) v(t) dt.<br />
<br />
∫ b<br />
a<br />
f (x) dx <br />
f f <br />
x <br />
v <br />
<br />
<br />
<br />
<br />
<br />
f [a, b] F f ∫ b<br />
a<br />
f (x) dx <br />
F(b) − F(a).<br />
a<br />
F(b) − F(a) <br />
F(b) − F(a) F(x)| b a ,<br />
F a b <br />
<br />
∫ b<br />
a<br />
f (x) dx F(x)| b a .<br />
<br />
F <br />
f d<br />
dx [ 1 3 x3 ] x 2 <br />
<br />
∫ 1<br />
x 2 dx 1 1<br />
3 x3 <br />
0<br />
0<br />
1 3 (1)3 − 1 3 (0)3<br />
1 3 .
∫ 4<br />
−1<br />
(2 − 2x) dx<br />
∫ π<br />
2<br />
sin(x) dx<br />
0<br />
∫ 1<br />
0 e x dx<br />
∫ 1<br />
<br />
−1 x5 dx<br />
∫ 2<br />
0 (3x3 − 2x 2 − e x ) dx<br />
<br />
<br />
<br />
f (x) <br />
sin(x) F(x) − cos(x) (x) x 2 G(x) 1 3 x3 <br />
f <br />
5 sin(x) − 4x 2 , x 2 sin(x),<br />
sin(x)<br />
x 2 , sin(x 2 ).<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
d<br />
[− cos(x)] sin(x),<br />
dx<br />
F(x) − cos(x) f (x) sin(x) <br />
f (x) sin(x) F(x) − cos(x) F f <br />
<br />
<br />
<br />
f (x) 5 sin(x) − 4x 2 <br />
− cos(x) sin(x) 1 3 x3 x 2 <br />
<br />
F(x) −5 cos(x) − 4 3 x3
f <br />
<br />
<br />
<br />
<br />
(x) x 2 G(x) 1 3 x3 <br />
G(x) 1 3 x3 + 7 G ′ (x) x 2 <br />
(x) x 2 <br />
<br />
G(x) 1 3 x3 + C,<br />
C <br />
+C G <br />
<br />
<br />
C <br />
<br />
∫ 1<br />
0<br />
x 2 dx.<br />
(x) x 2 G(x) <br />
1<br />
3 x3 + C. <br />
∫ 1<br />
0<br />
x 2 dx 1 3 x3 + C <br />
1<br />
<br />
0<br />
( ( 1 1<br />
3 (1)3 + C)<br />
−<br />
3 (0)3 + C)<br />
1 3 + C − 0 − C<br />
1 3 .<br />
C <br />
<br />
+C
F <br />
f <br />
<br />
f (x)<br />
kk <br />
x n n −1<br />
1<br />
x x > 0<br />
sin(x)<br />
cos(x)<br />
sec(x) tan(x)<br />
csc(x) cot(x)<br />
sec 2 (x)<br />
csc 2 (x)<br />
e x<br />
a x (a > 1)<br />
1<br />
1+x 2<br />
1<br />
√<br />
1−x 2<br />
F(x)<br />
<br />
<br />
<br />
<br />
∫ 1<br />
0<br />
∫ π/3<br />
0<br />
∫ 1<br />
0<br />
x 3 − x − e x + 2 dx<br />
(2 sin(t) − 4 cos(t) + sec 2 (t) − π) dt<br />
( √ x − x 2 ) dx<br />
<br />
<br />
<br />
<br />
v(t) <br />
[a, b] ∫ b<br />
a v(t) dt v(t) ≥ 0 [a, b] ∫ b<br />
a<br />
v(t) dt<br />
[a, b]<br />
f ∫ b<br />
a<br />
f (x) dx
y f (x) x [a, b] <br />
x <br />
<br />
f [a, b] f AVG[a,b] <br />
f AVG[a,b] 1 ∫ b<br />
b − a a<br />
f (x) dx.<br />
<br />
<br />
<br />
<br />
<br />
v V <br />
V(b) − V(a) <br />
∫ b<br />
a<br />
v(t) dt.<br />
V s v s ′ <br />
<br />
s(b) − s(a) <br />
∫ b<br />
a<br />
s ′ (t) dt.<br />
<br />
<br />
<br />
<br />
<br />
f <br />
<br />
f [a, b] f ′ f (b) −<br />
f (a) ∫ b<br />
a<br />
f ′ (x) dx. [a, b]<br />
[a, b]<br />
<br />
<br />
<br />
f <br />
f ′ f ′ f <br />
<br />
f f ′
4<br />
3<br />
2<br />
1<br />
3<br />
1<br />
3 4<br />
4<br />
3<br />
2<br />
1<br />
(2,4)<br />
(3,3)<br />
(1,3)<br />
(0,0)<br />
(4,0)<br />
-1<br />
-2<br />
-3<br />
1 2<br />
1<br />
y = f ′ (x)<br />
3<br />
-1<br />
-2<br />
-3<br />
1 2 3 4<br />
y = f (x)<br />
-4<br />
-4<br />
f ′ (x) 4 − 2x f (x) 4x − x 2 <br />
f f ′ <br />
f (1) − f (0) <br />
f (1) 3 f (0) 0 <br />
y f ′ (x) [0, 1] f (1) − f (0) ∫ 1<br />
0 f ′ (x) dx <br />
f ′ [0, 4] <br />
0 ∫ 4<br />
0 f ′ (x) dx 0 f (4) − f (0) 0 f (4) f (0)<br />
<br />
<br />
<br />
<br />
r(t) t r(t) <br />
r(t) 0.0069t 3 −0.125t 2 +11.079 y r(t)<br />
∫ 10<br />
4<br />
r(t) dt <br />
4 ≤ t ≤ 10
12<br />
gal/day<br />
10<br />
8<br />
y = r(t)<br />
6<br />
4<br />
2<br />
days<br />
2 4 6 8 10 12<br />
r(t) <br />
r(t) ≥ 0 ∫ 10<br />
4<br />
r(t) dt <br />
[4, 10] <br />
<br />
<br />
<br />
· .<br />
<br />
<br />
R(t)<br />
<br />
t R ′ (t) r(t) <br />
∫ 10<br />
4<br />
r(t) dt R(10) − R(4),<br />
<br />
<br />
<br />
r(t) 0.0069t 3 − 0.125t 2 + 11.079 <br />
∫ 10<br />
4<br />
0.0069t 3 − 0.125t 2 + 11.079 dt 0.0069 · 1<br />
4 t4 − 0.125 · 1<br />
3 t3 + 11.079t <br />
10<br />
≈ 44.282.<br />
4
4 ≤ t ≤ 10 <br />
r [4, 10] <br />
∫<br />
1 10<br />
r AVG[4,10] <br />
r(t) dt ≈ 44.282 7.380,<br />
10 − 4 4<br />
6<br />
<br />
<br />
c y c(t) <br />
0 ≤ t ≤ 10 c c(t) −0.05t 2 + t + 10 <br />
30 ≤ t ≤ 40 c(t) −0.05t 2 + 3t − 30<br />
15<br />
cal/min<br />
y = c(t)<br />
10<br />
5<br />
10 20 30<br />
min<br />
c(t) <br />
<br />
<br />
<br />
C(t) c(t) C(40) − C(0) <br />
<br />
<br />
<br />
<br />
<br />
0 ≤ t ≤ 40
f [a, b]<br />
F f <br />
∫ b<br />
a<br />
f (x) dx F(b) − F(a).<br />
f <br />
F [a, b]<br />
<br />
<br />
∫ b<br />
a<br />
f ′ (x) dx f (b) − f (a),<br />
f <br />
f [a, b] <br />
f [a, b]<br />
<br />
f (x) <br />
<br />
<br />
<br />
∫ b<br />
a<br />
f (x)dx <br />
∫ c<br />
b<br />
f (x)dx <br />
∫ c<br />
a<br />
f (x)dx <br />
∫ c<br />
a | f (x)|dx <br />
f (x)
∫ 0<br />
−3 f (x)dx <br />
A ∫ 5<br />
−3 f (x)dx <br />
A<br />
f (x) 4x + 4 [4, 9]<br />
<br />
f (x) (x)<br />
f (x)<br />
(x)<br />
<br />
f (x) 0 ≤ x ≤ 2<br />
<br />
<br />
(x) 0 ≤ x ≤ 2<br />
<br />
<br />
f (x) · (x) 0 ≤ x ≤ 2<br />
<br />
<br />
<br />
( f ) · ( f · )<br />
y f (x)
∫ 3<br />
−3<br />
f (x) dx ≈<br />
<br />
f <br />
<br />
∫ 6<br />
∫4.5<br />
4.5<br />
6<br />
f (x)dx <br />
∫ 7.5<br />
3<br />
(4 f (x) − 1)dx <br />
f (x)dx 4,<br />
∫ 4.5<br />
3<br />
f (x)dx 1,<br />
∫ 7.5<br />
6<br />
f (x)dx 1<br />
<br />
v 0 ≤ t ≤ 4 v(t) <br />
v(t) − 1 4 t3 + 3 2 t2 +1 v <br />
<br />
0 ≤<br />
t ≤ 4<br />
<br />
[12, 24]<br />
<br />
<br />
<br />
c <br />
t c <br />
<br />
[12, 24] <br />
15<br />
12<br />
9<br />
6<br />
3<br />
m/min<br />
y = v(t)<br />
min<br />
4 8 12 16 20 24<br />
f <br />
f (x) <br />
⎧⎪<br />
⎨<br />
⎪<br />
⎩<br />
<br />
<br />
−x 2 + 2x + 1, 0 ≤ x < 2<br />
−x + 3, 2 ≤ x < 3<br />
x 2 − 8x + 15, 3 ≤ x ≤ 5<br />
f x <br />
[2, 5]
f [0, 5]<br />
5 ≤ x ≤ 7 <br />
f f (x) (x) 5 ≤ x ≤ 7 ∫ 7<br />
0<br />
f (x) dx 0.<br />
<br />
<br />
<br />
c(h) h<br />
h 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000<br />
c 925 875 830 780 730 685 635 585 535 490 440<br />
m(h) <br />
h <br />
m(h) <br />
<br />
m(h) <br />
<br />
<br />
<br />
<br />
<br />
M 5 <br />
<br />
<br />
s(t) [a, b] <br />
s(b) − s(a)<br />
AV [a,b] .<br />
b − a<br />
<br />
v(t) [a, b] <br />
v AVG[a,b] 1 ∫ b<br />
v(t) dt.<br />
b − a a<br />
[a, b] <br />
[a, b]
f A(x) ∫ x<br />
0<br />
f (t) dt A<br />
<br />
f <br />
f <br />
y f ′ (x) f <br />
f <br />
<br />
f ′ f <br />
f f ′ <br />
f ′′ f <br />
f ′ f ′′ <br />
f <br />
<br />
<br />
<br />
<br />
<br />
f ′ f <br />
f <br />
[a, b] f ′ x
f <br />
f f <br />
<br />
<br />
<br />
f y f ′ (x) <br />
f ′ x ≤ 0 x ≥ 6 f ′ (x) 0 <br />
f (0) 1<br />
3<br />
y = f ′ (x)<br />
3<br />
1<br />
1<br />
-1<br />
1 3 5<br />
-1<br />
1 3 5<br />
-3<br />
-3<br />
y f ′ (x) y f (x)<br />
f f <br />
f <br />
f <br />
<br />
f (1) − f (0) <br />
∫ 1<br />
0<br />
f ′ (x) dx.<br />
f (1)
f (2) f (3) f (4) f (5) f (6)<br />
<br />
y f (x) <br />
f x < 0 x > 6<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f F <br />
a F(a) <br />
F(b) F(b) − F(a) ∫ b<br />
f (x) dx <br />
F(b) F(a) +<br />
∫ b<br />
a<br />
f (x) dx.<br />
a<br />
<br />
<br />
<br />
f F <br />
[a, b]<br />
F(b) − F(a) <br />
[a, b] ∫ b<br />
a<br />
f (x) dx<br />
f (x) x 2 f <br />
F(1) 2 a 1 b 2 <br />
<br />
F(2) F(1) +<br />
∫ 2<br />
1<br />
2 + 1 2<br />
3 x3 <br />
1<br />
( 8<br />
2 +<br />
3 − 1 )<br />
3<br />
13<br />
3 .<br />
x 2 dx<br />
f <br />
f <br />
F F
F <br />
<br />
f f <br />
<br />
y f (x) <br />
f <br />
F f F(0) −1 <br />
x ≤ 0 x ≥ 7 f (x) 0<br />
1<br />
y = f (x)<br />
1 2 3 4 5 6 7<br />
-1<br />
y f (x)<br />
F F <br />
F <br />
F <br />
F(x) x 1, 2,...,7<br />
F(−1) F(8)<br />
<br />
y F(x) <br />
F x < 0 x > 7 <br />
<br />
G<br />
f G(0) 0 <br />
G <br />
F <br />
<br />
<br />
f f
f <br />
3<br />
f<br />
G<br />
1<br />
2<br />
F<br />
-1<br />
1 3 5<br />
2 4<br />
H<br />
-2<br />
-3<br />
y f (x) f <br />
F f F(0) 1<br />
F(1) 1.5 F(2) 1.5 F(3) −0.5 F(4) −2<br />
F(5) −0.5 F(6) 1 F ′ f F <br />
<br />
<br />
F <br />
G f <br />
G(0) 3 G F f <br />
G F <br />
G(1) − G(0) ∫ 1<br />
0<br />
f (x) dx 0.5 F(1) − F(0) 0.5, G(0) 3<br />
G(1) G(0) + 0.5 3.5 F(1) F(0) + 0.5 1.5 F(0) 1 <br />
H(0) −1 <br />
<br />
<br />
G H f G − H <br />
<br />
G H f <br />
G ′ f H ′ f <br />
d<br />
dx [G(x) − H(x)] G′ (x) − H ′ (x) f (x) − f (x) 0<br />
<br />
G − H
f <br />
f (x) x 2 F(x) 1 3 x3 +C +C <br />
F f <br />
f F <br />
F(2) 3 <br />
x F(x) 1 3 x3 + C <br />
3 1 3 (2)3 + C,<br />
C 3 − 8 3<br />
1<br />
3 x3 + 1 3 .<br />
1 3 F(x) <br />
<br />
<br />
<br />
<br />
<br />
(x) |x| − 1 G(−1) 0 <br />
[−2, 2]<br />
h(x) sin(x) H(0) 1 <br />
[0, 4π]<br />
x ⎧⎪<br />
, 0 < x ≤ 1<br />
p(x) ⎨−(x − 2) 2 , 1 < x < 2 P(0) 1<br />
⎪<br />
⎩0 <br />
[−1, 3]<br />
<br />
<br />
F b F(a) <br />
a b f <br />
F(b) F(a) +<br />
∫ b<br />
a<br />
f (x) dx.<br />
F(b)<br />
(b, F(b)) F <br />
b
f A <br />
<br />
A(x) <br />
∫ x<br />
a<br />
f (t) dt.<br />
<br />
x <br />
A x <br />
a x <br />
t x <br />
A a x <br />
y f (t) <br />
f <br />
a 0 A(x) ∫ x<br />
0<br />
f (t) dt <br />
y = f (t)<br />
3<br />
1<br />
π<br />
2π<br />
-1<br />
x<br />
1<br />
A(x)<br />
π<br />
2π<br />
x<br />
f A(x) <br />
f (t) dt<br />
∫ x<br />
0<br />
A t 0 t x<br />
y f (t) <br />
y A(x) f A<br />
<br />
f <br />
A<br />
a <br />
a
A <br />
A(x) ∫ x<br />
1<br />
(t) dt<br />
3<br />
g<br />
3<br />
1<br />
1<br />
-1<br />
1 3 5<br />
-1<br />
1 3 5<br />
-3<br />
-3<br />
y (t) y A(x) A<br />
A(x) ∫ x<br />
1<br />
(t) dt<br />
A A <br />
<br />
A <br />
A <br />
A(0) A(1) A(2) A(3)<br />
A(4) A(5) A(6)<br />
<br />
y A(x) <br />
A x < 0 x > 6<br />
B A B B(x) ∫ x<br />
0<br />
(t) dt<br />
<br />
f F<br />
F F(a)
∫ b<br />
a<br />
f (x) dx a b <br />
F(3) F(3) F(a) + ∫ 3<br />
a<br />
f (x) dx <br />
F <br />
F ′ f F <br />
F<br />
F f <br />
G(x) F(x) + C f <br />
f <br />
<br />
<br />
f A(x) ∫ x<br />
a<br />
f (t) dt A <br />
f [a, x] A <br />
f <br />
<br />
f (x) <br />
∫ 0<br />
−5 f (x)dx <br />
A ∫ 7<br />
−5 f (x)dx <br />
A<br />
f (x) <br />
<br />
<br />
∫ 7<br />
0 f (x)dx ≈<br />
F f F(0) 40 F(7)<br />
F(7) ≈<br />
f ′ f f (5) 0
f <br />
f (0) <br />
<br />
f (7) <br />
<br />
∫ 7<br />
0 f ′ (x) dx <br />
<br />
f <br />
F ′ f F(0) 0 F(b) b <br />
<br />
b <br />
F(b)<br />
v <br />
A 1 7 6 A 2 8 3 <br />
s s(0) 0.5<br />
s(1) s(3) s(5) s(6)<br />
s s <br />
s s <br />
s <br />
v(t) −2 + 1 2 (t − 3)2 s
3<br />
v<br />
3<br />
s<br />
A 1<br />
A 2<br />
t<br />
1 6<br />
1<br />
-1<br />
2 4 6<br />
t<br />
-3<br />
-3<br />
v s<br />
<br />
<br />
<br />
c c <br />
15<br />
10<br />
cal/min<br />
c<br />
5<br />
10 20 30<br />
min<br />
10 20 30<br />
c C<br />
C c C
C(0) 0 C(t) t 5, 10, 15, 20, 25, 30<br />
C <br />
<br />
C 5 ≤ t ≤ 10<br />
f A B <br />
C A(x) ∫ x<br />
−1 f (t) dt B(x) ∫ x<br />
0 f (t) dt C(x) ∫ x<br />
1<br />
f (t) dt<br />
x −1, 0, 1,...,6 A(x)<br />
B(x) C(x)<br />
A B C<br />
A B C <br />
A <br />
f <br />
3<br />
3<br />
1<br />
f<br />
1<br />
-1<br />
1 3 5<br />
-1<br />
1 3 5<br />
-3<br />
-3<br />
f A B C
A(x) ∫ x<br />
1<br />
f (t) dt f <br />
<br />
<br />
<br />
<br />
<br />
<br />
f [a, b] F f F ′ f <br />
<br />
∫ b<br />
f (x) dx F(b) − F(a).<br />
a<br />
f <br />
f <br />
[a, b] F f<br />
<br />
<br />
<br />
<br />
∫ 4<br />
1<br />
x 2 dx,<br />
F(x) 1 3 x3 f (x) x 2 <br />
∫ 4<br />
1<br />
x 2 dx 1 3 x3 <br />
4<br />
1<br />
1 3 (4)3 − 1 3 (1)3<br />
21.<br />
<br />
F(b)− F(a) F f <br />
∫ b<br />
a<br />
f (x) dx <br />
F
F <br />
<br />
20<br />
10<br />
f (x)=x 2<br />
∫ 4<br />
1 x2 dx = 21<br />
20<br />
10<br />
(4, 64 3 )<br />
F(4) − F(1)=21<br />
F(x)=<br />
3 1x3<br />
(1, 1 3 )<br />
1 2 3 4<br />
1 2 3 4<br />
f (x) x 2 [1, 4] <br />
F(x) 1 3 x3 [1, 4] <br />
<br />
<br />
<br />
<br />
<br />
<br />
f AVG[a,b] 1<br />
b−a<br />
∫ b<br />
a<br />
f (x) dx<br />
<br />
A(x) ∫ x<br />
c<br />
f (t) dt <br />
<br />
A <br />
A(x) ∫ x<br />
c<br />
f (t) dt f <br />
c <br />
A f <br />
A <br />
A(x) <br />
∫ x<br />
1<br />
f (t) dt,
f (t) 4 − 2t<br />
A(1) A(2) <br />
A(x) <br />
∫ x<br />
1<br />
(4 − 2t) dt<br />
f A<br />
A ′ (x)<br />
f f (t) 4 − 2t f<br />
f (x) 4 − 2x <br />
A f <br />
<br />
f (t) 4 − 2t <br />
1 A <br />
f (t) cos(t) − t A(x) ∫ x<br />
2<br />
f (t) dt A<br />
<br />
A(x) <br />
∫ x<br />
2<br />
(cos(t) − t) dt<br />
sin(t) − 1 2 t2 <br />
x<br />
sin(x) − 1 2 x2 − (sin(2) − 2) .<br />
2<br />
A(x) (sin(2) − 2) <br />
A ′ (x) cos(x) − x,<br />
A ′ (x) f (x) f A <br />
f A(2) ∫ 2<br />
2<br />
f (t) dt 0 A <br />
f A(2) 0<br />
f A <br />
A(x) <br />
∫ x<br />
c<br />
f (t) dt,<br />
c A f <br />
A ′ (x) f (x) <br />
<br />
A ′ A(x + h) − A(x)<br />
(x) lim<br />
h→0 h<br />
∫ x+h<br />
c<br />
lim<br />
h→0<br />
f (t) dt − ∫ x<br />
c<br />
h<br />
f (t) dt
f (t) dt<br />
, <br />
h<br />
∫ x<br />
c<br />
f (t) dt+ ∫ x+h<br />
x<br />
f (t) dt <br />
f (t) dt h<br />
∫ x+h<br />
c<br />
∫ x+h<br />
x<br />
lim<br />
h→0<br />
∫ x+h<br />
x<br />
f (t) dt ≈ f (x) · h,<br />
<br />
<br />
∫ x+h<br />
A ′ x<br />
(x) lim<br />
h→0<br />
f (t) dt<br />
h<br />
lim<br />
h→0<br />
f (x) · h<br />
h<br />
f (x).<br />
A f A(c) ∫ c<br />
c<br />
f (t) dt 0. <br />
<br />
<br />
<br />
f c f <br />
A A(c) 0 A(x) ∫ x<br />
c<br />
f (t) dt<br />
f f <br />
<br />
1<br />
y = f (x)<br />
1 2 3 4 5 6 7<br />
-1<br />
y f (x) y A(x)<br />
A A(x) ∫ x<br />
2<br />
f (t) dt<br />
A f <br />
A(1) A(3) <br />
y A(x) <br />
A A <br />
A x 0, 1,...,7
A F <br />
<br />
<br />
B(x) ∫ x<br />
3 f (t) dt C(x) ∫ x<br />
1<br />
f (t) dt <br />
<br />
<br />
<br />
<br />
G <br />
G(x) <br />
∫ x<br />
c<br />
(t) dt<br />
G(c) 0 <br />
G ′ (x) (x) G<br />
<br />
d<br />
dx<br />
[∫ x<br />
]<br />
(t) dt (x).<br />
c<br />
<br />
G <br />
G ′ (x) (x) <br />
∫ x<br />
c<br />
(t) dt <br />
<br />
<br />
E(x) <br />
∫ x<br />
0<br />
e −t2 dt.<br />
E <br />
e −t2 <br />
<br />
f (t) 4 − 2t F(t) 4t − t 2 <br />
e −t2 <br />
<br />
<br />
E E <br />
x 0 <br />
erf(x) 2 √ π<br />
∫ x<br />
0 e−t2 dt 0 ≤ erf(x) < 1 <br />
x ≥ 0 lim x→∞ erf(x) 1
E E <br />
E ′ (x) d [∫ x<br />
]<br />
e −t2 dt e −x2 ,<br />
dx 0<br />
E E(0) 0 <br />
<br />
<br />
<br />
E E(0) 0 <br />
E E ′ (x) e −x2 <br />
x e −x2 > 0 E ′ (x) > 0 x E <br />
x →∞ E ′ (x) e −x2 → 0 <br />
E x →∞ x →−∞ <br />
E x <br />
<br />
E ′′ (x) −2xe −x2 E ′′ (0) 0 E ′′ (x) < 0 <br />
x > 0 E ′′ (x) > 0 x < 0 E x < 0 <br />
x > 0 x 0<br />
E x <br />
E(2) E(2) ≈ 0.8822<br />
E(3) E(3) ≈ 0.8862 <br />
x E 0.886 <br />
E <br />
f (t) e −t2 <br />
1<br />
1<br />
E(x)= ∫ x<br />
0 e−t2 dt<br />
f (t)=e −t2 -1<br />
-2 2<br />
-2 2<br />
-1<br />
f (t) e −t2 E(x) ∫ x<br />
0 e−t2 dt<br />
f E(0) 0
E f (t) e −t2 E(0) 0 <br />
y E(x) f (t) e −t2<br />
x E <br />
x f (t) e −t2<br />
x <br />
f (t) <br />
t F(x) ∫ x<br />
1+t 2 0<br />
f (t) dt<br />
f (t) <br />
t −10 ≤<br />
1+t 2<br />
t ≤ 10 <br />
F f <br />
F <br />
<br />
F <br />
f ′ (t) <br />
f ′ (t) <br />
1−t2 .<br />
(1+t 2 ) 2<br />
F(5) F(10) <br />
<br />
y F(x) <br />
<br />
f F
F <br />
f F F(x) ∫ x<br />
c<br />
f (t) dt<br />
F(x) ∫ x<br />
c<br />
f (t) dt <br />
∫<br />
F ′ (x) d x<br />
dx c<br />
f (t) dt f (x) <br />
<br />
<br />
F(x) <br />
∫ x<br />
π<br />
sin(t 2 ) dt,<br />
<br />
F ′ (x) sin(x 2 ).<br />
f <br />
<br />
[∫<br />
d x<br />
]<br />
f (t) dt f (x).<br />
dx<br />
a<br />
<br />
f f f t a<br />
t x x <br />
f (t) <br />
t t a t x <br />
<br />
∫ x<br />
a<br />
d f (t) dt?<br />
dt<br />
f (t) d dt<br />
<br />
∫ x<br />
a<br />
d x<br />
f (t) dt f (t) <br />
dt<br />
a<br />
f (x) − f (a).<br />
f (t)<br />
. <br />
f <br />
a x f f (a) <br />
f (a)
[∫ ]<br />
d x<br />
<br />
dx 4 e t2 dt<br />
∫ [ ]<br />
x d t<br />
4<br />
−2 dt 1+t dt 4<br />
[∫<br />
d 1<br />
<br />
dx x cos(t3 ) ]<br />
dt<br />
∫ x <br />
d<br />
3 dt ln(1 + t 2 ) dt<br />
[<br />
d ∫ ]<br />
x<br />
3<br />
<br />
dx 4<br />
sin(t 2 ) dt <br />
<br />
f A(x) ∫ x<br />
1<br />
f (t) dt <br />
f <br />
<br />
f c A(x) <br />
f (t) dt f A(c) 0<br />
∫ x<br />
c<br />
<br />
<br />
∫ x<br />
d f (t) dt f (x) − f (c)<br />
dt<br />
<br />
<br />
c<br />
d<br />
dx<br />
[∫ x<br />
c<br />
]<br />
f (t) dt f (x).<br />
(x) ∫ x<br />
2<br />
f (t) dt f (t)
(2) <br />
′ (3) <br />
<br />
<br />
<br />
<br />
x 0 ≤ x ≤ 8<br />
x <br />
<br />
∫<br />
d a<br />
cos(tan(t)) dt <br />
dx x<br />
F(9) <br />
F ′ (x) e −x2 /5 F(0) 2<br />
F(9) ≈<br />
F F(x) <br />
∫ x<br />
2 (t) dt. A 1 4.29 A 2 12.75 A 3 0.36<br />
A 4 1.79 A 2 x 0.5 x 2 <br />
6.06<br />
F <br />
F <br />
6<br />
15<br />
4<br />
2<br />
y = g(t)<br />
10<br />
5<br />
A 2<br />
A 1<br />
-2<br />
1 2 3 4<br />
A 4<br />
5 6<br />
A 3<br />
-4<br />
-1 1 2 3 4 5 6<br />
-5<br />
-10<br />
F
( ) 4πt<br />
R(t) 2 + 5 sin<br />
25<br />
<br />
S(t) <br />
15t<br />
1 + 3t<br />
R(t) S(t) t <br />
0 ≤ t ≤ 6 t 0 <br />
<br />
0 ≤ t ≤ 6 <br />
Y(x) <br />
x <br />
<br />
t 4 <br />
0 ≤ t ≤ 6 t <br />
<br />
<br />
<br />
<br />
c(h) h<br />
h 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000<br />
c 925 875 830 780 730 685 635 585 535 490 440<br />
<br />
m h y m(h) <br />
h <br />
m(h) <br />
m<br />
m(h) <br />
<br />
<br />
<br />
<br />
<br />
M(h) <br />
h <br />
M(6000) M(10000)
u <br />
<br />
<br />
<br />
F f <br />
∫ b<br />
a<br />
f (x) dx F(b) − F(a).<br />
<br />
<br />
<br />
∫ 1<br />
x 3 − √ x + 5 x dx,<br />
0<br />
f (x) x 3 − √ x + 5 x . <br />
F f F(x) 1 4 x4 − 2 3 x3/2 +<br />
ln(5) 1 5x <br />
<br />
∫ 1<br />
x 3 − √ x + 5 x dx 1<br />
0<br />
4 x4 − 2 3 x3/2 + 1<br />
1<br />
ln(5) 5x <br />
0<br />
( 1<br />
<br />
4 (1)4 − 2 3 (1)3/2 + 1 ) (<br />
ln(5) 51 − 0 − 0 + 1 )<br />
ln(5) 50<br />
− 5 12 + 4<br />
ln(5) .<br />
f <br />
∫ b<br />
a<br />
f (x) dx <br />
<br />
<br />
G(x) ∫ x<br />
a<br />
f (t) dt f
u <br />
x f u(x) <br />
d f (u(x)) f ′ (u(x)) · u ′ (x).<br />
dx<br />
c(x) f (u(x)) f<br />
u <br />
<br />
<br />
<br />
(x) <br />
′ (x)<br />
(x) e 3x<br />
h(x) sin(5x + 1)<br />
p(x) arctan(2x)<br />
q(x) (2 − 7x) 4<br />
r(x) 3 4−11x<br />
<br />
<br />
m M <br />
<br />
m(x) e 3x<br />
v(x) (2 − 7x) 3<br />
n(x) cos(5x + 1)<br />
w(x) 3 4−11x<br />
s(x) 1<br />
1+4x 2<br />
<br />
<br />
<br />
a(x) cos(πx)<br />
b(x) (4x + 7) 11<br />
c(x) xe x2<br />
+C
h(x) f (u(x)),<br />
f u(x) <br />
<br />
h(x) (5x − 3) 6 ,<br />
f f (u) u 6 u(x) 5x − 3<br />
f F(u) 1 7 u7 + C h <br />
H(x) 1 7 (5x − 3)7 · 1<br />
5 + C 1<br />
35 (5x − 3)7 + C.<br />
1 5<br />
<br />
u ′ (x) 5 H ′ (x) <br />
<br />
H ′ (x) 1<br />
35 · 7(5x − 3)6 · 5 (5x − 3) 6 h(x),<br />
H h<br />
<br />
<br />
h(x) f (ax + b) F f <br />
h <br />
H(x) 1 F(ax + b) + C.<br />
a<br />
<br />
<br />
d f (x)<br />
dx<br />
f (x) x <br />
<br />
∫<br />
f (x) dx<br />
f x <br />
h(x) (5x − 3) 6 h <br />
H <br />
∫<br />
(5x − 3) 6 dx 1 35 (5x − 6)7 + C.
d<br />
dx<br />
[□] x □<br />
∫ □ dx x □<br />
<br />
<br />
∫<br />
sin(8 − 3x) dx<br />
∫<br />
sec 2 (4x) dx<br />
∫ 1<br />
11x−9 dx<br />
<br />
∫<br />
csc(2x + 1) cot(2x + 1) dx<br />
∫ 1<br />
√<br />
1−16x 2 dx<br />
∫ 5 −x dx<br />
u<br />
<br />
<br />
<br />
(x) xe x2 h(x) e x2 ?<br />
<br />
+C <br />
<br />
<br />
<br />
∫<br />
d x<br />
5 5x 4 ,<br />
dx<br />
5x 4 dx x 5 + C.<br />
<br />
d f ((x)) f ′ ((x)) · ′ (x).<br />
dx<br />
<br />
∫<br />
f ′ ((x)) ′ (x) dx f ((x)) + C.<br />
<br />
<br />
f ′ ((x)) ′ (x) f <br />
(x)
′ (x) f ′ ((x)) ′ (x) <br />
<br />
<br />
u (x) u (x) <br />
du<br />
dx ′ (x) du ′ (x) dx <br />
u<br />
∫<br />
∫<br />
f ′ ((x)) ′ (x) dx f ′ (u) du.<br />
f ′ <br />
u <br />
f ′ ((x)) ′ (x) u u <br />
<br />
<br />
∫<br />
x 3 · sin(7x 4 + 3) dx<br />
<br />
x 3·sin(7x 4 +<br />
3) sin(7x 4 + 3) <br />
x 3 (7x 4 + 3) <br />
x 3 (7x 4 + 3) <br />
f (u) sin(u) <br />
<br />
u<br />
u sin(7x 4 + 3) u <br />
7x 4 + 3, du<br />
dx 28x3 . du 28x 3 dx <br />
x 3 dx 1<br />
28<br />
du <br />
<br />
∫<br />
sin(7x 4 + 3) · x 3 dx,<br />
u x u 7x 4 + 3 1<br />
28<br />
du <br />
x 3 dx <br />
∫<br />
∫<br />
sin(7x 4 + 3) · x 3 dx <br />
sin(u) ·<br />
1<br />
28 du.<br />
u <br />
u 7x 4 + 3 <br />
∫<br />
∫<br />
sin(7x 4 + 3) · x 3 1<br />
dx sin(u) ·<br />
28 du<br />
du<br />
dx ≈ Δu<br />
du<br />
Δx<br />
<br />
dx ′ (x) <br />
′ (x) ≈ Δu<br />
Δx Δu Δu ≈ ′ (x)Δx <br />
du ′ (x) dx
1 ∫<br />
28<br />
sin(u) du<br />
<br />
1 (− cos(u)) + C<br />
28<br />
− 1 28 cos(7x4 + 3) + C.<br />
<br />
[<br />
d<br />
− 1<br />
]<br />
dx 28 cos(7x4 + 3) − 1<br />
28 · (−1) sin(7x4 + 3) · 28x 3 sin(7x 4 + 3) · x 3 ,<br />
<br />
u <br />
sin(7x 4 + 3) x 3 x 2<br />
x 4 <br />
<br />
u u x 2 du 2xdx <br />
∫ ∫<br />
xe x2 dx e u · 1<br />
2 du<br />
1 ∫<br />
e u du<br />
2<br />
1 2 e u + C<br />
1 2 e x2 + C.<br />
<br />
∫<br />
e x2 dx,<br />
x e x2 u u x 2 <br />
u <br />
<br />
<br />
<br />
<br />
<br />
u du<br />
u<br />
u x
x <br />
<br />
∫ x 2<br />
5x 3 +1 dx<br />
∫<br />
e x sin(e x ) dx<br />
∫ cos( √ x)<br />
√ x<br />
dx<br />
u<br />
u <br />
f ((x)) ′ (x) <br />
<br />
<br />
<br />
∫ 5<br />
2<br />
xe x2 dx.<br />
<br />
<br />
∫ 5<br />
2<br />
xe x2 dx <br />
∫ x5<br />
x2<br />
xe x2 dx.<br />
u <br />
u x 2 <br />
du 2xdx x 2 u 2 2 4 x 5 u 5 2 25. <br />
u <br />
∫ x5<br />
x2<br />
∫ u25<br />
xe x2 dx e u · 1<br />
u4 2 du<br />
1 2 e u u25<br />
<br />
u4<br />
1 2 e25 − 1 2 e4 .<br />
∫ xe x2 dx, <br />
1 2 e x2 u <br />
<br />
∫ 5<br />
<br />
2<br />
xe x2 dx 1 2 e x2 <br />
5<br />
2<br />
1 2 e25 − 1 2 e4 ,
u<br />
∫ 2<br />
1<br />
x<br />
1+4x 2 dx<br />
∫ 1<br />
0 e−x (2e −x + 3) 9 dx<br />
∫ 4/π<br />
2/π<br />
cos( 1 x )<br />
x 2<br />
dx<br />
<br />
<br />
<br />
<br />
<br />
∫ f (x) dx<br />
f f F<br />
F ′ f <br />
∫<br />
d<br />
dx [F(x)] f (x) f (x) dx F(x) + C.<br />
f F F f <br />
u <br />
∫<br />
f ((x)) ′ (x) dx u (x) du ′ (x) dx<br />
∫<br />
∫<br />
f ((x)) ′ (x) dx <br />
f (u) du.<br />
x u <br />
(x) <br />
′ (x) <br />
′ (x) <br />
<br />
<br />
∫<br />
t 3 t 4 − 3 3<br />
dt <br />
F(x) f (x) <br />
<br />
f (x) 2x 3 sin(x 4 )<br />
F(x)
∫ ln 7 (z)<br />
dz <br />
z<br />
<br />
∫<br />
e 5x<br />
dx <br />
5 + e5x <br />
∫ 7e<br />
2 √ y<br />
√ dy y<br />
<br />
∫ 3π<br />
5π/2<br />
e sin(q) · cos q dq <br />
<br />
sin(x) cos(x)<br />
∫ tan(x) dx tan(x) <br />
sin(x)<br />
cos(x)<br />
u<br />
∫ tan(x) dx<br />
∫ cot(x) dx<br />
<br />
∫ sec 2 (x) + sec(x) tan(x)<br />
dx.<br />
sec(x) + tan(x)<br />
u sec(x) + tan(x)<br />
<br />
<br />
∫ sec(x) dx<br />
∫ csc(x) dx<br />
∫ x √ x − 1 dx.<br />
<br />
x <br />
√<br />
x − 1 u x − 1 x dx u<br />
x u<br />
√ u u 1/2 <br />
u <br />
∫ x 2√ x − 1 dx ∫ x √ x 2 − 1 dx
∫ sin 3 (x) dx<br />
u sin(x) <br />
<br />
sin 2 (x) + cos 2 (x) 1<br />
sin 3 (x) sin(x)·sin 2 (x) <br />
sin(x) <br />
u cos(x) <br />
<br />
∫ sin 3 (x) dx<br />
∫ cos 3 (x) dx<br />
<br />
<br />
<br />
r(t) 4 + sin(0.263t + 4.7) + cos(0.526t + 9.4).<br />
t r <br />
t <br />
r(t) <br />
<br />
∫ 24<br />
0<br />
r(t) dt
∫ x sin(x) dx ∫ xe x dx<br />
<br />
<br />
u dv <br />
<br />
u <br />
∫ x 3 sin(x 4 ) dx<br />
u <br />
<br />
u <br />
<br />
<br />
<br />
<br />
∫<br />
x sin(x) dx.<br />
<br />
f (x) x (x) sin(x) <br />
<br />
<br />
∫ x sin(x) dx <br />
<br />
<br />
<br />
<br />
f<br />
x <br />
d f (x) · (x) f (x) · ′ (x) + (x) · f ′ (x).<br />
dx
(x)<br />
′ (x)<br />
(x) x sin(x)<br />
h(x) xe x<br />
p(x) x ln(x)<br />
q(x) x 2 cos(x)<br />
r(x) e x sin(x)<br />
<br />
<br />
∫ xe x + e x dx<br />
∫ e x (sin(x) + cos(x)) dx<br />
∫ 2x cos(x) − x 2 sin(x) dx<br />
∫ x cos(x) + sin(x) dx<br />
∫ 1 + ln(x) dx<br />
<br />
<br />
<br />
<br />
<br />
∫<br />
x cos(x) dx.<br />
<br />
d<br />
[x sin(x)] x cos(x) + sin(x).<br />
dx<br />
<br />
<br />
∫ ( ) ∫<br />
∫<br />
d<br />
dx [x sin(x)] dx x cos(x) dx + sin(x) dx.<br />
<br />
<br />
<br />
∫<br />
x cos(x) dx
d f (x)(x) f (x) ′ (x) + (x) f ′ (x).<br />
dx<br />
x <br />
∫<br />
∫<br />
∫<br />
d f (x)(x) dx f (x) ′ (x) dx + (x) f ′ (x) dx.<br />
dx<br />
<br />
<br />
<br />
<br />
∫<br />
∫<br />
f (x)(x) f (x) ′ (x) dx + (x) f ′ (x) dx.<br />
<br />
<br />
<br />
f f (x) ′ (x) (x) f ′ (x) <br />
<br />
<br />
∫<br />
∫<br />
f (x) ′ (x) dx f (x)(x) − (x) f ′ (x) dx.<br />
<br />
u v u f (x) <br />
v (x) du f ′ (x) dx dv ′ (x) dx <br />
<br />
∫<br />
∫<br />
udv uv −<br />
vdu.<br />
<br />
u dv dv v ∫ vdu <br />
∫ udv <br />
<br />
<br />
∫<br />
x cos(x) dx
u dv <br />
u x dv cos(x) dx u cos(x) dv xdx <br />
∫ u dv ∫ <br />
vdu udv<br />
u x dv cos(x) dx <br />
du 1 dx v sin(x) <br />
<br />
∫<br />
∫<br />
x cos(x) dx x sin(x) − sin(x) · 1 dx.<br />
∫ sin(x) · 1 dx. <br />
<br />
∫<br />
x cos(x) dx x sin(x) − (− cos(x)) + C x sin(x) + cos(x) + C.<br />
<br />
<br />
<br />
∫ udv ∫ vdu <br />
u v <br />
∫ x cos(x) dx <br />
∫ sin(x)·1 dx x <br />
cos(x) <br />
<br />
+C<br />
<br />
<br />
<br />
<br />
∫<br />
te −t dt<br />
∫<br />
4x sin(3x) dx<br />
∫ z sec 2 (z) dz<br />
∫ x ln(x) dx<br />
<br />
<br />
<br />
<br />
u cos(x) dv xdx du − sin(x) dx <br />
v 1 2 x2 ∫ x cos(x) dx 1 2 x2 cos(x) − ∫ 1<br />
2<br />
x 2 (− sin(x)) dx <br />
x cos(x) 1 2 x2 sin(x)
∫<br />
arctan(x) dx.<br />
<br />
arctan(x) arctan(x)·1 <br />
arctan(x) 1 <br />
u arctan(x) dv 1 dx <br />
<br />
∫ t 3 sin(t 2 ) dt <br />
sin(t 2 ) <br />
t 3 t<br />
sin(t 2 ) u<br />
t 3 t · t 2 <br />
<br />
∫<br />
t · t 2 · sin(t 2 ) dt,<br />
z t 2 dz <br />
2t dt tdt 1 2<br />
dz z z <br />
u z<br />
<br />
∫<br />
∫<br />
t · t 2 · sin(t 2 ) dt z · sin(z) · 1<br />
2 dz.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ arctan(x) dx u <br />
arctan(x) dv 1 dx<br />
∫ ln(z) dz <br />
z t 2 ∫ t 3 sin(t 2 ) dt <br />
z <br />
∫ s 5 e s3 ds <br />
∫ e 2t cos(e t ) dt e 2t e t · e t .
∫ udv ∫ vdu <br />
<br />
<br />
<br />
∫ t 2 e t dt u t 2 dv e t dt <br />
du 2t dt v e t <br />
∫<br />
∫<br />
t 2 e t dt t 2 e t − 2te t dt.<br />
<br />
u 2t dv e t dt du 2 dt <br />
v e t <br />
∫<br />
( ∫ )<br />
t 2 e t dt t 2 e t − 2te t − 2e t dt .<br />
<br />
∫ 2te t dt <br />
<br />
∫<br />
t 2 e t dt t 2 e t − 2te t + 2e t + C.<br />
<br />
<br />
t 3 e t <br />
∫ e t cos(t) dt<br />
u e t cos(t) u cos(t) dv e t dt<br />
du − sin(t) dt v e t <br />
∫<br />
∫<br />
e t cos(t) dt e t cos(t) − e t (− sin(t)) dt,<br />
<br />
∫<br />
∫<br />
e t cos(t) dt e t cos(t) +<br />
e t sin(t) dt<br />
<br />
∫ e t sin(t) dt <br />
<br />
∫<br />
e t cos(t) dt
u sin(t) dv e t dt du cos(t) dt v e t <br />
∫<br />
( ∫<br />
e t cos(t) dt e t cos(t) + e t sin(t) −<br />
e t cos(t) dt<br />
)<br />
<br />
<br />
∫ e t cos(t) dt <br />
<br />
∫<br />
e t cos(t) dt <br />
∫<br />
2<br />
e t cos(t) dt e t cos(t) + e t sin(t),<br />
<br />
∫<br />
e t cos(t) dt 1 e t cos(t) + e t sin(t) + C.<br />
2<br />
<br />
C <br />
<br />
<br />
<br />
∫<br />
x 2 sin(x) dx<br />
∫ t 3 ln(t) dt<br />
∫<br />
e z sin(z) dz<br />
∫<br />
s 2 e 3s ds<br />
∫ t arctan(t) dt <br />
t<br />
2<br />
1 − 1 .<br />
1+t 2 1+t 2<br />
<br />
u <br />
<br />
∫ π/2<br />
0<br />
t sin(t) dt.<br />
∫ t sin(t) dt −t cos(t) +<br />
sin(t) + C,
∫ π/2<br />
0<br />
t sin(t) dt (−t cos(t) + sin(t)) π/2<br />
<br />
0<br />
<br />
(− π 2 cos( π 2 ) + sin( π )<br />
2 ) − (−0 cos(0) + sin(0))<br />
1.<br />
<br />
uv <br />
u t dv sin(t) dt <br />
du dt v − cos(t) <br />
∫ π/2<br />
0<br />
t sin(t) dt − t cos(t) π/2<br />
−<br />
0<br />
∫ π/2<br />
0<br />
(− cos(t)) dt<br />
− t cos(t) π/2<br />
+ sin(t) π/2<br />
<br />
0<br />
0<br />
<br />
(− π 2 cos( π 2 ) + sin( π )<br />
2 ) − (−0 cos(0) + sin(0))<br />
1.<br />
<br />
<br />
<br />
u <br />
<br />
<br />
<br />
<br />
<br />
∫<br />
∫<br />
e x2 dx x tan(x) dx,<br />
u <br />
<br />
e x2 x tan(x) <br />
<br />
<br />
F(x) ∫ x<br />
0 e t2 dt <br />
f (x) e x2 G(x) ∫ x<br />
0<br />
t tan(t) dt (x) x tan(x)<br />
F <br />
G u
∫ x sin(x) dx ∫ x ln(x) dx <br />
<br />
<br />
<br />
<br />
∫ f (x) ′ (x) dx u f (x) <br />
dv ′ (x) dx <br />
∫<br />
∫<br />
udv uv − vdu<br />
∫ f (x) ′ (x) dx ∫ vdu<br />
∫<br />
f ′ (x)(x) dx<br />
<br />
u dv <br />
∫ vdu <br />
∫ udv <br />
<br />
ln(x) <br />
u dv <br />
<br />
<br />
<br />
<br />
<br />
∫ x sin xdx<br />
∫ x 2<br />
dx<br />
1+x 3<br />
∫ x 2 e x3 dx<br />
∫ x 2 cos(x 3 ) dx<br />
∫ √<br />
1<br />
dx 3x+1<br />
<br />
<br />
<br />
∫<br />
3x cos(2x) dx +C<br />
<br />
∫<br />
(z + 1) e 4z dz
∫ 4<br />
0<br />
te −t dt <br />
f (t) te −2t F(x) ∫ x<br />
0<br />
f (t) dt<br />
F ′ (x)<br />
F <br />
F x > 0 <br />
∫ e 2x cos(e x ) dx<br />
e 2x e x · e x z e x <br />
z<br />
<br />
∫ e 2x cos(e 2x ) dx <br />
<br />
<br />
<br />
∫ e 2x sin(e x ) dx<br />
∫ e 3x sin(e 3x ) dx<br />
∫ xe x2 cos(e x2 ) sin(e x2 ) dx<br />
u<br />
<br />
<br />
u e 3x du 3e 3x dx<br />
<br />
<br />
<br />
∫ x 2 cos(x 3 ) dx<br />
∫ x 5 cos(x 3 ) dx x 5 x 2 · x 3 <br />
∫ x ln(x 2 ) dx<br />
∫ sin(x 4 ) dx<br />
∫ x 3 sin(x 4 ) dx<br />
∫ x 7 sin(x 4 ) dx
∫ √ a 2 + u 2 du<br />
<br />
<br />
u <br />
<br />
∫ xe x2 dx u <br />
∫<br />
xe x dx ∫ e x2 dx <br />
<br />
e x2 <br />
<br />
<br />
F(x) F ′ (x) e x2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
u <br />
<br />
<br />
<br />
u <br />
<br />
<br />
<br />
<br />
∫<br />
x 2 sin(x 3 ) dx ∫ ∫ ∫<br />
x 2 sin(x) dx sin(x 3 ) dx x 5 sin(x 3 ) dx<br />
∫ 1<br />
1+x 2 dx<br />
∫<br />
x<br />
1+x 2 dx<br />
∫ 2x+3<br />
1+x 2 dx<br />
∫<br />
e x<br />
1+(e x ) 2 dx
∫ x ln(x) dx<br />
∫ x √ 1 − x 2 dx<br />
∫ ln(x)<br />
x<br />
∫<br />
dx<br />
1<br />
√<br />
1−x 2 dx<br />
<br />
∫<br />
ln(1 + x 2 ) dx ∫ x ln(1 + x 2 ) dx<br />
∫<br />
√<br />
x<br />
dx ∫<br />
1−x 2<br />
1<br />
x √ 1−x 2 dx<br />
<br />
<br />
<br />
R(x) <br />
5x <br />
x 2 −x−2<br />
∫<br />
5x<br />
x 2 − x − 2 dx.<br />
R <br />
A<br />
x−2 + B<br />
x+1<br />
. <br />
5x<br />
(x − 2)(x + 1) A<br />
x − 2 +<br />
B<br />
x + 1 .<br />
(x − 2)(x + 1) <br />
5x A(x + 1) + B(x − 2).<br />
x <br />
x A B x −1<br />
5(−1) A(0) + B(−3),<br />
B 5 3 x 2 <br />
5(2) A(3) + B(0),<br />
A 10 3<br />
. <br />
∫<br />
5x<br />
x 2 − x − 2 dx ∫ 10/3<br />
x − 2 + 5/3<br />
x + 1 dx.<br />
<br />
∫<br />
5x 10<br />
x 2 dx <br />
− x − 2 3 ln |x − 2| + 5 ln |x + 1| + C.<br />
3<br />
R(x) P(x)<br />
Q(x)<br />
P<br />
Q <br />
<br />
A<br />
x − c ,<br />
A<br />
(x − c) n , Ax + B<br />
x 2 + k<br />
, <br />
Ax + B<br />
(x 2 + k) n<br />
P Q R
A B c k <br />
<br />
<br />
<br />
<br />
<br />
<br />
5x<br />
x 2 − x − 2 10<br />
3(x − 2) + 5<br />
3(x + 1) .<br />
<br />
<br />
<br />
<br />
<br />
∫<br />
1<br />
<br />
x 2 −2x−3 dx 1<br />
x 2 −2x−3 1/4<br />
x−3 − 1/4<br />
x+1<br />
<br />
<br />
∫ x 2 +1<br />
∫<br />
x 3 −x 2 dx x2 +1<br />
x 3 −x 2 − 1 x − 1 x 2 + 2<br />
x−1<br />
x−2<br />
x−2<br />
dx <br />
x 4 +x 2 x 4 +x 2<br />
1 x − 2 x 2 + −x+2<br />
1+x 2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
√ a 2 ± w 2 √ w 2 − a 2 . <br />
<br />
<br />
<br />
∫<br />
∫<br />
1<br />
√<br />
1 − x 2 dx,<br />
x<br />
√<br />
1 − x 2 dx, ∫ √1<br />
− x 2 dx.
arcsin(x) + C <br />
u u 1 − x 2 <br />
u<br />
<br />
(h)<br />
∫ √a 2 − u 2 du u 2<br />
√<br />
a 2 − u 2 + a2<br />
2 arcsin u a + C.<br />
a 1 u x du dx <br />
∫ √1<br />
− x 2 dx x √<br />
1 − x<br />
2<br />
2 − 1 arcsin x + C.<br />
2<br />
<br />
u <br />
u x <br />
∫ √9<br />
+ 64x 2 dx.<br />
a 3 u 8x <br />
+ du 8dx dx 1 8 du<br />
<br />
∫ √9<br />
∫ √9<br />
+ 64x 2 dx + u2 · 1<br />
8 du 1 ∫ √9<br />
+ u<br />
8<br />
2 du.<br />
<br />
∫ √9<br />
+ 64x 2 dx 1 ( u √<br />
u<br />
8 2<br />
2 + 9 + 9 2 ln |u + √ u 2 + 9| + C)<br />
1 8<br />
( 8x<br />
2<br />
√<br />
64x 2 + 9 + 9 2 ln |8x + √ 64x 2 + 9| + C)<br />
.<br />
1 8<br />
<br />
u<br />
<br />
∫ √<br />
x 2 + 4 dx<br />
∫<br />
x<br />
√<br />
x 2 +4 dx<br />
<br />
∫<br />
2<br />
√<br />
16+25x 2 dx<br />
∫ 1<br />
x 2√ 49−36x 2 dx
ax 2 +bx+c <br />
0 x a b c <br />
x −b±√ b 2 −4ac<br />
2a<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫<br />
x 2 dx x3<br />
3 + .<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫<br />
1<br />
16 − 5x 2 dx,<br />
<br />
<br />
<br />
1<br />
16 x − 5 3 x3 .
1<br />
<br />
16−5x 2<br />
<br />
∫ 1<br />
dx, <br />
16−5x 2<br />
<br />
<br />
<br />
<br />
∫<br />
√ √<br />
1<br />
5 5<br />
16 − 5x 2 dx 20 arctanh( 4 x).<br />
<br />
<br />
<br />
<br />
∫<br />
1<br />
16 − 5x 2 dx 1 (<br />
8 √ √ √ √ √ )<br />
log(4 5 + 5 x) − log(4 5 − 5 x) + .<br />
5<br />
<br />
<br />
<br />
∫<br />
e −x2 dx,<br />
<br />
<br />
<br />
∫<br />
e −x2 dx <br />
√ π<br />
erf(x) + .<br />
2<br />
(x) <br />
erf(x) √ 2 ∫ x<br />
e −t2 dt.<br />
π<br />
<br />
<br />
<br />
<br />
∫<br />
(1 + x)e x√ 1 + x 2 e 2x dx<br />
<br />
0
u xe x du (1 +<br />
x)e x dx <br />
∫<br />
(1 + x)e x√ 1 + x 2 e 2x dx <br />
∫ √1<br />
+ u 2 du,<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Bx+C<br />
x 2 +k<br />
k <br />
A<br />
(x−c) n<br />
n <br />
<br />
∫ √ a 2 + u 2 du a <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫<br />
2x<br />
x 2 − 25 dx<br />
<br />
∫<br />
2x<br />
x 2 −25 dx ∫ dx + ∫ dx <br />
<br />
+C
+C <br />
<br />
w x 2 − 25 <br />
∫<br />
2x<br />
x 2 −25 dx ∫ dw +C <br />
+C<br />
w <br />
+C <br />
<br />
∫<br />
1<br />
(x + 6)(x + 8) dx <br />
<br />
∫<br />
7x + 3<br />
x 2 − 3x + 2 dx <br />
<br />
∫ 6x 3 + 8x 2 + 2x + 6<br />
x 4 + 1x 2 dx<br />
<br />
<br />
a <br />
b <br />
c <br />
a<br />
x 2 + b x + cx + d<br />
x 2 + 1<br />
d <br />
<br />
+C<br />
<br />
25x − 10x 2 − 45<br />
(x − 5)(x 2 + 9) A<br />
x − 5 + Bx + C<br />
x 2 + 9<br />
A B C <br />
<br />
∫ 25x − 10x 2 − 45<br />
(x − 5)(x 2 + 9) dx
∫ x 3 +x+1<br />
dx<br />
x 4 −1<br />
∫<br />
<br />
<br />
x 5 +x 2 +3<br />
x 3 −6x 2 +11x−6 dx<br />
∫ x 2 −x−1<br />
(x−3) 3<br />
dx<br />
<br />
u <br />
<br />
<br />
∫ 1<br />
x √ dx<br />
9x 2 +25<br />
∫ √<br />
x 1 + x 4 dx<br />
∫<br />
<br />
√ e<br />
x<br />
4 + e 2x dx<br />
∫<br />
tan(x)<br />
√9−cos 2 (x) dx<br />
<br />
∫ √ x + √ 1 + x 2<br />
x<br />
dx.<br />
u <br />
u<br />
<br />
u dv
∫ 1<br />
0 e−x2 dx <br />
<br />
<br />
n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f [a, b] L n <br />
R n M n <br />
L n f (x 0 )Δx + f (x 1 )Δx + ···+ f (x n−1 )Δx <br />
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx <br />
M n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx <br />
∑n−1<br />
i0<br />
f (x i )Δx,<br />
n∑<br />
f (x i )Δx,<br />
i1<br />
n∑<br />
f (x i )Δx,<br />
i1<br />
<br />
<br />
<br />
x 0 a x i a + iΔx x n b Δx b−a<br />
n<br />
x i <br />
(x i−1 + x i )/2<br />
<br />
n <br />
<br />
<br />
<br />
f (x) 1<br />
20 (x −4)3 +7 [1, 8]
y = f (x)<br />
y = f (x)<br />
y = f (x)<br />
1 LEFT 8<br />
1 RIGHT 8<br />
1 MID 8<br />
y f (x) [1, 8] <br />
<br />
<br />
<br />
L n R n M n <br />
<br />
<br />
<br />
f (x) x y <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ 3<br />
0 x2 dx<br />
f (x) x 2 <br />
x 0 3 L 3 <br />
<br />
R 3 M 3 <br />
<br />
I <br />
∫ 3<br />
0 x2 dx
L 3 R 3 M 3 <br />
<br />
<br />
<br />
b 1 b 2 h<br />
f (x) x 2 [0, 3] <br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ b<br />
a<br />
f (x) dx <br />
n <br />
L n R n M n <br />
<br />
y f (x) <br />
<br />
<br />
y = f (x)<br />
D 1 D 2 D 3<br />
x 0 x 1 x 2 x 3<br />
∫ b<br />
a<br />
f (x) dx <br />
a x 0 b x 3
D 1 <br />
y f (x) [x 0 , x 1 ] <br />
f (x 0 ) f (x 1 ) <br />
x 1 − x 0 Δx b−a<br />
3<br />
<br />
<br />
D 1 1 2 ( f (x 0) + f (x 1 )) · Δx.<br />
D 2 D 3 T 3 <br />
∫ b<br />
a<br />
f (x) dx <br />
T 3 D 1 + D 2 + D 3<br />
1 2 ( f (x 0) + f (x 1 )) · Δx + 1 2 ( f (x 1) + f (x 2 )) · Δx + 1 2 ( f (x 2) + f (x 3 )) · Δx.<br />
<br />
<br />
T 3 1 2<br />
Δx <br />
f f <br />
T 3 1 ( f (x0 ) + f (x 1 ) + f (x 2 )) Δx + ( f (x 1 ) + 1 f (x2 ) + f (x 3 )) Δx.<br />
2<br />
2<br />
<br />
L 3 <br />
L 3 f (x 0 )Δx + f (x 1 )Δx + f (x 2 )Δx R 3 f (x 1 )Δx + f (x 2 )Δx +<br />
f (x 3 )Δx L 3 R 3 <br />
T 3 1 2 [L 3 + R 3 ] . <br />
<br />
<br />
<br />
T n ∫ b<br />
a<br />
f (x) dx n <br />
<br />
[ 1<br />
T n <br />
2 ( f (x 0) + f (x 1 )) + 1 2 ( f (x 1) + f (x 2 )) + ···+ 1 2 ( f (x n−1) + f (x n ))]<br />
Δx.<br />
<br />
∑n−1<br />
i0<br />
1<br />
2 ( f (x i) + f (x i+1 ))Δx.<br />
T n 1 2 [L n + R n ] .
∫ 2 1<br />
1<br />
dx<br />
x 2<br />
∫ 2<br />
1<br />
1<br />
x 2 dx <br />
<br />
∫ 2 1<br />
1<br />
dx T<br />
x 2 4 M 4 T 8 M 8 <br />
<br />
E T,4<br />
<br />
<br />
E T,4 <br />
∫ 2<br />
1<br />
1<br />
x 2 dx − T 4.<br />
<br />
<br />
∫ 2<br />
1<br />
E M,8 <br />
1 x 2 dx − M 8.<br />
E T,4 E T,8 E M,4 E M,8 <br />
<br />
<br />
f (x) 1 x 2<br />
<br />
<br />
f <br />
n L n R n M n<br />
∫ b<br />
a<br />
f (x) dx <br />
L n R n n <br />
n →∞ Δx b−a<br />
n<br />
→ 0<br />
n <br />
<br />
<br />
<br />
n
T 1 M 1 M 1<br />
∫ b<br />
a<br />
f (x) dx <br />
<br />
<br />
T 1 <br />
<br />
<br />
<br />
M 1 <br />
<br />
M 1 <br />
M 1<br />
∫ b<br />
a<br />
f (x) dx
n <br />
f [a, b] T n ∫ b<br />
a<br />
f (x) dx M n <br />
∫ b<br />
a<br />
f (x) dx f <br />
<br />
M n T n M 1 T 1<br />
f <br />
<br />
<br />
<br />
f (x) 1− x 2 ∫ 1<br />
0<br />
f (x) dx <br />
<br />
∫ 1<br />
(1 − x 2 ) dx x − x3<br />
1<br />
3<br />
2 0<br />
3 .<br />
0<br />
M 4 M 8 T 4 T 8 <br />
E M,4 E M,8 E T,4 E T,8 <br />
<br />
∫ 2<br />
1<br />
∫ 1<br />
1<br />
x 2 dx <br />
<br />
0 (1 − x2 ) dx 0.6 <br />
dx 0.5 <br />
x 2<br />
T 4 0.65625 −0.0104166667 0.5089937642 0.0089937642<br />
M 4 0.671875 0.0052083333 0.4955479365 −0.0044520635<br />
T 8 0.6640625 −0.0026041667 0.5022708502 0.0022708502<br />
M 8 0.66796875 0.0013020833 0.4988674899 −0.0011325101<br />
T 4 M 4 T 8 M 8 <br />
∫ 1<br />
0 (1 − x2 ) dx ∫ 2<br />
1<br />
dx<br />
1<br />
x 2<br />
∫ 2<br />
1<br />
1<br />
f [a, b] E T,4 ∫ b<br />
a<br />
f (x) dx − T 4 <br />
<br />
n 4 E T,4 <br />
T n M n n 4 n 8 <br />
<br />
<br />
T n M n <br />
<br />
<br />
E M,4 ≈− 1 2 E T,4 E M,8 ≈− 1 2 E T,8
E M,n ≈− 1 2 E T,n.<br />
<br />
<br />
<br />
<br />
<br />
T n 1 2 (L n + R n ).<br />
[a, b] <br />
L n R n ∫ b<br />
a<br />
f (x) dx <br />
L n R n <br />
L n R n <br />
M n T n <br />
<br />
M n T n <br />
<br />
S 2n 2M n + T n<br />
3<br />
. <br />
S 2n <br />
S 2n S n n <br />
n 2n <br />
<br />
<br />
S 8 2M 4+T 4<br />
3<br />
S 16 2M 8+T 8<br />
3
∫ 1<br />
∫ 2<br />
1<br />
1<br />
<br />
<br />
0 (1 − x2 ) dx 0.6 <br />
dx 0.5 <br />
x 2<br />
T 4 0.65625 −0.0104166667 0.5089937642 0.0089937642<br />
M 4 0.671875 0.0052083333 0.4955479365 −0.0044520635<br />
S 8 0.6666666667 0 0.5000298792 0.0000298792<br />
T 8 0.6640625 −0.0026041667 0.5022708502 0.0022708502<br />
M 8 0.66796875 0.0013020833 0.4988674899 −0.0011325101<br />
S 16 0.6666666667 0 0.5000019434 0.0000019434<br />
S 8 S 16 <br />
S 16 ∫ 2 1<br />
1<br />
dx<br />
x 2<br />
E S,16 0.0000019434 L 8 0.5491458502 <br />
E L,8 −0.0491458502 <br />
L n R n M n <br />
n T n S 2n <br />
∫ 1<br />
0 (1 − x2 ) dx <br />
∫ 1<br />
0 e−x2 dx <br />
e −x2 <br />
<br />
<br />
<br />
v <br />
<br />
t 0 0.3 0.6 0.9 1.2 1.5 1.8<br />
v(t) 100 99 96 90 80 50 0<br />
<br />
<br />
<br />
[0, 1.8]<br />
[0, 1.8] L 3 R 3 T 3
[0, 1.8] M 3 <br />
<br />
v<br />
<br />
<br />
<br />
<br />
<br />
[0, 1.8] <br />
<br />
0.3 0.6 0.9 1.2 1.5 1.8<br />
t<br />
<br />
<br />
L n R n T n M n S 2n <br />
<br />
<br />
<br />
<br />
f (x) 2 − x 2 (x) 2 − x 3 h(x) 2 − x 4 <br />
[0, 1] <br />
<br />
<br />
[0, 1] L 1 R 1 <br />
M 1 ∫ 1<br />
0 f (x) dx ∫ 1<br />
0 (x) dx ∫ 1<br />
0<br />
h(x) dx<br />
<br />
T 1 S 2 <br />
<br />
∫ 1<br />
0 f (x) dx ∫ 1<br />
0 (x) dx ∫ 1<br />
0<br />
h(x) dx <br />
<br />
f h L 1 R 1 M 1 T 1 <br />
S 2
2<br />
2<br />
2<br />
1<br />
1<br />
1<br />
<br />
<br />
f [a, b] L n ∫ b<br />
a<br />
f (x) dx<br />
f [a, b] M n <br />
<br />
L n R n T n M n <br />
<br />
<br />
f <br />
f <br />
<br />
<br />
<br />
∫ 5<br />
2 (5x3 − 2x 2 + 7x − 4) dx S 2n<br />
n <br />
<br />
<br />
<br />
∫ 1<br />
0 e−x2 dx <br />
<br />
<br />
L n R n M n <br />
n<br />
∫ b<br />
a<br />
f (x) dx <br />
<br />
T n 1 2 (L n + R n )<br />
S 2n 2n + 1 <br />
M n n T n n + 1 <br />
S 2n <br />
n
S n 2M n+T n<br />
3<br />
∫ b<br />
a<br />
f (x) dx<br />
S n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ 3<br />
0 e x dx<br />
∫ 3<br />
0 e x dx <br />
<br />
<br />
<br />
<br />
<br />
n 4 n 2<br />
<br />
<br />
<br />
<br />
<br />
n n 2 n 4 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f (x) <br />
∫ 3<br />
0<br />
f (x) dx
f <br />
<br />
f (x) (x) <br />
f <br />
f <br />
f <br />
f <br />
∫ b<br />
a<br />
<br />
f (x) dx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n 2
n 2 <br />
<br />
<br />
∫ 1<br />
0<br />
x tan(x) dx<br />
u <br />
<br />
L 4 R 4 M 4 T 4 S 4 <br />
∫ 1<br />
0<br />
x tan(x) dx<br />
<br />
f (x) <br />
f [3, 6]<br />
f [3, 6]<br />
f [3, 6]<br />
∫ 6<br />
3 f (x) dx L 4 7.23 R 4 6.75 M 4 7.05<br />
f [3, 6] <br />
f [3, 6] <br />
∫ 6<br />
3<br />
f (x) dx <br />
<br />
<br />
<br />
<br />
t 0 10 20 30 40 50 60<br />
r(t) 2000 2100 2400 3000 3900 5100 6500<br />
<br />
<br />
<br />
M n n <br />
M n <br />
<br />
S n <br />
n <br />
1<br />
60 S n 2000+2100+2400+3000+3900+5100+6500<br />
7
x y <br />
<br />
<br />
<br />
v <br />
a b <br />
∫ b<br />
a<br />
v(t) dt <br />
f [a, b] ∫ b<br />
a<br />
f (x) dx <br />
x x a x b<br />
<br />
<br />
<br />
<br />
f (x) 5 − (x − 1) 2 (x) <br />
4 − x<br />
f <br />
f <br />
<br />
<br />
y f (x) x f
y (x) x f <br />
f <br />
6<br />
4<br />
2<br />
1 2 3<br />
f <br />
<br />
<br />
<br />
f (x) (x − 1) 2 + 1 (x) x + 2 <br />
<br />
6<br />
6<br />
6<br />
4<br />
g<br />
4<br />
g<br />
4<br />
g<br />
2<br />
2<br />
2<br />
f<br />
f<br />
1 2 3<br />
1 2 3<br />
1 2 3<br />
f (x) (x − 1) 2 + 1 (x) x + 2 <br />
[0, 3]<br />
(x) x + 2 (0, 2) (3, 5)
y (x − 1) 2 + 1 y x + 2 x + 2 y <br />
x + 2 (x − 1) 2 + 1 x + 2 x 2 − 2x + 1 + 1 <br />
x 2 − 3x x(x − 3) 0,<br />
x 0 x 3 y x + 2 <br />
y <br />
[0, 3] <br />
∫ 3<br />
(x + 2) dx 21<br />
0<br />
2 ,<br />
f <br />
∫ 3<br />
0<br />
[(x − 1) 2 + 1] dx 6.<br />
<br />
A <br />
∫ 3<br />
0<br />
(x + 2) dx −<br />
∫ 3<br />
0<br />
[(x − 1) 2 + 1] dx 21 2 − 6 9 2 .<br />
<br />
<br />
<br />
x <br />
<br />
<br />
6<br />
4<br />
g<br />
g(x) − f (x)<br />
2<br />
g<br />
f<br />
△x<br />
f<br />
x<br />
1 2 3<br />
f (x) (x − 1) 2 + 1 (x) x + 2 <br />
[0, 3]
(x) − f (x) Δx <br />
<br />
A ((x) − f (x))Δx.<br />
[0, 3] <br />
A ≈<br />
n∑<br />
((x i ) − f (x i ))Δx,<br />
i1<br />
n →∞ <br />
A <br />
∫ 3<br />
0<br />
((x) − f (x)) dx.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ 3<br />
0<br />
(x) dx −<br />
∫ 3<br />
0<br />
f (x) dx <br />
∫ 3<br />
0<br />
((x) − f (x)) dx.<br />
<br />
y (x) y f (x) (a, (a)) (b, (b)) x <br />
a ≤ x ≤ b (x) ≥ f (x) A ∫ b<br />
a<br />
((x) − f (x)) dx.<br />
<br />
<br />
<br />
<br />
<br />
<br />
y √ x y 1 4 x<br />
y 12 − 2x 2 y x 2 − 8<br />
y f (x) cos(x) (x) sin(x) <br />
x f
y x 3 − x y x 2 <br />
<br />
<br />
<br />
x y 2 − 1 y x − 1 <br />
x x y +1 <br />
y + 1 y 2 − 1 y 2 − y − 2 0<br />
y −1 y 2 (0, −1) <br />
(3, 2)<br />
x<br />
x −1 x 0 <br />
<br />
<br />
<br />
2<br />
x = y 2 − 1<br />
2<br />
x = y 2 − 1<br />
2<br />
x = y 2 − 1<br />
1<br />
1<br />
1<br />
△y<br />
-1<br />
1 2 3<br />
y = x − 1<br />
-1<br />
1 2 3<br />
y = x − 1<br />
-1<br />
1 2 3<br />
x = y + 1<br />
x y 2 − 1 y x − 1 <br />
<br />
y <br />
y y −1 y 2 <br />
x y + 1 <br />
x y 2 − 1 Δy<br />
<br />
A [(y + 1) − (y 2 − 1)]Δy,<br />
y [−1, 2] <br />
<br />
A ≈<br />
n∑<br />
[(y i + 1) − (y 2 i<br />
− 1)]Δy.<br />
i1
A <br />
∫ y2<br />
[(y + 1) − (y 2 − 1)] dy.<br />
y−1<br />
<br />
y <br />
y <br />
Δy <br />
A 9 2 <br />
Δx <br />
<br />
x (y) x f (y) ((c), c) ((d), d) y <br />
c ≤ y ≤ d (y) ≥ f (y) <br />
A <br />
∫ yd<br />
yc<br />
((y) − f (y)) dy.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x y 2 x 6 − 2y 2 <br />
x 1 − y 2 x 2 − 2y 2 <br />
x y x 2 y 2 − x<br />
x y 2 − 2y y x
y<br />
f<br />
h<br />
△y<br />
x<br />
x 0 x 1 x 2 x 3 △x<br />
L slice<br />
y f (x) <br />
a x 0 b x 3 <br />
y f (x)<br />
x a x b L <br />
Δx L <br />
<br />
<br />
h L <br />
<br />
√<br />
L ≈ h (Δx) 2 + (Δy) 2 .<br />
<br />
<br />
(Δx) 2 <br />
√<br />
L ≈ (Δx) 2 + (Δy) 2<br />
<br />
√<br />
( )<br />
(Δx) 2 1 + (Δy)2<br />
(Δx) 2<br />
√<br />
1 + (Δy)2<br />
(Δx) 2 · Δx.<br />
n →∞ Δx → 0 Δy<br />
Δx → dy<br />
dx f ′ (x) <br />
√<br />
L ≈ 1 + f ′ (x) 2 Δx.
n →∞ <br />
<br />
f [a, b] L <br />
y f (x) x a x b <br />
∫ b √<br />
L 1 + f ′ (x) 2 dx.<br />
a<br />
<br />
<br />
<br />
y x 2 x −1 x 1<br />
y √ 4 − x 2 −2 ≤ x ≤ 2 <br />
<br />
<br />
y xe 3x [0, 1]<br />
<br />
<br />
y f (x) 0.1x 2 + 1 <br />
x y <br />
y f (x) <br />
t 4<br />
t 0 (0, f (0))<br />
<br />
<br />
x a x b<br />
A ((x) − f (x))Δx, <br />
<br />
A <br />
∫ b<br />
a<br />
((x) − f (x)) dx.<br />
<br />
Δx Δy <br />
<br />
y <br />
x
L y f (x) x a x b <br />
∫ b √<br />
L 1 + f ′ (x) 2 dx.<br />
a<br />
<br />
y x 1/2 y x 1/4 0 ≤ x ≤ 1<br />
<br />
y 7 sin x y 10 cos x [0,π] <br />
<br />
A <br />
x + y 2 42 x + y 0<br />
x y <br />
<br />
<br />
f (x) 9 √ x 3 x 5 x 8<br />
<br />
<br />
x y(y − 2) x −(y − 1)(y − 3)<br />
[ π 4 , 3π 4 ]<br />
x y 2 − y − 2 y 2x − 1<br />
y mx y x 2 − 1 m <br />
f (x) 1 − x 2 (x) ax 2 − a a <br />
a f <br />
f (x) 2−x 2 f <br />
∫<br />
[a, b] 1 b<br />
b−a a<br />
f (x) dx<br />
f (x) 2 − x 2 [0, √ 2] r<br />
y f (x) y r <br />
[0, √ 2] y f (x) <br />
y r y r y f (x)
y <br />
x<br />
x <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
V πr 2 h<br />
<br />
<br />
r 2 Δx<br />
<br />
<br />
V π · 2 2 · Δx.<br />
2<br />
y<br />
x<br />
3<br />
x<br />
Δx<br />
<br />
Δx → 0 <br />
V <br />
∫ 3<br />
0<br />
π · 2 2 dx.<br />
∫ 3<br />
0<br />
4π dx 12π 12π
y f (x) <br />
<br />
Δx <br />
x <br />
x 0 x 5 <br />
<br />
<br />
x<br />
<br />
<br />
3<br />
y<br />
x<br />
5<br />
x<br />
<br />
<br />
<br />
<br />
<br />
Δx<br />
<br />
<br />
<br />
V 1 3 πr2 h.<br />
<br />
<br />
<br />
<br />
<br />
<br />
(0, 2) (3, 2) x <br />
y 3 − 3 5<br />
x x 0 x 5<br />
x
R<br />
y 4 − x 2 x x<br />
y 4 − x 2 x (−2, 0) <br />
(2, 0) R x <br />
x <br />
<br />
<br />
x x −2<br />
x 2 <br />
Δx <br />
<br />
<br />
y 4 − x 2 <br />
V π(4 − x 2 ) 2 Δx,<br />
r<br />
h V πr 2 h<br />
y = 4 − x 2<br />
y<br />
x<br />
Δx<br />
<br />
<br />
<br />
V <br />
∫ 2<br />
−2<br />
π(4 − x 2 ) 2 dx.<br />
V 512<br />
15 π<br />
<br />
<br />
x x
y r(x) [a, b] <br />
x <br />
<br />
V <br />
∫ b<br />
a<br />
πr(x) 2 dx.<br />
<br />
<br />
<br />
R y 4 − x 2 y x + 2 x<br />
y 4 − x 2 y x + 2 <br />
y <br />
x + 2 4 − x 2 <br />
x 2 + x − 2 0,<br />
x −2 x 1 (−2, 0)<br />
(1, 1)<br />
R x <br />
<br />
y<br />
R(x)<br />
x<br />
r(x)<br />
<br />
r(x) R(x)
x x −2<br />
x 1 r(x) y x + 2 <br />
r(x) x + 2 R(x) y 4 − x 2 <br />
R(x) 4 − x 2 <br />
<br />
<br />
<br />
πR(x) 2 Δx − πr(x) 2 Δx π[R(x) 2 − r(x) 2 ]Δx,<br />
V π[(4 − x 2 ) 2 − (x + 2) 2 ]Δx.<br />
<br />
<br />
V <br />
∫ 1<br />
−2<br />
π[(4 − x 2 ) 2 − (x + 2) 2 ] dx.<br />
V 108<br />
5 π<br />
<br />
<br />
<br />
y R(x) y r(x) [a, b] <br />
R(x) ≥ r(x) x [a, b] <br />
x <br />
V <br />
∫ b<br />
a<br />
π[R(x) 2 − r(x) 2 ] dx.<br />
<br />
<br />
<br />
<br />
<br />
S x y √ x x 4<br />
S x<br />
S y y √ x y 2<br />
S x
S y √ x y x 3 S <br />
x<br />
S y 2x 2 + 1 y x 2 + 4 S<br />
x<br />
S y y √ x y 2<br />
S y <br />
y<br />
<br />
y <br />
Δy <br />
y <br />
<br />
R y √ x y x 4 y<br />
x 1 (1, 1)<br />
R y <br />
<br />
y<br />
R(y)<br />
r(y)<br />
x<br />
<br />
r(y) R(y)<br />
Δy
y y 0 <br />
y 1 y √ x <br />
x y x x y 2 r(y) <br />
y x 4 x <br />
y x 4 √ y <br />
V π[R(y) 2 − r(y) 2 ] π[ 4 √ y<br />
2 − (y 2 ) 2 ]Δy.<br />
y 0 <br />
y 1 <br />
V <br />
∫ y1<br />
y0<br />
π [ 4√ y<br />
2 − (y 2 ) 2] dy.<br />
V 7<br />
15 π<br />
<br />
<br />
<br />
<br />
<br />
S y y √ x y 2<br />
S y<br />
S x y √ x x 4<br />
S y<br />
S y 2x y x 3 <br />
S x<br />
S y 2x y x 3 <br />
S y<br />
S x (y − 1) 2 y x − 1 S<br />
y<br />
<br />
<br />
y 0 x 0
S y x 2 y x y −1<br />
<br />
<br />
<br />
(0, 0) (1, 1) <br />
y <br />
−1 <br />
<br />
<br />
<br />
y x 2 y x <br />
<br />
<br />
<br />
r(x) r(x) x 2 + 1 <br />
R(x) x + 1<br />
y<br />
x<br />
<br />
<br />
<br />
V π[R(x) 2 − r(x) 2 ]Δx π (x + 1) 2 − (x 2 + 1) 2 Δx.<br />
<br />
V <br />
∫ 1<br />
0<br />
π (x + 1) 2 − (x 2 + 1) 2 dx 7 15 π.<br />
<br />
<br />
<br />
<br />
<br />
S y 2x y x 3 <br />
S y −2<br />
S y 4<br />
S x −1<br />
S x 5
Δy y <br />
x <br />
x y <br />
<br />
<br />
<br />
<br />
<br />
y e 2x , y 0, x −2, x 0 x <br />
<br />
<br />
<br />
y x 6 y 1 y y<br />
<br />
<br />
y x 6 y 1 y x<br />
<br />
<br />
y x 6 y 1 y y −5<br />
<br />
<br />
y 4 <br />
<br />
y x 2 , y 1<br />
<br />
x −5<br />
y x 2 , x y 2
f (x) 3 cos( x3<br />
4<br />
) <br />
y x f (x) 0 R<br />
f x y<br />
f <br />
R <br />
R <br />
<br />
R x <br />
<br />
<br />
R y <br />
<br />
<br />
y sin(x) y cos(x) <br />
<br />
<br />
R y y sin(x) y cos(x)<br />
x <br />
<br />
R<br />
<br />
R x<br />
<br />
R y<br />
<br />
R y 2<br />
<br />
R x −1<br />
R y 1 + 1 2 (x − 2)2 y 1 2 x2 <br />
x 0<br />
<br />
<br />
<br />
R y −1<br />
<br />
R y<br />
<br />
R
v(t) <br />
t <br />
<br />
∫ b<br />
n∑<br />
v(t) dt ≈ v(t i )Δt,<br />
a<br />
v(t) t<br />
i1<br />
· .<br />
∫ b<br />
a<br />
v(t) dt <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
c(x) 200 + 100e −0.1x <br />
x <br />
∫ 2<br />
0 (200 + 100e−0.1x ) dx<br />
c(x) · Δx<br />
<br />
<br />
∫ 2<br />
n∑<br />
c(x) dx ≈ c(x i )Δx?<br />
0<br />
i1
∫ 2<br />
0 c(x) dx ∫ 2 <br />
0 200 + 100e<br />
−0.1x<br />
dx <br />
<br />
B <br />
x <br />
B(x) B(x) 0.5 + 1 <br />
(x+1) 2<br />
B(x) · Δx<br />
<br />
<br />
∫ 36<br />
n∑<br />
B(x) dx ≈ B(x i )Δx?<br />
12<br />
∫ 72<br />
0<br />
B(x) dx ∫ ( )<br />
72<br />
0 0.5 +<br />
1<br />
(x+1) dx <br />
2<br />
<br />
i1<br />
<br />
<br />
<br />
<br />
3 <br />
3 <br />
1500<br />
0.0023 3 .<br />
3 <br />
<br />
d m V<br />
d m , m d · V.<br />
V<br />
<br />
m d · V <br />
· <br />
A l · w
Δt <br />
v(t)Δt v(t) <br />
· f<br />
Δx <br />
f (x) Δx f (x)Δx <br />
<br />
ft/sec<br />
y<br />
y = v(t)<br />
y = f (x)<br />
v(t)<br />
sec<br />
f (x)<br />
x<br />
△t<br />
△x<br />
v(t)Δt <br />
f (x)Δx<br />
<br />
<br />
<br />
<br />
b x 0<br />
2 ρ(x)<br />
<br />
x ρ(x) <br />
x<br />
Δx<br />
2 ρ(x)<br />
3 <br />
x
Δx <br />
Δx <br />
2 1Δx 3 <br />
<br />
<br />
≈ ρ(x)<br />
<br />
3 · 1Δx 3 ρ(x) · Δx .<br />
<br />
n∑ ∫ b<br />
ρ(x i )Δx ≈ ρ(x) dx,<br />
i1<br />
0 b <br />
<br />
<br />
x <br />
ρ(x) <br />
ρ(x) · Δx Δx <br />
· <br />
<br />
<br />
ρ(x) <br />
M x a x b <br />
M <br />
∫ b<br />
a<br />
0<br />
ρ(x) dx.<br />
<br />
<br />
2 <br />
ρ(x) 2e −0.2x x <br />
ρ(x) <br />
<br />
<br />
<br />
<br />
<br />
3
x <br />
ρ(x) 400+ 200 3 <br />
1+x 2<br />
<br />
<br />
x <br />
<br />
2 <br />
ρ(x) 1<br />
25 (x − 15)2 <br />
z <br />
<br />
<br />
<br />
n a 1 a 2 ... a n <br />
<br />
a 1 + a 2 + ···+ a n<br />
n<br />
f [a, b] <br />
[a, b] <br />
∫<br />
1 b<br />
f (x) dx.<br />
b − a a<br />
<br />
<br />
<br />
,<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
3.3 + 3.7 + 2.7 + 2.7<br />
4<br />
3.1.
3.3 · 5 + 3.7 · 4 + 2.7 · 3 + 2.7 · 3<br />
5 + 4 + 3 + 3<br />
3.16.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x 0 <br />
x 1 0 x 2 6<br />
x <br />
<br />
x 1 0 <br />
x 2 2x 3 4 x 4 6 x <br />
x <br />
x 1 0 x 2 2 x 3 3 x 4 6<br />
<br />
x 1 0 x 2 2 x 3 4 x 4 6 <br />
x <br />
<br />
<br />
x 3 5 x <br />
<br />
x 3 4 <br />
x <br />
<br />
<br />
<br />
<br />
<br />
x
n m 1 ... m n <br />
x 1 ... x n <br />
x x 1m 1 + x 2 m 2 + ···+ x n m n<br />
m 1 + m 2 + ···+ m n<br />
.<br />
<br />
<br />
<br />
ρ(x) <br />
<br />
Δx x i m i <br />
m i ≈ ρ(x i )Δx<br />
n <br />
x ≈ x 1 · ρ(x 1 )Δx + x 2 · ρ(x 2 )Δx + ···+ x n · ρ(x n )Δx<br />
.<br />
ρ(x 1 )Δx + ρ(x 2 )Δx + ···+ ρ(x n )Δx<br />
<br />
x ≈<br />
∑ n<br />
i1 x i · ρ(x i )Δx<br />
∑ n<br />
i1 ρ(x i)Δx . <br />
<br />
<br />
n →∞ <br />
<br />
<br />
ρ(x) x a x b <br />
<br />
∫ b<br />
a<br />
xρ(x) dx<br />
x ∫ b<br />
a ρ(x) .<br />
dx
x <br />
x <br />
<br />
<br />
ρ(x) 4 + 0.1x x 0 <br />
ρ x <br />
M <br />
<br />
<br />
x <br />
<br />
p(x) 4e 0.020732x <br />
x 0 ρ(x) p(x) <br />
<br />
<br />
<br />
p(x) 4e 0.020732x <br />
<br />
D V m m <br />
D · V.<br />
<br />
ρ(x) <br />
x a x b <br />
m <br />
∫ b<br />
a<br />
ρ(x) dx.<br />
m 1 ,...,m n <br />
x 1 ,...,x n x <br />
x <br />
∑ n<br />
i1 x im i<br />
∑ n<br />
i1 m i<br />
.<br />
<br />
ρ(x) <br />
x a x b <br />
x <br />
∫ b<br />
a<br />
xρ(x) dx<br />
∫ b<br />
a ρ(x) .<br />
dx
x <br />
<br />
<br />
x <br />
δ(x) 4 + 5x <br />
Dx Δx<br />
Σ<br />
<br />
<br />
<br />
δ(x) 7 + sin(x) x x 0 x π. <br />
<br />
<br />
<br />
<br />
δ(x) ( ( 350 2 + sin 4 √ )) x + 0.175 , x <br />
0 ≤ x ≤ 20<br />
<br />
Dx Δx<br />
Σ<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a ρ(x) 10e −0.1x x <br />
ρ <br />
a<br />
<br />
<br />
<br />
x
ρ(x) 1<br />
1+x 2 p(x) e −0.1x <br />
<br />
<br />
f (x) ρ(x) + p(x)<br />
<br />
<br />
∫ 10<br />
0<br />
xf(x) dx <br />
<br />
<br />
<br />
<br />
y f (x) 2xe −1.25x + (30 − x)e −0.25(30−x) <br />
x 0 ...30 y 0 ...3 <br />
x y <br />
x <br />
x y <br />
x <br />
<br />
<br />
<br />
<br />
x <br />
x ρ(x) <br />
0.6π f (x) 2 f <br />
x 0 <br />
x 30
y = f (x)<br />
y<br />
y = v(t)<br />
ρ(x)<br />
f (x)<br />
v(t)<br />
t<br />
△x<br />
a<br />
△x<br />
b<br />
a<br />
△t<br />
b<br />
<br />
y f (x) y v(t) <br />
y ρ(x)<br />
<br />
<br />
<br />
A l · w <br />
y f (x) x [a, b] <br />
Δx A f (x)Δx <br />
<br />
A <br />
∫ b<br />
a<br />
f (x) dx.<br />
<br />
y v(t) [a, b] v(t)
d r · t <br />
r Δt v(t) <br />
d v(t)Δt <br />
Δt <br />
d <br />
∫ b<br />
a<br />
v(t) dt.<br />
<br />
M D · V <br />
<br />
<br />
<br />
x y ρ(x) <br />
Δx M ρ(x)Δx Δx → 0 <br />
<br />
M <br />
∫ b<br />
a<br />
ρ(x) dx.<br />
A l·w d r·t<br />
M D · V <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
W F · d 20 · 4 80 .<br />
<br />
<br />
<br />
W F · d <br />
d <br />
<br />
<br />
B h <br />
B(h) <br />
B(0) 20 B(50) 12
h <br />
B(h)<br />
B(5)Δh Δh 2 <br />
<br />
h 5 h 7<br />
<br />
h 5 h 7 <br />
B(22)Δh Δh 0.25 <br />
<br />
W B(h)Δh <br />
h Δh<br />
∫ 50<br />
0<br />
B(h) dh <br />
<br />
<br />
W F · d F <br />
<br />
<br />
h B(h) <br />
12 + 8e −0.1h <br />
<br />
<br />
<br />
h Δh <br />
W B(h)Δh (12 + 8e −0.1h )Δh.<br />
Δh <br />
<br />
W <br />
∫ 50<br />
0<br />
B(h) dh <br />
∫ 50<br />
0<br />
(12 + 8e −0.1h ) dh.<br />
<br />
<br />
<br />
W ∫ 50<br />
0 (12 + 8e−0.1h ) dh ≈ 679.461
x F(x)<br />
a b <br />
W <br />
∫ b<br />
a<br />
F(x) dx.<br />
<br />
<br />
<br />
<br />
<br />
F(h) h <br />
<br />
<br />
<br />
<br />
B(h) <br />
h <br />
<br />
<br />
h B(h) 25 + 15e −0.05h <br />
<br />
h 0 h 100<br />
<br />
<br />
<br />
x <br />
F(x) kx k <br />
<br />
<br />
<br />
F(1/3) 5 k
y +<br />
(0,1.5)<br />
Δx<br />
(4,0.75)<br />
x +<br />
<br />
<br />
<br />
<br />
<br />
x
y <br />
<br />
<br />
<br />
<br />
<br />
<br />
y f (x) (0, 1.5) (4, 0.75) x <br />
f (x) 1.5−0.1875x<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
V π f (x) 2 Δx π(1.5 − 0.1875x) 2 Δx.<br />
3 <br />
<br />
F 62.4 · V 62.4π(1.5 − 0.1875x) 2 Δx.<br />
x <br />
x <br />
<br />
<br />
d x + 9.<br />
<br />
W F · d 62.4π(1.5 − 0.1875x) 2 Δx · (x + 9),<br />
<br />
x 0 x 4 <br />
Δx → 0 <br />
∫ 4<br />
W 62.4π(1.5 − 0.1875x) 2 (x + 9) dx,<br />
0<br />
W <br />
10970.5π
3 <br />
<br />
F ma m a <br />
9.81 3 3 <br />
<br />
<br />
<br />
x + y +<br />
<br />
<br />
3 <br />
<br />
<br />
3 <br />
<br />
<br />
<br />
<br />
3
P F , F P · A,<br />
A<br />
P F A <br />
F PA <br />
A<br />
<br />
<br />
<br />
<br />
<br />
<br />
1 × 1 d<br />
V 1 · 1 · d 3 <br />
62.4d <br />
d P 62.4d 2 <br />
d<br />
P 62.4d d<br />
F PA <br />
<br />
<br />
<br />
<br />
<br />
<br />
x
45<br />
x<br />
y +<br />
x − 5 y = f (x)<br />
△x<br />
x + (25,30)<br />
<br />
<br />
<br />
<br />
<br />
x <br />
y f (x) <br />
f y f (x) 45 − 3 5<br />
x <br />
<br />
A 2 f (x)Δx 2(45 − 3 5 x)Δx.<br />
<br />
x <br />
<br />
(x − 5) <br />
P 62.4d <br />
P 62.4(x − 5).<br />
<br />
F PA <br />
F P · A 62.4(x − 5) · 2(45 − 3 5 x)Δx.<br />
x<br />
x 5
x 30 <br />
F <br />
∫ x30<br />
x5<br />
62.4(x − 5) · 2(45 − 3 x) dx.<br />
5<br />
F ≈ 1.248 × 10 6 <br />
<br />
<br />
x + y +<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
3
W F · d <br />
<br />
F P · A <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
δ 1080 / 3 . <br />
<br />
<br />
<br />
<br />
50 3 <br />
<br />
<br />
<br />
<br />
δ 62.4 3 <br />
<br />
<br />
<br />
3
1000 3 <br />
<br />
<br />
<br />
9.8 2 <br />
f (x) 3 cos( x3<br />
4<br />
) <br />
y x f (x) 0 R<br />
f x y x<br />
y <br />
x<br />
y R <br />
x <br />
<br />
<br />
<br />
x<br />
y R<br />
x <br />
<br />
<br />
h <br />
<br />
<br />
<br />
3
t a t b t <br />
<br />
∫ b<br />
a<br />
0.3e −0.3t dt.<br />
<br />
∫ 3<br />
2<br />
0.3e −0.3t dt −e −0.3t <br />
3<br />
2<br />
−e −0.9 + e −0.6<br />
≈ 0.1422.<br />
t 2 t 3 <br />
<br />
<br />
<br />
<br />
p(t) 0.25e −0.25t <br />
t a<br />
t b <br />
∫ b<br />
a<br />
p(t) dt.
F(b) 0 b <br />
F(b) <br />
F(b) <br />
lim b→∞ F(b) <br />
<br />
<br />
<br />
<br />
<br />
b <br />
<br />
∫ b<br />
0<br />
0.3e −0.3t dt<br />
<br />
<br />
∫ b<br />
lim 0.3e −0.3t dt,<br />
b→∞ 0<br />
<br />
∫ ∞<br />
0<br />
0.3e −0.3t dt.
y<br />
y<br />
b<br />
t<br />
···<br />
t<br />
p(t) 0.3e −0.3t [0, b] <br />
b →∞··· <br />
<br />
<br />
∞ <br />
<br />
∫ ∞<br />
1<br />
∫<br />
1<br />
0<br />
x 2 dx,<br />
−∞<br />
∫<br />
1<br />
∞<br />
1 + x 2 dx, <br />
−∞<br />
e −x2 dx<br />
∞ <br />
<br />
∫ ∞<br />
0<br />
f (x) dx <br />
∫ ∞<br />
∫ b<br />
f (x) dx lim f (x) dx.<br />
0<br />
b→∞ 0<br />
∫ b<br />
0<br />
f (x) dx <br />
<br />
<br />
<br />
∫ ∞<br />
1<br />
1<br />
x dx ∫ ∞<br />
1<br />
1<br />
x 3/2 dx<br />
∫ ∞<br />
1<br />
1<br />
x dx<br />
∫ 10 1<br />
1 x dx ∫ 1000 1<br />
1 x<br />
dx <br />
∫ 100000 1<br />
1 x<br />
dx
∫ b<br />
1<br />
b<br />
<br />
∫ ∞<br />
1<br />
1<br />
x 3/2 dx<br />
∫ b<br />
lim<br />
b→∞ 1<br />
1<br />
x dx.<br />
∫ 10<br />
1<br />
∫ 100000<br />
1<br />
<br />
∫ b<br />
1<br />
b<br />
<br />
1<br />
x<br />
dx <br />
1<br />
dx ∫ 1000 1<br />
x 3/2 1<br />
dx<br />
x 3/2<br />
1<br />
dx <br />
x 3/2<br />
<br />
∫ b<br />
lim<br />
b→∞ 1<br />
1<br />
dx.<br />
x3/2 1<br />
x 3/2 dx <br />
y 1 x y 1 x <br />
x 3/2<br />
0 ...10 x <br />
<br />
∫ ∞ 1<br />
1 x dx ∫ ∞ 1<br />
1<br />
dx <br />
x 3/2<br />
x ≥ 1<br />
<br />
f <br />
[1, ∞] f (x) 1 1<br />
x<br />
f (x) <br />
∫<br />
x 3/2<br />
b<br />
lim b→∞ 1<br />
f (x) dx <br />
<br />
f (x) x ≥ a ∫ ∞<br />
a<br />
<br />
∫ b<br />
lim f (x) dx<br />
b→∞ a<br />
∫ ∞<br />
a<br />
f (x) dx <br />
f (x) dx<br />
<br />
f lim x→∞ f (x) 0
f 0 x →∞ ∫ ∞<br />
a<br />
<br />
f (x) dx <br />
<br />
<br />
<br />
<br />
<br />
∫ ∞<br />
1<br />
∫ ∞<br />
1<br />
x 2 dx<br />
0<br />
e −x/4 dx<br />
∫ ∞<br />
2<br />
9<br />
(x+5) 2/3 dx<br />
<br />
∫ ∞<br />
4<br />
3<br />
(x+2) 5/4 dx<br />
∫ ∞<br />
0<br />
xe −x/4 dx<br />
∫ ∞ 1<br />
<br />
1 x<br />
dx p <br />
p<br />
<br />
<br />
<br />
<br />
∫ 1<br />
0<br />
1<br />
√ x<br />
dx,<br />
f (x) √ 1<br />
x<br />
x 0 f [0, 1]<br />
<br />
<br />
y<br />
y<br />
f (x)= 1 √ x<br />
1<br />
f (x)= 1 √ x<br />
1<br />
a<br />
x<br />
x<br />
f (x) √ 1<br />
x<br />
[a, 1] <br />
a → 0 +
∫ 1<br />
√<br />
1<br />
0 x<br />
dx 0 a a <br />
<br />
∫ 1<br />
∫<br />
1<br />
1<br />
1<br />
√ dx lim √ dx,<br />
0 x a→0 +<br />
a x<br />
∫ 1<br />
a<br />
1<br />
√ x<br />
dx <br />
<br />
<br />
<br />
∫ 1<br />
∫ 1<br />
0<br />
0<br />
∫<br />
1<br />
1<br />
√ dx lim<br />
x a→0 +<br />
a<br />
lim 2√ x 1<br />
a→0 + a<br />
1<br />
√ x<br />
dx<br />
lim<br />
a→0 + 2√ 1 − 2 √ a<br />
2,<br />
1<br />
√ x<br />
dx <br />
<br />
<br />
∫ 3<br />
1<br />
1<br />
(x − 2) 2 dx.<br />
<br />
<br />
1<br />
<br />
(x−2) 2<br />
− 1<br />
x−2<br />
1 3 <br />
<br />
1<br />
f (x) <br />
(x−2) 2<br />
<br />
<br />
y<br />
y = 1<br />
(x−2) 2<br />
1 2 3<br />
x<br />
1<br />
f (x) <br />
x 2<br />
(x−2) 2
−2 <br />
<br />
f (x) 1 x 2 <br />
(x−2) 2<br />
∫ 3<br />
1<br />
∫<br />
1<br />
a<br />
dx lim<br />
(x − 2) 2 a→2 − 1<br />
∫<br />
1<br />
3<br />
dx + lim<br />
(x − 2) 2 b→2 + b<br />
1<br />
(x − 2) 2 dx<br />
<br />
a 2 <br />
∫ 2<br />
1<br />
∫<br />
1<br />
a<br />
dx lim<br />
(x − 2) 2 a→2 − 1<br />
lim<br />
a→2 − − 1<br />
1<br />
(x − 2) 2 dx<br />
a<br />
(x − 2)<br />
1<br />
lim − 1<br />
a→2 − (a − 2) + 1<br />
1 − 2<br />
∞,<br />
1<br />
a−2 →−∞ a ∫ 2<br />
∫ 3<br />
2<br />
∫ 3<br />
1<br />
1<br />
1<br />
1<br />
(x−2) 2 dx<br />
dx <br />
(x−2) 2 1<br />
dx <br />
(x−2) 2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∫ 1<br />
0<br />
∫ 2<br />
1<br />
x 1/3 dx<br />
0 e−x dx<br />
∫ 4<br />
1<br />
1<br />
√<br />
4−x<br />
dx<br />
<br />
∫ 2<br />
−2<br />
1<br />
x 2 dx<br />
∫ π/2<br />
0<br />
tan(x) dx<br />
<br />
∫ 1<br />
0<br />
1<br />
√<br />
1−x 2 dx<br />
<br />
∫ b<br />
a<br />
f (x) dx a b ±∞ <br />
f x c c <br />
a ≤ c ≤ b
∫ ∞<br />
∫ b<br />
f (x) dx lim f (x) dx,<br />
a<br />
b→∞ a<br />
<br />
<br />
<br />
<br />
∫ ∞<br />
1<br />
1<br />
x p dx<br />
p <br />
p > 1 0 < p ≤ 1 <br />
∫ 1 1<br />
0 x<br />
dx 0 < p < 1 p ≥ 1<br />
p<br />
<br />
<br />
∫ 3<br />
−8<br />
0 x √ x dx<br />
<br />
<br />
<br />
∫ ∞<br />
2<br />
1x 2 e −x3 dx <br />
<br />
∫ 1<br />
−∞<br />
e 5x<br />
1+e 5x dx <br />
<br />
∫ 1<br />
−1<br />
1<br />
v dv <br />
y 1<br />
<br />
<br />
cos 2 (t)<br />
t 0 t π/2 <br />
<br />
<br />
∫ ∞<br />
e<br />
∫ ∞<br />
e<br />
ln(x)<br />
x<br />
dx<br />
1<br />
x ln(x) dx
∫ ∞<br />
e<br />
∫ ∞<br />
e<br />
∫ 1<br />
<br />
1<br />
x(ln(x)) 2 dx<br />
1<br />
x(ln(x)) p dx p <br />
ln(x)<br />
x<br />
0<br />
dx<br />
∫ 1<br />
0<br />
ln(x) dx<br />
<br />
∫ ∞ 1<br />
1<br />
dx <br />
1+x 3<br />
1<br />
<br />
1+x 3<br />
<br />
x > 0 1 + x 3 > x 3 <br />
<br />
∫ b<br />
1<br />
1 + x 3 < 1 x 3 .<br />
∫<br />
1<br />
b<br />
1 1 + x 3 dx < 1<br />
1 x 3 dx<br />
b > 1 b →∞ ∫ ∞<br />
1<br />
1<br />
1+x 3 dx<br />
∫ ∞ 1<br />
1<br />
dx <br />
x 3<br />
<br />
∫ ∞ 1<br />
1<br />
dx <br />
1+x 3<br />
<br />
<br />
x 2 + x + 1 > x 2 x ≥ 1 ∫ ∞<br />
∫ ∞ 1<br />
1<br />
dx<br />
x 2<br />
x > 1 ln(x) < x <br />
∫ b<br />
2<br />
∫<br />
1<br />
b<br />
x dx <<br />
b > 2 ∫ b<br />
√<br />
x 4 +1<br />
x 4<br />
2<br />
2<br />
1<br />
ln(x) dx<br />
1<br />
ln(x)<br />
dx <br />
1<br />
1<br />
dx <br />
x 2 +x+1<br />
> 1 x > 1 <br />
<br />
∫ √ ∞<br />
1<br />
1 x · x 4 + 1<br />
x 4 dx
s(t)<br />
s t v(t) <br />
4t + 1 <br />
[0, 4]
s(4) t <br />
4 s(4) <br />
s(4)<br />
s(0) 7 s(2)<br />
<br />
s(0) 3 s(2)<br />
v(t) 4t + 1 <br />
s(t) 2t 2 + t − 4 <br />
s(t) <br />
v(t) 4t + 1 <br />
s(0) s(t)<br />
<br />
<br />
<br />
dy<br />
dx x sin x<br />
y(x) <br />
<br />
<br />
t <br />
<br />
ds<br />
dt<br />
4t + 1.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A(t) <br />
t <br />
dA/dt 0.03A
dA<br />
dt 0.03A.<br />
ds<br />
dt<br />
4t + 1<br />
t <br />
s(t) 4t + 1 <br />
A <br />
A(t)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
◦ <br />
<br />
<br />
◦ <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
m <br />
ma m, <br />
dv<br />
dt ,<br />
v 9.8
6<br />
v<br />
6<br />
v<br />
5<br />
5<br />
4<br />
4<br />
3<br />
3<br />
2<br />
2<br />
1<br />
1<br />
1 2 3<br />
t<br />
1 2 3<br />
t<br />
<br />
<br />
<br />
dv/dt v 0.5, 1.0, 1.5, <strong>2.0</strong> 2.5 <br />
<br />
dv/dt
1.5<br />
dv<br />
dt<br />
1.0<br />
0.5<br />
1 2 3 4 5<br />
v<br />
-0.5<br />
-1.0<br />
-1.5<br />
<br />
dv/dt v 3.5, 4.0, 4.5 5.0
ds<br />
dt<br />
4t + 1,<br />
s(t) s(t) 2t 2 + t <br />
s(t) 2t 2 + t + 4 <br />
<br />
<br />
2x 2 − 2x 2x + 6 x 3 <br />
x □<br />
2□ 2 − 2□ 2□ + 6.<br />
x 3 <br />
3 □ <br />
<br />
2□ 2 − 2□ 2 · 3 2 − 2 · 3 12,<br />
<br />
2□ + 6 2 · 3 + 6 12.<br />
x 3 <br />
<br />
dv<br />
dt<br />
d□<br />
dt<br />
1.5 − 0.5v, <br />
1.5 − 0.5□.<br />
v(t) 3 − 2e −0.5t v <br />
dv<br />
dt d□<br />
dt d dt [3 − 2e−0.5t ] −2e −0.5t · (−0.5) e −0.5t<br />
<br />
1.5 − 0.5v 1.5 − 0.5□ 1.5 − 0.5(3 − 2e −0.5t ) 1.5 − 1.5 + e −0.5t e −0.5t .<br />
dv<br />
dt 1.5 − 0.5v t v 3 − 2e−0.5t
dv<br />
dt<br />
1.5 − 0.5v.<br />
<br />
v(t) 1.5t − 0.25t 2 <br />
v(t) 3 + 2e −0.5t <br />
v(t) 3<br />
v(t) 3+ Ce −0.5t C <br />
<br />
<br />
<br />
<br />
C v(t) <br />
3 + Ce −0.5t <br />
<br />
C <br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
v<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
1 2 3<br />
t<br />
<br />
dv<br />
dt<br />
1.5 − 0.5v<br />
C v(0) v(0) <br />
3 + C <br />
<br />
<br />
<br />
v(t)<br />
<br />
<br />
<br />
<br />
dv<br />
1.5 − 0.5v, v(0) 0.5<br />
dt
v <br />
v v <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dv<br />
1.5 − 0.5v<br />
dt<br />
<br />
d 2 y<br />
dt 2 −10y<br />
<br />
<br />
<br />
dv<br />
1.5 − 0.5v<br />
dt<br />
dv/dt <br />
<br />
dy<br />
1.5t − 0.5y,<br />
dt
d 2 y<br />
dx 2<br />
9y<br />
dy<br />
dx 3y<br />
d 2 y<br />
dx 2<br />
−9y<br />
dy<br />
dx −3y<br />
y e −3x y e 3x<br />
y e 3x<br />
y 3 sin(x)<br />
y sin(3x) y 3 sin(x)<br />
y e −3x<br />
y sin(3x)<br />
k y cos(kt) <br />
k <br />
d 2 y<br />
+ 4y 0.<br />
dt2 A k <br />
dy<br />
dt<br />
k(A − y)<br />
T(t) <br />
T t <br />
<br />
dT<br />
dt − 1 15 T + 5.<br />
T(0) 105 <br />
dT<br />
dt | T105 <br />
T t 0<br />
t 1<br />
dT/dt T
5<br />
4<br />
3<br />
dT<br />
dt<br />
2<br />
1<br />
-1<br />
30 60 90 120<br />
T<br />
-2<br />
-3<br />
T T T T <br />
<br />
T(t) 75 + 30e −t/15 <br />
T(0) 105 <br />
P(t)<br />
P <br />
<br />
f (P) <br />
dP<br />
dt f (P)<br />
dP<br />
dt<br />
P<br />
1 2 3 4
P <br />
P <br />
P(0) 3 <br />
0 < P(0) < 1 <br />
<br />
1 < P(0) < 3 <br />
<br />
3 < P(0) <br />
<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dt y − t.<br />
<br />
<br />
y(t) t + 1 + 2e t<br />
y(t) t + 1<br />
y(t) t + 2<br />
h <br />
<br />
d 2 h<br />
dt 2 −kh<br />
k <br />
<br />
h(t) 4 sin(3t)
f a <br />
f a <br />
f (a) f ′ (a)<br />
<br />
f (x) ≈ f (a) + f ′ (a)(x − a).<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dt<br />
t − 2, y(0) 1.<br />
<br />
y(t) t 0 <br />
t 0 −0.25 ≤ t ≤ 0.25 <br />
<br />
y(t) <br />
t 1, 2, 3
y(t)<br />
0 ≤ t ≤ 3 <br />
y(t)<br />
<br />
y(t) <br />
<br />
<br />
<br />
<br />
<br />
3<br />
2<br />
1<br />
-1<br />
-2<br />
y<br />
1 2 3<br />
t<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dt<br />
t − 2. t 0 <br />
dy/dt 0 − 2 −2 <br />
y <br />
y <br />
t 0 −2<br />
3<br />
2<br />
1<br />
y<br />
-1<br />
1 2 3<br />
t<br />
-2<br />
<br />
t 0
t 1 <br />
dy/dt 1 −<br />
2 −1 <br />
y <br />
y t 1 −1<br />
3<br />
2<br />
1<br />
-1<br />
-2<br />
y<br />
1 2 3<br />
t<br />
<br />
t 1<br />
t 2 dy/dt 0<br />
t 3 dy/dt 1 <br />
<br />
<br />
3<br />
2<br />
y<br />
1<br />
-1<br />
-2<br />
1 2 3<br />
t<br />
<br />
t 2 t 3
3<br />
2<br />
1<br />
-1<br />
-2<br />
y<br />
1 2 3<br />
t<br />
<br />
<br />
<br />
<br />
<br />
y(0) 1 <br />
<br />
<br />
3<br />
2<br />
1<br />
-1<br />
-2<br />
y<br />
1 2 3<br />
t
3<br />
y<br />
3<br />
y<br />
2<br />
2<br />
1<br />
1<br />
-1<br />
1 2 3<br />
t<br />
-1<br />
1 2 3<br />
t<br />
-2<br />
-2<br />
3<br />
y<br />
3<br />
y<br />
2<br />
2<br />
1<br />
1<br />
-1<br />
1 2 3<br />
t<br />
-1<br />
1 2 3<br />
t<br />
-2<br />
-2<br />
<br />
<br />
y(0) 1<br />
dy<br />
dt<br />
t−2<br />
<br />
<br />
y(0) <br />
dy<br />
dt<br />
t − 2
dy<br />
dt −1 (y − 4).<br />
2<br />
dy<br />
dt<br />
y <br />
y y y y <br />
3<br />
2<br />
dy<br />
dt<br />
1<br />
y<br />
-1 1 2 3 4 5 6 7<br />
-1<br />
-2<br />
<br />
<br />
<br />
y(0) 0 y(0) 2<br />
y(0) 4 y(0) 6.<br />
y(t) 4+2e −t/2 <br />
<br />
y(0) 6.<br />
<br />
<br />
<br />
y(0) 4<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
-1 1 2 3 4 5 6 7<br />
-1<br />
t<br />
<br />
dy<br />
dt<br />
− 1 2<br />
(y − 4)<br />
<br />
<br />
<br />
dy/dt f (y) dy/dt 0 f (y)
dy<br />
dt<br />
f (y) − 1 2<br />
(y − 4) <br />
f (y) − 1 2<br />
(y − 4) 0 <br />
y 4<br />
y f (y) 0 dy<br />
dt<br />
f (y) <br />
<br />
<br />
dy<br />
dt −1 y(y − 4).<br />
2<br />
dy<br />
dt<br />
y <br />
y y y y <br />
2<br />
dy<br />
dt<br />
1<br />
y<br />
-2 -1 1 2 3 4 5 6<br />
-1<br />
-2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y(0) <br />
−1, 0, 1,...,5<br />
6<br />
5<br />
y<br />
4<br />
3<br />
2<br />
1<br />
t<br />
-1<br />
-1<br />
1 2 3 4 5 6 7<br />
-2<br />
<br />
dy<br />
dt<br />
− 1 2<br />
y(y − 4)
y y <br />
y lim t→∞ y(t) y<br />
y <br />
y <br />
<br />
y(t) <br />
y(0) <br />
<br />
dy/dt <br />
f (y) y f (y) 0 <br />
dy/dt f (y) y <br />
y <br />
y <br />
dy<br />
dt<br />
= f (y)<br />
dy<br />
dt<br />
= f (y)<br />
y<br />
y<br />
y<br />
y<br />
dy<br />
dt<br />
<br />
y<br />
dy<br />
dt<br />
y
y
y x<br />
y x 0<br />
y ′ 0 <br />
<br />
y ′ <br />
<br />
<br />
y ′ 2xy + 2xe −x2<br />
y ′ 2 sin(3x) + 1 + y<br />
y ′ 2y − 2<br />
y ′ (2x + y)<br />
−<br />
(2y)<br />
<br />
<br />
x ′ (t) x 4 − 5x 3 − 2x 2 + 24x + 0<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dt t − y.
y(0) −4, −3,...,4<br />
<br />
y(0) <br />
−1 <br />
<br />
<br />
<br />
<br />
t y <br />
<br />
4<br />
3<br />
y<br />
2<br />
1<br />
t<br />
-4 -3 -2 -1<br />
-1<br />
1 2 3 4<br />
-2<br />
-3<br />
-4<br />
<br />
P(t) P <br />
<br />
f (P) <br />
dP<br />
dt f (P)<br />
dP<br />
dt<br />
P<br />
1 2 3 4
P(0) > 0<br />
<br />
<br />
P(0) > 1 <br />
P(0) < 1 <br />
<br />
<br />
<br />
P(t) P <br />
t <br />
<br />
dP<br />
f (P) P(6 − P).<br />
dt<br />
f (P) P(6 − P) <br />
<br />
<br />
<br />
P(t) <br />
dP/dt <br />
P <br />
<br />
<br />
<br />
h <br />
<br />
<br />
<br />
<br />
y(t) t <br />
<br />
<br />
<br />
C <br />
<br />
M(y) C<br />
y<br />
2 + y
M(y) C 1 <br />
M(y) y y <br />
C 1 <br />
<br />
<br />
C 60
dy<br />
dt 1 (y + 1), y(0) 0.<br />
2<br />
<br />
<br />
y(t) t 0 <br />
<br />
t 0<br />
<br />
<br />
0 ≤ t ≤ 2 <br />
y(2) <br />
t 2<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
t<br />
1 2 3 4 5 6 7<br />
<br />
<br />
y(2) y(2) <br />
y(t) t 2
t 2<br />
2 ≤ t ≤ 4 <br />
y(4) t 4<br />
y(6)<br />
<br />
<br />
<br />
<br />
Δt<br />
<br />
<br />
<br />
<br />
<br />
m Δy<br />
Δt<br />
<br />
<br />
<br />
y<br />
Δy<br />
Δy mΔt.<br />
Δt<br />
t<br />
<br />
dy<br />
t − y, y(0) 1.<br />
dt<br />
<br />
<br />
<br />
y(t) t − 1 + 2e −t <br />
<br />
(t i , y i ) y i ≈ y(t i ) Δt 0.2<br />
y(0) 1 (t 0 , y 0 ) (0, 1) <br />
Δt 0.2 (t 1 , y 1 ) t 1 t 0 + Δt 0.2 <br />
<br />
e
m dy<br />
dt<br />
0 − 1 −1.<br />
(0,1)<br />
<br />
Δt 0.2<br />
<br />
Δy mΔt −1 · 0.2 −0.2.<br />
y(0.2) ≈<br />
y 1 y 0 + Δy 1 − 0.2 0.8 <br />
<br />
<br />
1.2<br />
0.8<br />
0.4<br />
y<br />
(t 0 ,y 0 )<br />
(t 1 ,y 1 )<br />
t<br />
0.4 0.8 1.2<br />
<br />
(t 1 , y 1 ) (0.2, 0.8) <br />
<br />
m dy<br />
dt<br />
0.2 − 0.8 −0.6.<br />
(0.2,0.8)<br />
Δt t 2 <br />
t 1 + Δ 0.4 <br />
Δy −0.6 · 0.2 −0.12.<br />
y 2 y 1 +Δy 0.8−<br />
0.12 0.68 y(0.2) ≈ 0.68 <br />
<br />
<br />
1.2<br />
0.8<br />
0.4<br />
y<br />
(t 0 ,y 0 )<br />
(t 1 ,y 1 )<br />
(t 2 ,y 2 )<br />
t<br />
0.4 0.8 1.2
(t i , y i ) <br />
y(t) y(t)<br />
<br />
1.2<br />
y<br />
1.2<br />
y<br />
0.8<br />
0.8<br />
0.4<br />
0.4<br />
0.4 0.8 1.2<br />
t<br />
0.4 0.8 1.2<br />
t<br />
<br />
<br />
<br />
<br />
<br />
<br />
(t i , y i ) <br />
<br />
<br />
t i y i dy/dt Δy<br />
0.0000 1.0000<br />
m dy/dt dy/dt<br />
Δy y Δy <br />
mΔt<br />
t i y i dy/dt Δy<br />
0.0000 1.0000 −1.0000 −0.2000<br />
t i Δt y i Δy <br />
t i y i dy/dt Δy<br />
0.0000 1.0000 −1.0000 −0.2000<br />
0.2000 0.8000
t i y i dy/dt Δy<br />
0.0000 1.0000 −1.0000 −0.2000<br />
0.2000 0.8000 −0.6000 −0.1200<br />
0.4000 0.6800 −0.2800 −0.0560<br />
0.6000 0.6240 −0.0240 −0.0048<br />
0.8000 0.6192 0.1808 0.0362<br />
1.0000 0.6554 0.3446 0.0689<br />
1.2000 0.7243 0.4757 0.0951<br />
Δt 0.2<br />
<br />
dy<br />
dt<br />
2t − 1, y(0) 0<br />
Δt 0.2 t i 0.2, 0.4, 0.6, 0.8<br />
1.0 (t i , y i ) <br />
<br />
t i y i dy/dt Δy<br />
0.0000 0.0000<br />
0.2000<br />
0.4000<br />
0.6000<br />
0.8000<br />
1.0000<br />
0.2<br />
-0.2<br />
-0.6<br />
y<br />
0.4 0.8 1.2<br />
t<br />
-1.0<br />
<br />
<br />
<br />
<br />
<br />
t i <br />
y 5 <br />
∫ 1<br />
0 (2t−<br />
1) dt
y(0) 1 <br />
<br />
dy<br />
dt<br />
6y − y 2 <br />
<br />
<br />
8 y<br />
t<br />
6<br />
4<br />
2<br />
2 4 6 8<br />
<br />
<br />
<br />
y(0) <br />
1<br />
y(0) 1 Δt 0.2 <br />
t i 0.2, 0.4, 0.6, 0.8 1.0 <br />
(t i , y i )
t i y i dy/dt Δy<br />
0.0 1.0000<br />
0.2<br />
0.4<br />
0.6<br />
0.8<br />
1.0<br />
8<br />
6<br />
4<br />
2<br />
y<br />
0.4 0.8 1.2<br />
t<br />
<br />
Δt 0.2<br />
<br />
<br />
y(0) <br />
6<br />
<br />
<br />
<br />
<br />
dy<br />
dt<br />
y, y(0) 1<br />
<br />
<br />
t 0 <br />
y(t) e t y(1) e <br />
Δt E Δt <br />
<br />
Δt 0.2 <br />
y(1) ≈ E 0.2 2.4883 <br />
y(1) − E 0.2 e − 2.4883 ≈ 0.2300.<br />
Δt
Δt E Δt <br />
0.200 2.4883 0.2300<br />
0.100 2.5937 0.1245<br />
0.050 2.6533 0.0650<br />
0.025 2.6851 0.0332<br />
0.2<br />
Error<br />
0.1<br />
0.1 0.2<br />
Δt<br />
<br />
Δt <br />
<br />
Δt.<br />
Δt <br />
<br />
t <br />
Δt <br />
K<br />
y(t) − E Δt ≈ KΔt<br />
<br />
<br />
t<br />
y(t) t Δt<br />
Δt<br />
<br />
y ′ −x − y<br />
Δx 0.1 y x 1.4 <br />
<br />
y(1) 1 y(1.4) ≈<br />
Δx 0.1 y x 2.4 <br />
y(1) 0 y(2.4) ≈
y ′ −3y y(0) 1.5<br />
<br />
<br />
Δx 0.2 x 0.2, 0.4,...,1<br />
<br />
x <br />
y ≈ <br />
<br />
<br />
y 1.5e −3x y ′ −3y y(0) 1.5<br />
<br />
B 1100 t 0 <br />
Δt 1 B(1) ≈<br />
Δt 0.5 B(1) ≈<br />
dB<br />
dt 0.04B<br />
Δt 0.25 B(1) ≈<br />
B <br />
<br />
<br />
<br />
<br />
<br />
T(t) T r <br />
dT<br />
dt −k(T − T r),<br />
k <br />
<br />
◦ <br />
◦ <br />
k 0.5 k 0.1 <br />
<br />
<br />
<br />
<br />
T r 70 + 10 sin t.<br />
Δt 0.1 <br />
0 ≤ t ≤ 50
Δt E Δt <br />
t <br />
y(t) − E Δt ≈ KΔt<br />
K<br />
<br />
<br />
Δt<br />
<br />
y(t) − E Δt KΔt.<br />
dy/dt y y(0) 1 <br />
Δt 0.2 Δt 0.1<br />
y(1) − 2.4883 0.2K<br />
y(1) − 2.5937 0.1K.<br />
y(1) K <br />
y(1) y(1) <br />
e 2.71828 ....<br />
E 0.05 2.6533 E 0.025 2.6851 <br />
<br />
<br />
<br />
dy<br />
dt<br />
t − y, y(0) 0.<br />
y(0.3) E 0.1 E 0.05 <br />
<br />
y(0.3) 0.0408<br />
<br />
<br />
Δt <br />
<br />
<br />
<br />
dy/dt y y(0) 1 <br />
Δt 0.2 0 ≤ t ≤ 0.2 t 0
y(0.2) ≈ y 1 1 + 1(0.2) 1.2<br />
0 ≤ t ≤ 0.2 <br />
t 0 t 0.2 <br />
<br />
(1 + 1.2)/2 1.1 <br />
y(1) y 1 1 + 1.1(0.2) 1.22<br />
<br />
t i y i (t i+1 , y i+1 ) <br />
0.0 1.0000 1.2000 1.1000<br />
0.2 1.2200 1.4640 1.3420<br />
0.4 1.4884 1.7861 1.6372<br />
.<br />
.<br />
.<br />
.<br />
.<br />
.<br />
<br />
.<br />
.<br />
y(1) e<br />
Δt 0.1 y(1)<br />
Δt <br />
Δt
dy<br />
dt ty.<br />
t y t <br />
y dy/dt <br />
y<br />
1 dy<br />
y dt t.<br />
<br />
dy/dt <br />
<br />
dy<br />
dt t − y,<br />
<br />
y + dy<br />
dt t.<br />
<br />
y dy<br />
dt
(y) dy<br />
dt<br />
h(t)<br />
<br />
dy<br />
dt −3y<br />
<br />
dy<br />
ty − y<br />
dt<br />
<br />
dy<br />
dt t + 1<br />
<br />
dy<br />
dt t2 − y 2 <br />
<br />
+C <br />
∫<br />
xdx x2<br />
2 + C?<br />
f <br />
∫<br />
∫<br />
f ′ (x) dx xdx.<br />
f <br />
<br />
<br />
<br />
y <br />
dy<br />
dt t<br />
y 2 .<br />
<br />
<br />
y 2 dy<br />
dt t.
t <br />
∫<br />
y 2 dy ∫<br />
dt dt tdt.<br />
<br />
t y dy dy<br />
dt<br />
dt<br />
∫ ∫<br />
y 2 dy tdt.<br />
<br />
<br />
C C<br />
<br />
y 3<br />
3 t2<br />
2 + C.<br />
<br />
y 3 /3 t 2 /2 <br />
<br />
y t <br />
y(t) 3 √<br />
3<br />
2 t2 + 3C.<br />
3C <br />
C <br />
<br />
y(t) 3 √<br />
3<br />
2 t2 + C<br />
C<br />
C <br />
<br />
<br />
<br />
dy<br />
dt t , y(0) 2,<br />
y2 y(t) 3 √<br />
3<br />
2 t2 + C C <br />
C y(0) 2 <br />
2 y(0) 3 √<br />
3<br />
2 02 + C 3 √<br />
C,<br />
dy/dt
C 2 3 8 <br />
y(t) 3 √<br />
3<br />
2 t2 + 8.<br />
dy<br />
dt<br />
<br />
(y) · h(t) <br />
<br />
1 dy<br />
(y) dt h(t),<br />
t <br />
y y t<br />
<br />
<br />
dy<br />
dt 3y.<br />
<br />
<br />
1 dy<br />
y dt 3.<br />
t<br />
∫<br />
∫<br />
1 dy<br />
y dt dt <br />
3 dt,<br />
∫ 1<br />
y dy ∫<br />
3 dt.<br />
<br />
ln |y| 3t + C.<br />
y <br />
<br />
|y| e 3t+C e 3t e C .<br />
C e C <br />
e x x y <br />
<br />
+ −C <br />
y(t) Ce 3t .
y 0 <br />
<br />
y y 0 <br />
<br />
C 0<br />
<br />
<br />
P(t) t <br />
<br />
<br />
P(t)<br />
<br />
<br />
k <br />
<br />
105 ◦ <br />
75 ◦ <br />
dT<br />
dt<br />
−k(T − 75),<br />
k <br />
<br />
<br />
k<br />
<br />
t →∞ <br />
<br />
<br />
80 ◦ <br />
<br />
<br />
<br />
dy<br />
dt<br />
− (2 − t)y 2 − t
1 dy<br />
<br />
t dt<br />
e t2 −2y<br />
y ′ 2y + 2 y(0) 2<br />
y ′ 2y 2 y(−1) 2<br />
<br />
dy<br />
dt<br />
−2ty<br />
t 2 +1 y(0) 4<br />
<br />
<br />
<br />
<br />
<br />
t <br />
y<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dx x2 y (x, y) (1, 3)<br />
<br />
<br />
dy<br />
dt<br />
0.5(y − 250)<br />
y 70 t 0<br />
y <br />
<br />
y 5 t 1<br />
y <br />
dy<br />
dt y2 (5 + t),<br />
u<br />
du<br />
dt e2u+8t .
u(0) 13.<br />
u <br />
<br />
<br />
dy<br />
dx 45yx4<br />
y 5<br />
y(x) <br />
<br />
<br />
M(t) k <br />
<br />
M 0 M(t)<br />
<br />
k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dy<br />
dt − t y , y(0) 8<br />
<br />
t <br />
y <br />
<br />
y <br />
<br />
h(t) <br />
<br />
<br />
k
k −1 k −10 <br />
<br />
<br />
k<br />
<br />
<br />
<br />
t <br />
<br />
<br />
P(t) <br />
( )<br />
dP P<br />
dt −P ln −P(ln P − ln 3).<br />
3<br />
P(t) 0 ≤ P ≤ 6<br />
<br />
P(t) P(0) 1 <br />
P(t) <br />
P(t) P(0) 6 <br />
P(t)
5 · 10 12 <br />
5 · 10 10 <br />
<br />
3 · 10 −8 <br />
<br />
P(t) P <br />
t <br />
P(t)<br />
P(t) <br />
5 · 10 10 <br />
3 · 10 −8 <br />
(<br />
5 · 10 10 m3 <br />
<br />
)(<br />
3 · 10 −8 m3 <br />
m 3 <br />
)<br />
1.5 · 10 3 .<br />
P(t) <br />
P 5 · 10 12 <br />
<br />
P<br />
.<br />
5 · 1012 5 · 10 10 <br />
<br />
(<br />
P m 3 )(<br />
)<br />
<br />
5 · 10 12 m 3 5 · 10 10 m3 <br />
<br />
P .<br />
<br />
100<br />
P <br />
<br />
dP<br />
dt 1.5 · 103 −<br />
P<br />
100<br />
1<br />
100 (1.5 · 105 − P).
P(t) <br />
<br />
dP/dt P<br />
<br />
dP/dt<br />
P<br />
1.5 · 10 5 P<br />
t<br />
dP<br />
dt<br />
P dP<br />
dt<br />
1<br />
100 (1.5 · 105 − P)<br />
<br />
dP<br />
1<br />
100 (1.5 · 105 − P)<br />
dt<br />
<br />
P 1.5 · 10 5 <br />
1.5 · 10 5 <br />
<br />
<br />
<br />
dP<br />
dt 1<br />
100 (1.5 · 105 − P), P(0) 0.<br />
<br />
1 dP<br />
1.5 · 10 5 − P dt 1<br />
100 .<br />
t<br />
∫<br />
∫<br />
1 dP<br />
1.5 · 10 5 − P dt dt <br />
1<br />
100 dt,<br />
<br />
∫<br />
∫<br />
dP<br />
1.5 · 10 5 − P <br />
1<br />
100 dt
− ln |1.5 · 10 5 − P| 1<br />
100 t + C<br />
<br />
−1 <br />
1.5 · 10 5 − P Ce −t/100 . <br />
C P 0 t 0<br />
1.5 · 10 5 − 0 Ce 0 C.<br />
C 1.5 · 10 5 <br />
C <br />
P <br />
P<br />
P(t) 1.5 · 10 5 (1 − e −t/100 ).<br />
P <br />
<br />
<br />
<br />
1.5 · 10 5<br />
<br />
t<br />
P(t) <br />
<br />
dP<br />
dt<br />
1<br />
100 (1.5 · 105 − P)<br />
<br />
<br />
<br />
dP<br />
dt<br />
P <br />
<br />
<br />
<br />
1.5 · 10 5
t 0 <br />
dP<br />
dt − 1<br />
100 P, P(0) 1.5 · 105 .<br />
P(t) <br />
1.5 · 10 5 e −t/100 <br />
T P(T) 0.75 · 10 5 <br />
0.75 · 10 5 1.5 · 10 5 e −T/100 ,<br />
<br />
<br />
1<br />
2 e−T/100 .<br />
( ) 1<br />
T −100 ln ≈ 69.3<br />
2<br />
<br />
<br />
A(t)<br />
t<br />
A t<br />
<br />
A<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
M(t) <br />
k <br />
M 0
M(t) <br />
k<br />
<br />
<br />
<br />
<br />
<br />
M(t) M(t) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2860 0.04 <br />
6 <br />
<br />
<br />
y(0) <br />
<br />
S <br />
<br />
S ′ f (t, S) <br />
<br />
<br />
<br />
<br />
S(t) <br />
<br />
<br />
S(78) <br />
<br />
50 1000 0.025 <br />
10
220 <br />
5 1100 <br />
t t<br />
<br />
<br />
5 <br />
1510 <br />
320 <br />
58 <br />
<br />
<br />
<br />
100 88 <br />
<br />
<br />
k <br />
0.05 <br />
<br />
S(t) k<br />
S(t) <br />
k 1820000 47 <br />
k
≈ 9.8 <br />
<br />
k <br />
<br />
<br />
k<br />
v(t)<br />
<br />
k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
S(t) t <br />
t<br />
<br />
<br />
S<br />
<br />
S(t) <br />
t<br />
S(t)
dP<br />
dt 1 2 P.<br />
<br />
4<br />
P<br />
3<br />
2<br />
1<br />
t<br />
2 4<br />
dP<br />
<br />
dt<br />
1 2<br />
P
P(0) dP<br />
dt<br />
1 2 P<br />
<br />
dP<br />
dt 1 P(3 − P).<br />
2<br />
<br />
4<br />
P<br />
3<br />
2<br />
1<br />
t<br />
2 4<br />
<br />
<br />
P(0) <br />
<br />
<br />
<br />
<br />
<br />
<br />
5.932 6.008 6.084 6.159 6.234 6.456 6.531 6.606 6.681 6.756 6.831
P(t) t <br />
dP<br />
dt kP<br />
k <br />
P ′ (0) <br />
t 0 <br />
P(0)<br />
k <br />
<br />
k <br />
dP<br />
dt<br />
kP, P(0) 6.084.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k <br />
dP<br />
dt<br />
kP k <br />
k dP/dt<br />
P .<br />
k
P<br />
<br />
k dP<br />
dt<br />
P t 0 <br />
<br />
<br />
P<br />
0.015 per capita growth rate<br />
0.014<br />
0.013<br />
0.012<br />
0.011<br />
0.010<br />
6.0 6.2 6.4 6.6 6.8 7.0<br />
P<br />
<br />
<br />
<br />
0.015 per capita growth rate<br />
2 4 6 8 10 12 14 16 18 20<br />
0.03<br />
per capita growth rate<br />
0.014<br />
0.013<br />
0.012<br />
0.011<br />
0.010<br />
6.0 6.2 6.4 6.6 6.8 7.0<br />
0.02<br />
0.01<br />
P<br />
-0.01<br />
P
dP/dt<br />
P<br />
0.025 − 0.002P.<br />
P <br />
dP<br />
dt<br />
P(0.025 − 0.002P).<br />
dP/dt P <br />
dP<br />
dt<br />
P <br />
0.10<br />
dP<br />
dt<br />
0.05<br />
2 4 6 8 10 12 14 16 18 20<br />
P<br />
-0.05<br />
-0.10<br />
dP<br />
dt<br />
P dP<br />
dt<br />
P(0.025 − 0.002P)<br />
dP<br />
dt<br />
P(0.025 − 0.002P) <br />
<br />
<br />
P 0<br />
P 12.5 <br />
P(0) > 0 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dP<br />
dt<br />
kP(N − P).
P 0 1 − P N<br />
0 P N<br />
P N <br />
<br />
<br />
<br />
1 dP<br />
P(N − P) dt k,<br />
∫<br />
∫<br />
1<br />
P(N − P) dP kdt.<br />
<br />
1<br />
P(N − P) 1 [ ]<br />
1<br />
N P + 1<br />
.<br />
N − P<br />
<br />
∫ [ ]<br />
1 1<br />
N P + 1<br />
N − P<br />
∫<br />
dP <br />
kdt.<br />
N 1 N<br />
<br />
1<br />
(ln |P| − ln |N − P|) kt + C.<br />
N<br />
<br />
N <br />
<br />
ln P<br />
<br />
N − P<br />
kNt + C.<br />
<br />
e C C C <br />
<br />
P<br />
N − P CekNt .<br />
C P<br />
P(0) P 0 C P 0<br />
N−P 0<br />
<br />
P<br />
N − P P 0<br />
N − P 0<br />
e kNt .<br />
P (N − P)(N − P 0 )<br />
<br />
P(N − P 0 ) P 0 (N − P)e kNt<br />
P 0 Ne kNt − P 0 Pe kNt .
P <br />
P 0 Ne kNt P(N − P 0 ) + P 0 Pe kNt<br />
P <br />
P(N − P 0 + P 0 e kNt ).<br />
P 0 Ne kNt<br />
N − P 0 + P 0 e kNt .<br />
1 P 0<br />
e −kNt <br />
P(t) <br />
N<br />
( N−P0<br />
)<br />
P 0<br />
e −kNt + 1 .<br />
<br />
<br />
dP<br />
dt kP(N − P), P(0) P 0,<br />
<br />
N<br />
P(t) ( N−P0<br />
)<br />
P 0<br />
e −kNt + 1 . <br />
<br />
<br />
k 0.002, N 12.5, P 0 6.084.<br />
<br />
P(t) <br />
<br />
12.5<br />
1.0546e −0.025t + 1 ,<br />
15<br />
12<br />
P<br />
9<br />
6<br />
3<br />
t<br />
40 80 120 160 200
N k P 0 <br />
<br />
<br />
dP<br />
dt<br />
kP(N − P)<br />
dP<br />
dt<br />
P <br />
dP<br />
dt<br />
N<br />
P<br />
N/2<br />
dP<br />
dt<br />
P<br />
P <br />
P
N<br />
P(t) ( N−P0<br />
)<br />
P 0<br />
e −kNt + 1 .<br />
P(0) P 0 lim t→∞ P(t) N<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
P dP/dt 3P − 3P 2 <br />
<br />
<br />
P <br />
P(0) <br />
<br />
P(0) <br />
<br />
P(0) <br />
<br />
<br />
<br />
<br />
<br />
P <br />
P <br />
P <br />
P P
P P <br />
<br />
<br />
<br />
P →<br />
<br />
<br />
<br />
P <br />
<br />
dP/dt P <br />
dP/dt <br />
P <br />
dP/dt <br />
P <br />
<br />
dP/dt <br />
P <br />
<br />
dP/dt <br />
P <br />
dP/dt <br />
<br />
P <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
P <br />
<br />
P ′ (1980) ≈<br />
P ′ (1985) ≈<br />
P ′ (1990) ≈<br />
<br />
P
P <br />
L <br />
<br />
L <br />
<br />
<br />
75<br />
P <br />
1 + 316.75e −0.699t .<br />
<br />
L <br />
<br />
<br />
t k 2<br />
<br />
<br />
P <br />
<br />
<br />
<br />
<br />
P <br />
dP<br />
dt<br />
− .<br />
<br />
<br />
<br />
dP<br />
dt aP2 − bP a, b > 0.<br />
<br />
dP/dt P dP/dt <br />
dP/dt < 0 P <br />
dP/dt > 0 P <br />
<br />
<br />
<br />
dP/dt <br />
P b/a
P(0) > b/a P <br />
P →<br />
P(0) b/a P <br />
P →<br />
P(0) < b/a P <br />
P →<br />
<br />
p(t) t <br />
<br />
dp<br />
0.2p(1 − p)<br />
dt<br />
p <br />
p(0) 0.1<br />
p(t) <br />
p(t)<br />
p <br />
<br />
b(t) <br />
b t <br />
<br />
<br />
db/dt<br />
b<br />
b <br />
db<br />
dt <br />
<br />
<br />
<br />
<br />
<br />
dP<br />
0.1P(10 − P),<br />
dt<br />
P t
1, 1, 2, 3, 5, 8,... <br />
<br />
<br />
1, 3, 6, 10, 15, 21, 28, 36, 45, 55,...
$5000 <br />
8% <br />
0.08<br />
12 · P P <br />
<br />
<br />
( ) 0.08<br />
5000<br />
12<br />
$33.33 $5033.33 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
P n <br />
n <br />
(<br />
P n 5000 1 + 0.08 ) n<br />
<br />
12<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
0 $0.00 $5000.00<br />
1 $33.33 $5033.33<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
12
s 1 , s 2 ,...,s n ...<br />
<br />
s 1 , s 2 , s 3 ,... <br />
f<br />
s 1 s 2 s 3 ...<br />
f f (n) s n n <br />
<br />
<br />
s 1 , s 2 , s 3 ,...<br />
{s n } s n s(n) n <br />
n <br />
<br />
<br />
<br />
<br />
s n n 1, 2, 3,... s n <br />
<br />
(n, s n ) n <br />
<br />
<br />
<br />
n <br />
<br />
n <br />
<br />
<br />
x →∞
{s n } n →∞ <br />
s n n 1, 1 2 , 1 3 ,... s n <br />
<br />
x →∞ {s n } n →∞<br />
<br />
s n n 2, 3 2 , 4 3 , 5 4 ,... s n <br />
x →∞ <br />
{s n } n →∞ <br />
<br />
<br />
<br />
f L x →∞<br />
<br />
<br />
{s n } n →∞<br />
<br />
1+n<br />
2+n <br />
n →∞ <br />
{s n } <br />
n {s n } n <br />
<br />
L <br />
lim<br />
n→∞ s n L<br />
{s n } {s n } <br />
<br />
<br />
{s n } L s k <br />
k ≥ n L n <br />
n →∞ <br />
x →∞
1+2n<br />
<br />
3n−2<br />
<br />
5+3 n<br />
10+2 n<br />
<br />
10 n<br />
n! ! n! n(n − 1)(n − 2) ···(2)(1) <br />
n 0! <br />
<br />
<br />
<br />
<br />
{s n } L <br />
s k k ≥ n L n <br />
<br />
<br />
n →∞<br />
s n n(n + 1) − 1<br />
s n 1/(n + 1)<br />
s n 3 − 1/n<br />
s n n sin(n)/(n + 1)<br />
s n (n + 1)/n<br />
±∞<br />
<br />
∞<br />
<br />
<br />
s n n ≥ 1 <br />
s n <br />
<br />
n →∞ <br />
<br />
4n+8<br />
n<br />
4 n
4n+8<br />
n 2<br />
sin n<br />
4n<br />
<br />
<br />
f (t) sin t t <br />
f (t) Δt <br />
s n f (nΔt) f <br />
sin(1/10) sin(2/10) sin(3/10)<br />
<br />
f (t) (t − 0.5) 2 .<br />
Δt 0.25 <br />
<br />
<br />
<br />
<br />
ln(n)<br />
n<br />
<br />
<br />
<br />
f <br />
f (x) ln(x)<br />
x<br />
<br />
f (x) [0, 10] <br />
{ }<br />
ln(n)<br />
<br />
n <br />
lim x→∞ f (x) L <br />
f (x) ∞ ∞<br />
x <br />
lim x→∞ f (x) <br />
ln(n)<br />
lim x→∞ f (x) lim n→∞ n<br />
<br />
<br />
<br />
P r% <br />
r<br />
12<br />
r <br />
8% 0.08<br />
12<br />
P <br />
( ) r<br />
P<br />
12<br />
<br />
P <br />
)<br />
)<br />
P 1 P + P P<br />
<br />
( r<br />
12<br />
(<br />
1 + r<br />
12
P 1 <br />
P 2 <br />
<br />
(<br />
P 2 P 1 1 + r ) (<br />
P 1 + r ) 2<br />
<br />
12 12<br />
P 3 <br />
P<br />
P n <br />
P <br />
P n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A <br />
t A 2<br />
t + 5 <br />
A 0 100 <br />
A 1 A 1 <br />
A 2 A 2 <br />
A 3 A 3 <br />
A 4 A 4 <br />
A n 5n A n <br />
{A n }
f f (x) sin(4x) [0, 10] <br />
f <br />
1.0<br />
0.5<br />
2 4 6 8 10<br />
x<br />
-0.5<br />
-1.0<br />
f (x) sin(4x) [0, 10]<br />
f [0, 10] <br />
f <br />
<br />
[0, 10] <br />
<br />
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f [0, 10]
n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Q(n) <br />
(n + 1) <br />
<br />
Q(1) 5 × 0.08 <br />
Q(2) (5 + Q(1)) × 0.08 <br />
Q(2) (5 × 0.08) (1 + 0.08) <br />
Q(3) (5 + Q(2)) × 0.08 <br />
Q(3) (5 × 0.08) 1 + 0.08 + 0.08 2 <br />
Q(4) (5 + Q(3)) × 0.08 <br />
Q(4) (5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3 <br />
<br />
Q(n) n
Q(n) n <br />
Q(n) Q(n)<br />
n <br />
Q(1) 0.40<br />
Q(2) <br />
Q(3) <br />
Q(4) <br />
Q(5) <br />
Q(6) <br />
Q(7) <br />
Q(8) <br />
Q(9) <br />
Q(10) <br />
Q(n) n<br />
<br />
<br />
(5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3 + ···+ 0.08 n−1 <br />
<br />
<br />
a + ar + ar 2 + ···+ ar n−1<br />
<br />
a 5 × 0.08 r 0.08 r <br />
<br />
a r r 1 <br />
S n a + ar + ar 2 + ···+ ar n−1 <br />
S n n<br />
<br />
S n r <br />
rS n S n <br />
S n − rS n a − ar n
S n S n <br />
n <br />
<br />
S n <br />
S n a + ar + ar 2 + ···+ ar n−1 <br />
<br />
a r r 1 S n <br />
<br />
S n a + ar + ar 2 + ···+ ar n−1 a(1 − rn )<br />
<br />
<br />
1 − r<br />
<br />
<br />
Q(n) (5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3 + ···+ 0.08 n−1 <br />
Q(n) a 5 × 0.08 0.4 r 0.08 <br />
( 1 − 0.08<br />
n )<br />
Q(n) 0.4<br />
1<br />
1 − 0.08 2.3 (1 − 0.08n ) <br />
n 0.08 n <br />
1<br />
lim Q(n) lim<br />
n→∞ n→∞ 2.3 (1 − 0.08n ) 1<br />
2.3 ≈ 0.435<br />
1<br />
2.3 <br />
<br />
n <br />
n <br />
n <br />
<br />
<br />
∞∑<br />
a + ar + ar 2 + ··· ar n <br />
n0<br />
<br />
r <br />
n + 1 ar n n ar n−1 r<br />
<br />
<br />
<br />
N 0.121212
N 0.12 + 0.0012 + 0.000012 + ···<br />
( ) ( )( ) ( )( ) 12 12 1 12 1 2<br />
+<br />
+<br />
+ ···<br />
100 100 100 100 100<br />
a 12<br />
100 r 1<br />
100 <br />
<br />
<br />
<br />
r 1 a <br />
S a + ar + ar 2 + ···ar n−1 + ···<br />
n <br />
S n a + ar + ar 2 + ···+ ar n−1 <br />
<br />
S n a 1 − rn<br />
1 − r <br />
n S n S<br />
lim n→∞ r n |r| > 1 |r| < 1 <br />
|r| < 1 S n <br />
S S<br />
N <br />
12<br />
<br />
100 + 12<br />
1<br />
<br />
100 100 + 12<br />
1<br />
2<br />
100 100 + ··· a <br />
12<br />
100 r 1<br />
100 <br />
N 12 100<br />
<br />
1<br />
1 − 1<br />
100<br />
12<br />
100<br />
<br />
( ) 100<br />
4<br />
99 33 <br />
<br />
S n <br />
∞∑<br />
ar k<br />
<br />
k0<br />
∑n−1<br />
S n a + ar + ar 2 + ···+ ar n−1 ar k <br />
S n n <br />
<br />
k0
∞∑<br />
a + ar + ar 2 + ··· ar n <br />
n0<br />
<br />
a r r 0<br />
n S n <br />
S n a + ar + ar 2 + ···+ ar n−1 <br />
|r| < 1 S n<br />
<br />
a 1−rn<br />
1−r<br />
S <br />
S lim S n lim a 1 − rn<br />
n→∞ n→∞ 1 − r a<br />
1 − r<br />
<br />
n 0 <br />
n 0 <br />
<br />
<br />
∞∑ ( ) 1 k ( ) ( ) 1 1 2 ( ) 1 3<br />
(2) (2) + (2) + (2) + ···<br />
3 3 3 3<br />
k1<br />
(2) 1<br />
3 <br />
<br />
a r −1 < r < 1 <br />
∞∑<br />
ar k ar 3 + ar 4 + ar 5 + ···<br />
k3<br />
ar 3 <br />
a r −1 <<br />
r < 1 n <br />
∞∑<br />
ar k ar n + ar n+1 + ar n+2 + ···<br />
kn
ar n <br />
<br />
<br />
∞∑<br />
ar k<br />
a r r 0<br />
∑ ∞<br />
k0 ark n <br />
k0<br />
∑n−1<br />
S n ar k <br />
k0<br />
n <br />
S n a 1 − rn<br />
1 − r <br />
|r| < 1 ∑ ∞<br />
k0 ark <br />
a<br />
1−r <br />
<br />
7 th <br />
−3, −16.5, −90.75, ...<br />
<br />
<br />
1 + 1 3 + 1 9 + ... + 1 + ...<br />
3n−1 <br />
<br />
∞∑ ( 3 n + 5 n )<br />
9 n<br />
n1<br />
<br />
<br />
−16 + 4 − 4 4 + 4<br />
16 − 4<br />
64 + 4<br />
256 −···<br />
∑14<br />
n<br />
<br />
n3<br />
1<br />
2
1 $0.01 $0.01<br />
2 $0.02 $0.03<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
<br />
n <br />
n <br />
h <br />
<br />
<br />
3 4 h n <br />
n h 0 h <br />
<br />
h 1 h<br />
h 2 h<br />
h 3 h<br />
h n h
H <br />
t <br />
H − 1 2 t2 <br />
<br />
H<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
1 6 <br />
<br />
<br />
<br />
<br />
( )( )( ) ( ) 5 5 1 5 2 ( ) 1<br />
<br />
6 6 6 6 6<br />
<br />
<br />
( )( )( )( )( ) ( ) 5 5 5 5 1 5 4 ( ) 1<br />
<br />
6 6 6 6 6 6 6<br />
<br />
<br />
<br />
<br />
<br />
( )<br />
1 5 2 ( ) ( ) 1 5 4 ( ) 1<br />
6 + + + ···<br />
6 6 6 6<br />
<br />
<br />
k <br />
n <br />
<br />
<br />
<br />
P
P <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
P <br />
0.75P(1 + 0.75 + 0.75 2 + ···)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
P <br />
r <br />
<br />
<br />
<br />
M <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
r<br />
<br />
P r<br />
M <br />
P <br />
<br />
r r<br />
12 <br />
P
0.0r<br />
12 P M <br />
M − r<br />
12<br />
P <br />
P 1 <br />
<br />
( ( ) ) ( r<br />
P 1 P − M − P 1 + r )<br />
P − M<br />
12<br />
12<br />
P 1 <br />
<br />
P 1 r <br />
<br />
P 2 <br />
P 2 ()P 1 − ()M<br />
<br />
P 2 <br />
1 + r 2 <br />
12 P − 1 + 1 +<br />
r<br />
12 M<br />
P 3 <br />
P 3 <br />
(<br />
1 + r ) 3<br />
P −<br />
12<br />
[<br />
1 +<br />
(<br />
1 + r ) (<br />
+ 1 + r ) 2<br />
]<br />
M<br />
12 12<br />
<br />
n <br />
P n <br />
(<br />
1 + r ) n<br />
P −<br />
12<br />
⎡<br />
∑n−1<br />
⎢<br />
⎣ k0<br />
(<br />
1 + r ) k ⎤ ⎥⎥⎥⎥⎦<br />
M <br />
12<br />
<br />
<br />
<br />
∑n−1<br />
k0<br />
(<br />
1 + r ) k<br />
<br />
12<br />
P n <br />
P n <br />
P(t) t <br />
(<br />
P(t) P − 12M )(<br />
1 + r ) 12t<br />
+ 12M <br />
<br />
r 12 r
M 0 <br />
M P(t) 0 <br />
P(t) 0 M <br />
M <br />
12t<br />
rP 1 + r<br />
( 12<br />
<br />
12 1 +<br />
r 12t<br />
) <br />
12 − 1
n <br />
<br />
<br />
<br />
<br />
<br />
N 0.1212121212 ··· 12<br />
100 + 12<br />
100 · 1<br />
100 + 12<br />
100 · 1<br />
100 2 + ···<br />
N <br />
N 4<br />
33 <br />
<br />
<br />
e<br />
<br />
e π ln(2) <br />
<br />
<br />
f (x) e x <br />
f x 0 e <br />
L(x) L(1) L(1) ≈ f (1) e<br />
e x e <br />
<br />
e <br />
e x x <br />
f (x) f ′ (x) <br />
x 0 P 2 (x) f (x) e x <br />
x 0 P 2 (0) f (0) P ′ 2 (0) f ′ (0) P ′′<br />
2 (0) f ′′ (0)<br />
P 2 (x) 1+x+ x2<br />
2 P 2(0) f (0) P ′ 2 (0) f ′ (0) P ′′<br />
2 (0) f ′′ (0)<br />
P 2 (x) e P 2 (1) ≈ f (1)
e <br />
f <br />
f x <br />
P 3 (x) 1+x + x2<br />
2 + x3<br />
6 P 3(0) f (0) P ′ 3 (0) f ′ (0) P ′′<br />
3 (0) f ′′ (0)<br />
P ′′′<br />
3 (0) f ′′′ (0) P 3 (x) e <br />
P 2 (x) <br />
<br />
e <br />
e 2.5 <br />
<br />
<br />
e<br />
1 + 1 2<br />
1 + 1 + 1 2<br />
2.5<br />
1 + 1 + 1 2 + 1 6<br />
2.6<br />
1 + 1 + 1 2 + 1 6 + 1<br />
24<br />
2.7083<br />
1 + 1 + 1 2 + 1 6 + 1<br />
24 + 1<br />
120<br />
2.716<br />
e <br />
1 + 1 + 1 2 + 1 6 + 1<br />
24 + 1<br />
120 + ···+ 1 n!<br />
<br />
n<br />
<br />
<br />
e ≈ 1 + 1 + 1 2 + 1 6 + 1 24 + 1<br />
120 + ···+ 1 n∑<br />
n! 1<br />
k! <br />
n n <br />
<br />
e <br />
∞∑ 1<br />
e <br />
k! <br />
k0<br />
<br />
1<br />
n!<br />
<br />
<br />
0! 0! 1<br />
k0
a 1 + a 2 + ···+ a n + ···<br />
∞∑<br />
a k <br />
k1<br />
a 1 a 2 ... <br />
<br />
{a n } n≥1 <br />
∑<br />
<br />
k≥1<br />
a k<br />
∞∑<br />
a k <br />
k1<br />
a 1 +a 2 +···+a n +···<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∞∑<br />
k1<br />
1<br />
k 2 <br />
<br />
<br />
n n <br />
<br />
<br />
2∑<br />
k1<br />
1<br />
k 2
1∑<br />
k1<br />
2∑<br />
k1<br />
3∑<br />
k1<br />
4∑<br />
k1<br />
5∑<br />
k1<br />
1<br />
k 2 1<br />
1<br />
k 2 <br />
1<br />
k 2 <br />
1<br />
k 2 <br />
1<br />
k 2 <br />
6∑<br />
k1<br />
7∑<br />
k1<br />
8∑<br />
k1<br />
9∑<br />
k1<br />
10∑<br />
k1<br />
1<br />
k 2 <br />
1<br />
k 2 <br />
1<br />
k 2 <br />
1<br />
k 2 <br />
1<br />
k 2 <br />
∑ ∞<br />
k1<br />
1<br />
k 2<br />
n S n ∑ n<br />
k1<br />
{S n } <br />
{S n } <br />
∑ ∞<br />
k1<br />
1<br />
k 2 <br />
<br />
n <br />
<br />
<br />
n ∑ ∞<br />
k1 a k S n ∑ n<br />
k1 a k<br />
n S n n <br />
<br />
S n a 1 + a 2 + ···+ a n <br />
<br />
S 1 , S 2 ,...,S n ,...<br />
<br />
<br />
<br />
∞∑ 1<br />
k 2<br />
k1<br />
<br />
<br />
1<br />
k 2
∞∑<br />
k1<br />
a k<br />
{S n } <br />
S n <br />
n∑<br />
a k <br />
k1<br />
lim n→∞ S n S S ∑ ∞<br />
k1 a k <br />
∞∑<br />
k1<br />
a k lim<br />
n→∞<br />
S n S<br />
<br />
<br />
∞∑<br />
k1<br />
<br />
<br />
∞∑<br />
a k<br />
a k<br />
k1<br />
a k k <br />
m <br />
∞∑<br />
a k (a 1 + a 2 + ···+ a m ) +<br />
k1<br />
∞∑<br />
km+1<br />
a 1 + a 2 + ···+ a m ∑ ∞<br />
k1 a k <br />
∑ ∞<br />
km+1 a k <br />
<br />
∑<br />
a k<br />
<br />
<br />
<br />
<br />
a k
∑ ∞<br />
k0 ark a r r 1 <br />
n S n <br />
S n 1 − rn<br />
1 − r <br />
|r| < 1 <br />
r <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∑ a k <br />
{a n } <br />
∑ ∞<br />
k1 a k L<br />
n S n ∑ ∞<br />
k1 a k<br />
(n − 1) S n−1 ∑ ∞<br />
k1 a k<br />
n (n − 1) <br />
∑ ∞<br />
k1 a k<br />
∑ ∞<br />
k1 a k L lim n→∞ S n <br />
lim n→∞ S n−1 <br />
lim n→∞ a n <br />
lim n→∞ (S n − S n−1 )<br />
<br />
∑ ∞<br />
k1 a k {a k } k <br />
<br />
{a k } n <br />
∑ ∞<br />
k1 a k <br />
<br />
lim k→∞ a k 0 ∑ a k
∑ k<br />
k+1<br />
∑ (−1) k <br />
∑ 1<br />
k<br />
<br />
<br />
<br />
<br />
<br />
∑ a k <br />
lim k→∞ a k 0 ∑ a k <br />
lim k→∞ a k 0 <br />
<br />
<br />
<br />
∞∑<br />
k1<br />
1<br />
k <br />
<br />
<br />
1∑<br />
k1<br />
2∑<br />
k1<br />
3∑<br />
k1<br />
4∑<br />
k1<br />
k1<br />
1<br />
k 1<br />
1<br />
k 1.5<br />
1<br />
k 1.833333333<br />
1<br />
k <strong>2.0</strong>83333333<br />
6∑<br />
k1<br />
7∑<br />
k1<br />
8∑<br />
k1<br />
9∑<br />
k1<br />
k1<br />
1<br />
k 2.450000000<br />
1<br />
k 2.592857143<br />
1<br />
k 2.717857143<br />
1<br />
k 2.828968254<br />
5∑ 1<br />
∑10<br />
k 2.283333333 1<br />
k 2.928968254<br />
<br />
∑ ∞<br />
k1<br />
∑ ∞ 1<br />
k1 k<br />
<br />
<br />
∑ 1000<br />
n1 1 k ≈ 7.485470861<br />
<br />
<br />
1<br />
k
∑ ∞ 1<br />
k1 k<br />
<br />
n S n<br />
∑ ∞ 1<br />
k1 k n∑ 1<br />
S n <br />
k<br />
k1<br />
1 + 1 2 + 1 3 + ···+ 1 n<br />
( ) ( ) ( )<br />
1 1 1<br />
1(1) + (1) + (1) + ···+ (1) <br />
2 3 n<br />
S n <br />
<br />
<br />
1 m<br />
<br />
<br />
1.00<br />
0.75<br />
0.50<br />
a k<br />
0.25<br />
k<br />
0.00<br />
1 2 3 4 5 6 7 8 9 10<br />
<br />
<br />
<br />
f f (x) 1 x<br />
<br />
<br />
<br />
<br />
S 9 <br />
<br />
n <br />
n ∑ ∞ 1<br />
k1 k<br />
<br />
∫ ∞ 1<br />
1 x<br />
dx <br />
∫ ∞<br />
1<br />
1<br />
x<br />
dx <br />
∑ ∞<br />
k1<br />
1<br />
k <br />
f
a k f (k) k ∑ ∞<br />
k1 a k<br />
<br />
n∑<br />
S n <br />
a k<br />
k1<br />
f (x) <br />
a k <br />
f <br />
S n ><br />
∫ n<br />
1<br />
f (x) dx<br />
n <br />
∞∑<br />
a k ><br />
k1<br />
∫ ∞<br />
1<br />
f (x) dx<br />
∫ ∞<br />
1<br />
f (x) dx ∑ ∞<br />
k1 a k<br />
1.00<br />
a k<br />
0.75<br />
0.50<br />
0.25<br />
k<br />
0.00<br />
1 2 3 4 5 6 7 8 9 10<br />
1.00<br />
a k<br />
0.75<br />
0.50<br />
0.25<br />
k<br />
0.00<br />
1 2 3 4 5 6 7 8 9 10<br />
<br />
f [1, n] <br />
<br />
∫ ∞<br />
∞∑<br />
f (x) dx > a k <br />
1<br />
k2
∫ ∞<br />
1<br />
f (x) dx ∑ ∞<br />
k2 a k ∑ ∞<br />
<br />
<br />
f f x <br />
c a k f (k) k<br />
∫ ∞<br />
c<br />
∫ ∞<br />
c<br />
k1 a k<br />
f (x) dx ∑ ∞<br />
k1 a k <br />
f (x) dx ∑ ∞<br />
k1 a k <br />
<br />
<br />
<br />
<br />
∑ 1<br />
k<br />
p <br />
p<br />
p p 1 <br />
p <br />
∫ ∞ 1<br />
1<br />
dx ∑ ∞ 1<br />
x 2<br />
k1<br />
<br />
k 2<br />
<br />
∫ ∞ 1<br />
1 x<br />
dx p > 1 p <br />
p<br />
∑ ∞ 1<br />
k1 k<br />
<br />
p<br />
∫ ∞ 1<br />
x<br />
dx p < 1 <br />
p<br />
p ∑ ∞<br />
k1<br />
1<br />
1<br />
k<br />
p<br />
<br />
p ∑ ∞<br />
k1<br />
1<br />
k p <br />
<br />
<br />
p <br />
<br />
∑ k 2 +1<br />
<br />
k 4 +2k+2<br />
<br />
p<br />
∑ k+1<br />
<br />
k 3 +2<br />
k
k <br />
k+1 k 100 k 1000 <br />
k 3 +2<br />
k+1<br />
k<br />
k <br />
k 3 +2 k 3<br />
a k k+1<br />
k 3 +2 b k k <br />
k 3<br />
<br />
a k<br />
lim <br />
k→∞ b k<br />
a k b k k<br />
<br />
∑ k<br />
<br />
k 3<br />
∑ k+1<br />
k 3 +2 <br />
<br />
<br />
∑ a k ∑ b k <br />
∑ a k <br />
k a k b k <br />
k ∑ a k ∑ b k <br />
c <br />
b k<br />
lim c<br />
k→∞ a k<br />
b k ≈ ca k k <br />
∑ ∑ ∑<br />
b k ≈ ca k c a k <br />
<br />
∑ a k ∑ b k <br />
<br />
<br />
∑ a k ∑ b k <br />
b k<br />
lim c<br />
k→∞ a k<br />
c ∑ a k ∑ b k <br />
<br />
∑ p(k)<br />
q(k)<br />
<br />
p(k) m q(k) l <br />
∑ p(k)<br />
q(k)<br />
∑ k m<br />
<br />
k l
∑ 3k 2 + 1<br />
5k 4 + 2k + 2 <br />
∑ 1<br />
<br />
k 2<br />
<br />
<br />
<br />
<br />
<br />
<br />
∞∑<br />
k1<br />
2 k<br />
3 k − k <br />
<br />
∑ a k a k+1<br />
a k <br />
2k<br />
3 k −k <br />
∑ 2 k<br />
<br />
3 k −k<br />
<br />
<br />
a k<br />
k<br />
5<br />
10<br />
20<br />
21<br />
22<br />
23<br />
24<br />
25<br />
a k+1<br />
a k<br />
∑ 2 k<br />
3 k −k<br />
a k+1<br />
a k<br />
k
∑ 2 k<br />
3 k −k<br />
<br />
k ∑ 2 k<br />
<br />
3 k −k<br />
<br />
∑<br />
ak <br />
a k+1<br />
≈ r<br />
a k<br />
k a k+1 ≈ ra k k ∑ a k <br />
∑ ar k k r <br />
−1 < r < 1 ∑ a k <br />
a k+1<br />
lim r<br />
k→∞ a k<br />
r |r| < 1 <br />
<br />
∑ a k <br />
|a k+1 |<br />
lim<br />
k→∞ |a k |<br />
0 ≤ r < 1 ∑ a k <br />
1 < r ∑ a k <br />
r 1 <br />
r<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p <br />
p <br />
<br />
n
∑ k<br />
2 k<br />
∑ k 3 +2<br />
k 2 +1<br />
∑ 10 k<br />
k!<br />
∑ k 3 −2k 2 +1<br />
k 6 +4<br />
<br />
<br />
<br />
∞∑<br />
a 1 + a 2 + ···+ a n + ···<br />
a k k<br />
n S n ∑ ∞<br />
k1 a k n <br />
<br />
n∑<br />
S n a 1 + a 2 + ···+ a n a k <br />
{S n } ∑ ∞<br />
k1 a k <br />
<br />
◦ ∑ ∞<br />
k1 a k {S n } <br />
<br />
<br />
∞∑<br />
k1<br />
k1<br />
k1<br />
a k lim<br />
n→∞<br />
S n <br />
◦ ∑ ∞<br />
k1 a k {S n } <br />
a k<br />
<br />
<br />
A n 80<br />
8 n<br />
<br />
<br />
∞∑<br />
(A n ) <br />
n1<br />
{A n } <br />
<br />
<br />
∞∑<br />
n1<br />
10<br />
n + 2 s n <br />
s n <br />
n∑<br />
i1<br />
10<br />
i + 2 .
s 4 s 8<br />
s 4 <br />
s 8 <br />
<br />
2n<br />
a n <br />
10n + 7 .<br />
<br />
<br />
∞ −∞ <br />
∞∑ 2n<br />
<br />
10n + 7 <br />
n1<br />
{<br />
2n<br />
<br />
10n + 7<br />
}<br />
<br />
<br />
∫<br />
∞ −∞ <br />
∞<br />
3 dx<br />
1 x 2 + 1 <br />
∞∑ 3<br />
<br />
n 2 <br />
+ 1<br />
n1<br />
<br />
b n<br />
n!<br />
b<br />
∑ 10 k<br />
k!<br />
<br />
∑ b k<br />
k!<br />
b<br />
<br />
b n<br />
n! <br />
<br />
b n<br />
n! <br />
<br />
∑ a k a n → 0 n →∞<br />
<br />
n√<br />
an → r<br />
n a n ≈ r n n<br />
∑ a k n <br />
∑ a k <br />
r ∑ a k<br />
n√ a n → r n →∞
∑ n<br />
k0 a k ∑ n<br />
k0 b k <br />
n∑ n∑<br />
a k + b k <br />
k0 k0<br />
n∑<br />
(a k + b k )<br />
k0<br />
<br />
a n 1 + 1<br />
2<br />
b n n −1 n<br />
∑ ∞<br />
k0 a k ∑ ∞<br />
k0 b k <br />
∑ ∞<br />
k0 (a k + b k ) <br />
<br />
∞∑ ∞∑ ∞∑<br />
a k + b k (a k + b k )<br />
k0<br />
k0<br />
∑ ∞<br />
k0 a k ∑ ∞<br />
k0 b k<br />
∑ ∞<br />
k0 (a k + b k ) <br />
<br />
<br />
∑ a k ∑ b k ∑ (a k + b k )<br />
<br />
k0<br />
∑ ∑ ∑<br />
(a k + b k ) a k + b k <br />
A n B n n ∑ ∞<br />
k1 a k ∑ ∞<br />
k1 b k <br />
<br />
n∑<br />
A n + B n (a k + b k )<br />
<br />
k1<br />
∞∑<br />
(a k + b k ) <br />
k1<br />
∞∑ ∞∑<br />
a k + b k <br />
k1 k1<br />
<br />
<br />
∑ ∞ 2 k +3 k<br />
k0<br />
<br />
5 k
∑ 1<br />
k 2 ∑ 1<br />
k 2 + k <br />
∑ 1<br />
k 2 p p 2 > 1 ∑ 1<br />
k 2<br />
∑ 1<br />
k 2<br />
∑ 1<br />
k 2 +k a k 1 b<br />
k 2 k 1<br />
k 2 +k <br />
S n n ∑ 1<br />
S 1 T 1 <br />
<br />
<br />
<br />
T<br />
k 2 n n ∑ 1<br />
k 2 +k <br />
S 2 S 1 + a 2 T 2 T 1 + b 2 <br />
a 2 b 2 S 2 T 2 <br />
<br />
S 3 S 2 + a 3 T 3 T 2 + b 3 <br />
a 3 b 3 S 3 T 3 <br />
a n b n S n T n <br />
<br />
∑ 1<br />
∑ 1<br />
k 2 k 2 +k ∑ 1<br />
<br />
k 2 +k<br />
<br />
<br />
<br />
<br />
<br />
k <br />
0 ≤<br />
0 ≤ b k ≤ a k<br />
∑ ∑<br />
b k ≤ a k<br />
∑ a k ∑ b k <br />
∑ b k ∑ a k <br />
<br />
<br />
∑ 1<br />
k−1 <br />
<br />
∑<br />
k<br />
k 3 +1
n <br />
<br />
<br />
<br />
<br />
<br />
2 − 4 3 + 8 9 −···+ 2 (<br />
− 2 3<br />
) n<br />
+ ···<br />
a 2 r − 2 3 <br />
<br />
S <br />
a<br />
1 − r 2<br />
1 − <br />
− 2 6 5 <br />
3<br />
<br />
<br />
<br />
e <br />
ln(2) <br />
<br />
f (x) ln(1 + x)<br />
f x 0 ln(2)<br />
L(x) f (x) <br />
L(1) ≈ f (1) ln(2)<br />
ln(1 + x) ln(2)<br />
<br />
ln(2) <br />
ln(1 + x) x
x 0 P 2 (x) f (x) <br />
ln(1 + x) x 0 P 2 (0) f (0) P ′ 2 (0) f ′ (0) <br />
P ′′<br />
2 (0) f ′′ (0)<br />
P 2 (x) x − x2<br />
2 P 2(0) f (0) P ′ 2 (0) f ′ (0) P ′′<br />
2 (0) f ′′ (0)<br />
P 2 (x) ln(2) P 2 (1) ≈ f (1)<br />
ln(2) <br />
f <br />
f x P 3 (x) <br />
x − x2<br />
2 + x3<br />
3 P 3(0) f (0) P ′ 3 (0) f ′ (0) P ′′<br />
3 (0) f ′′ (0) <br />
P ′′′<br />
3 (0) f ′′′ (0) P 3 (x)<br />
ln(2)<br />
ln(1 + x) <br />
ln(2) <br />
<br />
<br />
<br />
ln(2) <br />
1 − 1 2<br />
1 2 <br />
<br />
ln(2) <br />
<br />
ln(2) <br />
<br />
∞∑<br />
(−1) k+1 1 k<br />
k1<br />
<br />
<br />
(−1) k+1 <br />
1 1<br />
1 − 1 2<br />
0.5<br />
1 − 1 2 + 1 3<br />
0.83<br />
1 − 1 2 + 1 3 − 1 4<br />
0.583<br />
1 − 1 2 + 1 3 − 1 4 + 1 5<br />
0.783<br />
<br />
<br />
ln(2) ≈ 0.6931471806 <br />
ln(2) <br />
ln(2)
∞∑<br />
(−1) k a k <br />
a k > 0 k<br />
k0<br />
<br />
∞∑<br />
(−1) k+1 a k <br />
k1<br />
k 1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∞∑<br />
(−1) k+1 1 k <br />
k1<br />
<br />
<br />
<br />
<br />
<br />
1∑<br />
(−1) k+1 1 6∑<br />
k <br />
(−1) k+1 1 k <br />
k1<br />
2∑<br />
(−1) k+1 1 k <br />
k1<br />
3∑<br />
(−1) k+1 1 k <br />
k1<br />
4∑<br />
(−1) k+1 1 k <br />
k1<br />
k1<br />
7∑<br />
(−1) k+1 1 k <br />
k1<br />
8∑<br />
(−1) k+1 1 k <br />
k1<br />
9∑<br />
(−1) k+1 1 k <br />
k1<br />
5∑<br />
(−1) k+1 1 ∑10<br />
k (−1) k+1 1 k <br />
k1<br />
k1<br />
<br />
∑ ∞<br />
k1 (−1)k+1 1 k
lim k→∞ a k 0 ∑ a k <br />
<br />
∞∑<br />
(−1) k+1 a k<br />
k1<br />
a k <br />
n S n <br />
n∑<br />
S n (−1) k+1 a k <br />
<br />
k1<br />
S 2 a 1 − a 2 a 1 > a 2 0 < S 2 < S 1 <br />
S 3 S 2 + a 3 S 2 < S 3 a 3 < a 2 S 3 < S 1 0 < S 2 < S 3 < S 1 <br />
S 4 S 3 − a 4 S 4 < S 3 a 4 < a 3 S 2 < S 4 0 < S 2 < S 4 < S 3 < S 1 <br />
S 5 S 4 + a 5 S 4 < S 5 a 5 < a 4 S 5 < S 3 0 < S 2 < S 4 < S 5 < S 3 < S 1 <br />
<br />
n <br />
<br />
a n<br />
S 2 S 4 S 6<br />
... S n−1 S n S 5 S 3 S 1<br />
<br />
<br />
(n − 1) S n−1<br />
n S n <br />
|S n − S n−1 | a n <br />
{a n } <br />
<br />
<br />
<br />
<br />
<br />
∑<br />
(−1) k a k <br />
{a k } k →∞
{a k } <br />
<br />
<br />
∑ ∞ (−1) k<br />
k1 k 2 +2<br />
<br />
∑ ∞<br />
k1<br />
(−1) k+1 2k<br />
k+5<br />
∑ ∞ (−1) k<br />
k2 ln(k)<br />
<br />
<br />
n S n <br />
S <br />
S <br />
∞∑<br />
(−1) k a k <br />
k1<br />
S <br />
|S − S n | < |S n+1 − S n | a n+1 <br />
a n+1 <br />
S n S <br />
<br />
<br />
<br />
S <br />
∞∑<br />
(−1) k+1 a k<br />
k1<br />
S n <br />
n∑<br />
(−1) k+1 a k<br />
k1<br />
n <br />
∞∑<br />
(−1) k+1 a k − S n ≤ a n+1 <br />
k1
S 100 <br />
∞∑ (−1) k+1<br />
<br />
k1<br />
k<br />
<br />
S ∑ ∞<br />
k1<br />
|S 100 − S| < a 101 <br />
(−1) k+1<br />
k<br />
<br />
<br />
a 101 1<br />
101 ≈ 0.0099<br />
<br />
<br />
∞∑ (−1) k+1<br />
S <br />
ln(2)<br />
k<br />
S ≈ 0.693147 <br />
S 100 <br />
∑100<br />
k1<br />
k1<br />
(−1) k+1<br />
k<br />
≈ 0.6881721793<br />
S S 100 0.0049750013 <br />
0.0099<br />
<br />
<br />
∞∑ (−1) k+1<br />
<br />
k1<br />
k 4<br />
<br />
<br />
1 − 1 4 − 1 9 + 1 16 + 1 25 + 1 36 − 1 49 − 1 64 − 1 81 − 1 + ··· <br />
100
1 − 1 4 − 1 9 + 1<br />
16 + 1 25 + 1 36 − 1<br />
49 − 1 64 − 1 81 − 1<br />
100 + ···<br />
<br />
<br />
∞∑<br />
k1<br />
1<br />
k 2 <br />
1 − 1 4 − 1 9 + 1<br />
16 + 1 25 + 1 36 − 1<br />
49 − 1 64 − 1 81 − 1<br />
100 + ···<br />
<br />
∞∑<br />
− 1 k 2 <br />
<br />
k1<br />
1 − 1 4 − 1 9 + 1<br />
16 + 1 25 + 1 36 − 1<br />
49 − 1 64 − 1 81 − 1<br />
100 + ···<br />
<br />
<br />
∑ a k <br />
∑ |a k | <br />
∑ a k ∑ |a k | ∑ a k <br />
<br />
p ∑ 1<br />
<br />
k 2<br />
∑ (−1) k+1 1 k<br />
<br />
∑ 1<br />
k<br />
<br />
<br />
∑ a k <br />
∑ a k ∑ |a k | <br />
<br />
∑ a k ∑ |a k |<br />
∑ a k
∑ (−1) k ln(k)<br />
k<br />
<br />
<br />
<br />
<br />
∑ (−1) k ln(k)<br />
<br />
k 2<br />
<br />
ln(k) < √ k <br />
k p<br />
{a n } <br />
∑ a k ∑ (−1) k a k <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∑ ar k r −1 < r < 1 <br />
|r| ≥ 1<br />
∞∑<br />
<br />
k0<br />
ar k <br />
a n ∑ a k <br />
lim n→∞ a n <br />
0 <br />
f [c, ∞) a k f (k)<br />
k ≥ c<br />
∫ ∞<br />
c<br />
f (t) dt ∑ a k <br />
∫ ∞<br />
c<br />
f (t) dt ∑ a k <br />
a<br />
1−r
f (x) <br />
<br />
0 ≤ a k ≤ b k k<br />
∑ b k ∑ a k <br />
∑ a k ∑ b k <br />
<br />
<br />
a n b n <br />
a k<br />
lim L<br />
k→∞ b k<br />
L ∑ a k ∑ b k <br />
<br />
<br />
<br />
a k 0 k <br />
|a k+1 |<br />
lim<br />
k→∞ |a k |<br />
r<br />
r < 1 ∑ a k <br />
r > 1 ∑ a k <br />
r 1 <br />
<br />
<br />
a k ≥ 0 k <br />
lim<br />
k→∞<br />
r < 1 ∑ a k <br />
r > 1 ∑ a k <br />
r 1 <br />
k√<br />
ak r<br />
<br />
a k k <br />
a n lim<br />
n→∞<br />
a n 0 <br />
∑ (−1) k+1 a k <br />
a n <br />
{a n }
S n <br />
<br />
n∑<br />
(−1) k+1 a k n <br />
k1<br />
∞∑<br />
(−1) k+1 a k a n > 0 n a n<br />
k1<br />
lim S n S |S − S n | < a n+1 <br />
n→∞<br />
S n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
∑ ∞<br />
k3<br />
∑ ∞<br />
k1<br />
∑ ∞<br />
k0<br />
∑ ∞<br />
k0<br />
∑ ∞<br />
k1<br />
<br />
∑ ∞<br />
k1<br />
2<br />
√<br />
k−2<br />
k<br />
1+2k<br />
2k 2 +1<br />
k 3 +k+1<br />
100 k<br />
k!<br />
2 k<br />
5 k<br />
k 3 −1<br />
k 5 +1<br />
∑ ∞<br />
k2<br />
∑ ∞<br />
k2<br />
∑ ∞<br />
k1<br />
∑ ∞<br />
k2<br />
3 k−1<br />
7 k<br />
1<br />
k k<br />
(−1) k+1<br />
√<br />
k+1<br />
1<br />
k ln(k)<br />
n <br />
n S n <br />
∑ ∞ (−1) n<br />
n2 ln(n)<br />
<br />
<br />
<br />
<br />
<br />
∑<br />
(−1) k a k<br />
a k k<br />
∑ ∞<br />
k1 (−1)k a k {S n } <br />
<br />
n∑<br />
S n (−1) k a k <br />
k1<br />
<br />
n
∑ ∞<br />
k1 (−1)k a k <br />
{a n } n <br />
n − 1 S n−1 n S n <br />
∑ ∞<br />
k1 (−1)k a k |S n − S n−1 | a n <br />
S <br />
|S − S n | < a n <br />
n ∑ ∞<br />
k1 (−1)k a k <br />
a n <br />
<br />
∞ ∑<br />
∞ ∑<br />
n1<br />
n1<br />
(−1) n<br />
3 n <br />
(−1) n<br />
3n<br />
<br />
<br />
∞∑<br />
a n 0.45 − (0.45)3 + (0.45)5 − (0.45)7 + ...<br />
3! 5! 7!<br />
n1<br />
<br />
<br />
<br />
∞∑<br />
a n 1 − (0.4)2 + (0.4)4 − (0.4)6 + (0.4)8 − ...<br />
2! 4! 6! 8!<br />
n1<br />
<br />
<br />
<br />
∞∑<br />
a n 1 − 1 10 + 1<br />
100 − 1<br />
1000 + ...<br />
n1<br />
<br />
<br />
<br />
<br />
π<br />
4 1 − 1 3 + 1 5 − 1 ∞∑<br />
7 + ··· (−1) k 1<br />
2k + 1 <br />
<br />
π<br />
k0
S n n <br />
π 4 S 100 <br />
π 4 <br />
<br />
<br />
<br />
∑ (−1) k+1 a k S <br />
S n <br />
S n +S n+1<br />
2<br />
S S n <br />
S n+S n+1<br />
2<br />
S n + 1 2 (−1)n+2 a n+1 <br />
n 20 ln(2) <br />
∞∑<br />
ln(2) (−1) k+1 1 k <br />
k1<br />
S 20 S 20 155685007<br />
232792560 <br />
<br />
<br />
1 − 1 + 1 2 − 1 4 + 1 3 − 1 9 + 1 4 − 1 16 + ···<br />
{ }<br />
1<br />
n −<br />
1<br />
n 2<br />
n n <br />
<br />
( 1<br />
(1 − 1) +<br />
2 − 1 ) ( 1<br />
+<br />
4 3 − 1 ) ( 1<br />
+<br />
9 4 − 1 )<br />
+ ···<br />
16<br />
<br />
{a n } <br />
<br />
<br />
∑ ∞ (−1) k+1<br />
k1 k<br />
<br />
<br />
<br />
<br />
∞∑ (−1) k+1<br />
ln(2)<br />
k<br />
k1
∑ ∞<br />
k1<br />
1<br />
2k <br />
∑ ∞ 1<br />
k1 2k+1 <br />
<br />
∑ ∞ 1<br />
kC 2k ∑ ∞ 1<br />
kC 2k+1<br />
C <br />
<br />
<br />
a + b b + a<br />
a b <br />
<br />
<br />
<br />
<br />
S <br />
∑ ∞ (−1) k+1<br />
k1 k<br />
S <br />
S <br />
<br />
S<br />
S <br />
P 1<br />
P 1 <br />
∑n 1<br />
k1<br />
1<br />
2k + 1 1 + 1 3 + 1 5 + ···+ 1<br />
2n 1 + 1<br />
S<br />
<br />
S 1 P 1 <br />
S 1 <br />
N 1<br />
<br />
∑m 1<br />
1<br />
N 1 −<br />
2k −1 2 − 1 4 − 1 6 −···− 1<br />
2m 1<br />
k1<br />
S 2 S 1 + N 1 ≤ S<br />
S 2 <br />
P 2<br />
P 2 <br />
∑n 2<br />
kn 1 +1<br />
1<br />
2k + 1 1<br />
2(n 1 + 1) + 1 + 1<br />
2(n 1 + 2) + 1 + ···+ 1<br />
2n 2 + 1<br />
<br />
S 3 S 2 + P 2 ≥ S
S 3 <br />
N 2 −<br />
∑m 2<br />
km 1 +1<br />
1<br />
2k − 1<br />
2(m 1 + 1) − 1<br />
2(m 1 + 2) −···− 1<br />
2m 2<br />
<br />
S 4 S 3 + N 2 ≤ S<br />
{S n }<br />
<br />
lim n→∞ S n S
1 + 1 2 + 1 4 + ···+ 1 ∞∑<br />
2 + ··· 1<br />
k 2 <br />
k<br />
<br />
x <br />
r 1 2<br />
x <br />
k0<br />
1 + x + x 2 + ···+ x k + ···<br />
∞∑<br />
x k <br />
k0<br />
<br />
<br />
r |r| < 1 <br />
a<br />
1−r<br />
<br />
|x| < 1<br />
1 + x + x 2 + ···+ x k + ··· 1<br />
1 − x <br />
1<br />
1−x<br />
<br />
<br />
k <br />
<br />
1 + x + x 2 + x 3 ≈ 1<br />
1 − x<br />
x <br />
<br />
<br />
<br />
<br />
x
e <br />
ln(2) <br />
<br />
e x f (x) e x <br />
P 1 (x) f (x) x 0 <br />
P 1 P 1 (x) <br />
f (x) x 0 f P 1 x 0<br />
<br />
f (x) e x <br />
<br />
f x 0 <br />
P 1 (x) <br />
P 2 (x) P 1 (x) + c 2 x 2<br />
c 2 c 2 <br />
P 2 (x) f (x) x 0<br />
P 1 (x) f (x) 0 <br />
P 1 (0) f (0) P ′ 1 (0) f ′ (0) <br />
c 2 <br />
P 2 (0) f (0) P ′ 2 (0) f ′ (0) P ′′<br />
2 (0) f ′′ (0)<br />
P 2 (x) P 1 (x) + c 2 x 2 <br />
P 2 (0) P 2 (0) f (0)<br />
P ′ 2 (0) P′ 2 (0) f ′ (0)<br />
P ′′<br />
2 (x) c 2 P ′′<br />
2 (0) f ′′ (0)<br />
P ′′<br />
2 (0) f ′′ (0) <br />
P 2 P 2 f x 0<br />
<br />
<br />
<br />
<br />
x
f x a <br />
P 1 (x) f <br />
x a f a P 1 (x) <br />
f (a, f (a)) <br />
P 1 (x) f (a) + f ′ (a)(x − a)<br />
<br />
<br />
P 2 (x) P 1 (x) + c 2 (x − a) 2<br />
P 2 (x) f (x) x a <br />
P 2 (x)<br />
P 2 (x) P 1 (x) + c 2 (x − a) 2<br />
P 2 (a) P 1 (a) f (a)<br />
P ′ 2 (x) P′ 1 (x) + 2c 2(x − a)<br />
P ′ 2 (a) P′ 1 (a) f ′ (a)<br />
P ′′<br />
2 (x) 2c 2 P ′′<br />
2 (a) 2c 2<br />
P 2 (x) f (x) P 1 (x) P 2 (x) f (x) <br />
x a <br />
P ′′<br />
2 (a) f ′′ (a)<br />
<br />
<br />
2c 2 f ′′ (a)<br />
c 2 f ′′ (a)<br />
<br />
2<br />
P 2 (x) f x 0 <br />
P 2 (x) f (a) + f ′ (a)(x − a) + f ′′ (a)<br />
(x − a) 2 <br />
2!<br />
<br />
<br />
<br />
P 3 (x) P 2 (x) + c 3 (x − a) 3 <br />
P 4 (x) P 3 (x) + c 4 (x − a) 4 <br />
P 5 (x) P 4 (x) + c 5 (x − a) 5 <br />
P n (x) P n−1 (x) + c n (x − a) n
n P n (x) <br />
n f x a <br />
k n<br />
P (k)<br />
n (a) f (k) (a)<br />
<br />
P n (x) c 0 + c 1 (x − a) + c 2 (x − a) 2 + ···+ c n (x − a) n <br />
<br />
P (0)<br />
n (a) c 0<br />
P (1)<br />
n (a) c 1<br />
P (2)<br />
n (a) 2c 2<br />
P (3)<br />
n (a) (2)(3)c 3<br />
P (4)<br />
n (a) (2)(3)(4)c 4<br />
P (5)<br />
n (a) (2)(3)(4)(5)c 5<br />
<br />
P (k)<br />
n (a) (2)(3)(4) ···(k − 1)(k)c k k!c k <br />
P (k)<br />
n (a) f (k) (a) k!c k f (k) (a) <br />
c k f (k) (a)<br />
k!<br />
k c k n<br />
f <br />
<br />
n f x a <br />
P n (x) f (a) + f ′ (a)(x − a) + f ′′ (a)<br />
(x − a) 2 + ···+ f (n) (a)<br />
(x − a) n<br />
2!<br />
n!<br />
n∑ f (k) (a)<br />
(x − a) k <br />
k!<br />
k0<br />
n f (x) x a <br />
(a) f (k) (a) k 0 ...n<br />
P (k)<br />
n<br />
f (x) e x <br />
n f x 0
f ′ (x) e x f ′′ (x) e x f ′′′ (x) e x <br />
f (0) f ′ (0) f ′′ (0) f ′′′ (0) 1<br />
f (x) e x x 0 <br />
P 3 (x) f (0) + f ′ (0)(x − 0) + f ′′ (0)<br />
2!<br />
1 + x + x2<br />
2 + x3<br />
6 <br />
(x − 0) 2 + f ′′′ (0)<br />
(x − 0) 3<br />
3!<br />
f f (k) (x) e x k<br />
k n f (x) x 0 <br />
f (k) (0)<br />
(x − 0) k 1 k!<br />
k! xk <br />
n f (x) e x x 0 <br />
P n (x) 1 + x + x2<br />
2! + ···+ 1 n∑ x k<br />
n! xn <br />
k! <br />
k0<br />
n a 0<br />
e x <br />
n∑ x k<br />
k! <br />
k0<br />
<br />
<br />
<br />
f (x) 1<br />
1−x <br />
f (x) x 0 <br />
1<br />
P 4 (x) <br />
1−x<br />
<br />
f (k) (0)<br />
f (x) cos(x)<br />
f (x) x 0 <br />
P 4 (x) cos(x) <br />
f (k) (0) <br />
k k <br />
f (x) sin(x)
f (x) x 0 <br />
P 4 (x) sin(x) <br />
f (k) (0) <br />
k k <br />
n n <br />
<br />
P 2 (x) <br />
sin(x) P 2 (x) x <br />
<br />
<br />
P 4 (x) sin(x) <br />
<br />
P 4 (x) x − x3<br />
3! <br />
f (k) (0) <br />
<br />
P 5 (x) x − x3<br />
3! + x(5)<br />
5! <br />
P 7 (x) x − x3<br />
3! + x(5)<br />
− x(7)<br />
5! 7! <br />
P 9 (x) x − x3<br />
3! + x(5)<br />
− x(7)<br />
+ x(9)<br />
5! 7! 9!
3<br />
2<br />
y<br />
3<br />
2<br />
y<br />
1<br />
x<br />
-4 -2 2 4<br />
-1<br />
-2<br />
1<br />
x<br />
-4 -2 2 4<br />
-1<br />
-2<br />
-3<br />
-3<br />
3<br />
2<br />
y<br />
3<br />
2<br />
y<br />
1<br />
x<br />
-4 -2 2 4<br />
-1<br />
-2<br />
-3<br />
1<br />
x<br />
-4 -2 2 4<br />
-1<br />
-2<br />
-3<br />
x 0 f (x) sin(x)<br />
P 1 (x) x <br />
<br />
<br />
{P n (x)} <br />
f <br />
<br />
f <br />
f (x) <br />
∞∑<br />
k0<br />
f (k) (a)<br />
(x − a) k <br />
k!
f x a <br />
f x a T f (x) <br />
T f (x) <br />
∞∑<br />
k0<br />
f (k) (a)<br />
(x − a) k <br />
k!<br />
a 0 <br />
f n <br />
e x e x <br />
∞∑<br />
k0<br />
x k<br />
k! <br />
<br />
<br />
<br />
f (x) 1<br />
1−x<br />
f (x) cos(x) <br />
<br />
f (x) sin(x) <br />
<br />
n f (x) 1<br />
1−x<br />
x 0<br />
x <br />
<br />
<br />
<br />
e x e x <br />
x<br />
<br />
cos(x) <br />
cos(x) x cos(x)<br />
1<br />
<br />
1−x <br />
x <br />
1<br />
1−x e x
cos(x) <br />
1<br />
1−x <br />
e x sin(x) cos(x) 1<br />
<br />
1−x<br />
<br />
<br />
<br />
e x cos(x) sin(x) <br />
1<br />
1<br />
x <br />
1−x<br />
<br />
1−x <br />
1<br />
1<br />
x <br />
1−x<br />
<br />
1−x<br />
(−1, 1) x <br />
f (x) x f <br />
x<br />
f <br />
f (x) x <br />
e x <br />
x x <br />
<br />
∞∑ x k<br />
<br />
k!<br />
k0<br />
<br />
<br />
<br />
x <br />
<br />
∞∑<br />
x k<br />
∞∑<br />
k!<br />
|x| k<br />
k! <br />
<br />
k0<br />
a k+1<br />
lim<br />
k→∞ a k<br />
lim<br />
k0<br />
|x| k+1<br />
(k+1)!<br />
k→∞ |x| k<br />
k<br />
|x| k+1 k!<br />
lim<br />
k→∞ |x| k (k + 1)!<br />
|x|<br />
lim<br />
k→∞ k + 1<br />
0<br />
x x <br />
x
e x x <br />
e x x <br />
<br />
<br />
x <br />
<br />
∞∑<br />
c k (x − a) k <br />
k0<br />
x a a k |c k (x − a) k | <br />
a<br />
lim k+1<br />
k→∞ a k<br />
< 1<br />
<br />
a k+1<br />
|x − a| |c k+1|<br />
<br />
a k |c k |<br />
<br />
a k+1<br />
lim lim |x − a| c k+1<br />
<br />
k→∞ a k k→∞ c k<br />
c k f (k) (a)<br />
k!<br />
<br />
c k+1<br />
lim<br />
k→∞ c k<br />
L<br />
<br />
a k+1<br />
lim<br />
k→∞ a k<br />
|x − a| · L<br />
L L <br />
L x <br />
<br />
L 0 (−∞, ∞)<br />
L x a<br />
L x <br />
<br />
|x − a| · L < 1<br />
x <br />
|x − a| < 1 L <br />
(<br />
a − 1 L , a + 1 )<br />
<br />
L
|x − a| · L 1 a ± 1 L <br />
<br />
x <br />
x a <br />
<br />
<br />
<br />
<br />
<br />
f (x) 1<br />
1−x<br />
x 0<br />
<br />
f (x) cos(x) x 0<br />
<br />
f (x) sin(x) x 0<br />
x <br />
f <br />
f (x) <br />
f <br />
<br />
<br />
sin(x) <br />
<br />
n P n (x) <br />
f (x) <br />
f <br />
<br />
P n (x) f P n (x)<br />
f (x) <br />
P n (x) f (x) <br />
<br />
a E n (x) f (x) <br />
P n (x) <br />
E n (x) f (x) − P n (x)<br />
|E n (x)| P n f <br />
P (k)<br />
n (0) f (k) (0)
0 ≤ k ≤ n <br />
E (k)<br />
n (0) 0<br />
0 ≤ k ≤ n P n (x) n<br />
<br />
E (n+1)<br />
n<br />
x<br />
P (n+1)<br />
n (x) 0<br />
(x) f (n+1) (x) − P (n+1)<br />
n (x) <br />
E (n+1)<br />
n (x) f (n+1) (x)<br />
f (x) c P n (c) <br />
| f (n+1) (t)| M [0, c] <br />
f (n+1) (t) <br />
≤ M<br />
0 ≤ t ≤ c <br />
<br />
E(n+1) n<br />
t c <br />
(t) <br />
<br />
f (n+1) (t) <br />
≤ M<br />
− M ≤ E (n+1)<br />
n (t) ≤ M <br />
[0, c] t 0 t x <br />
∫ x ∫ x<br />
∫ x<br />
−M dt≤ E (n+1)<br />
n (t) dt ≤ Mdt<br />
0<br />
x [0, c] E (n)<br />
n (0) 0 <br />
x [0, c]<br />
0<br />
−Mx ≤ E (n)<br />
n (x) ≤ Mx<br />
<br />
<br />
x [0, c]<br />
∫ x<br />
n <br />
0<br />
−Mt dt ≤<br />
−M x2<br />
∫ x<br />
0<br />
E (n)<br />
n (t) dt ≤<br />
0<br />
∫ x<br />
2 ≤ E(n−1) n (x) ≤ M x2<br />
2<br />
−M<br />
xn+1<br />
(n + 1)! ≤ E n(x) ≤ M xn+1<br />
(n + 1)!<br />
0<br />
Mt dt
x [0, c] <br />
|E n (x)| ≤ M |x|n+1<br />
(n + 1)!<br />
x [0, c] E n <br />
a 0 <br />
a <br />
P n (x) f n + 1 <br />
M f (n+1) (x) ≤ M <br />
[a, c] P n (x) n f (x) x a <br />
<br />
Pn (c) − f (c) |c − a|n+1<br />
≤ M<br />
(n + 1)! <br />
<br />
<br />
P 10 (x) sin(x) <br />
sin(2)<br />
f (x) sin(x) c 2 a 0 n 10 <br />
<br />
M f (x) sin(x) ± sin(x) ± cos(x) <br />
f (n+1) (x) <br />
≤ 1<br />
n x M <br />
<br />
P10 (2) − f (2) |2 − 0|11<br />
≤ (1)<br />
(11)!<br />
211<br />
(11)! ≈ 0.00005130671797<br />
P 10 (2) sin(2) <br />
<br />
P 10 (2) ≈ 0.9093474427 sin(2) ≈ 0.9092974268<br />
<br />
P n (x) n sin(x) x 0<br />
n P n (2) sin(2) <br />
<br />
sin(x) sin(x) x<br />
<br />
f (x) sin(x) <br />
f (n+1) (x) <br />
≤ 1
n x M 1 <br />
|P n (x) − sin(x)| ≤ |x|n+1<br />
(n + 1)!<br />
<br />
x<br />
∑ ∞ x k<br />
k0 k!<br />
x<br />
<br />
lim<br />
n→∞<br />
x n+1<br />
(n + 1)! 0<br />
x n →∞ <br />
<br />
x<br />
lim |P n(x) − sin(x)| 0<br />
n→∞<br />
sin(x) <br />
∞∑<br />
n0<br />
(−1) n x 2n+1<br />
(2n + 1)!<br />
<br />
cos(x) cos(x) <br />
x<br />
e x <br />
e x e x <br />
x<br />
e x e x <br />
x<br />
e x e x <br />
x <br />
e x x <br />
e x e x x<br />
P n (x) n e x <br />
n P n (5) e 5
n <br />
x a f <br />
P n (x) f (a) + f ′ (a)(x − a) + f ′′ (a)<br />
(x − a) 2 + ···+ f (n) (a)<br />
(x − a) n<br />
2!<br />
n!<br />
n∑ f (k) (a)<br />
(x − a) k <br />
k!<br />
k0<br />
x a f <br />
∞∑<br />
k0<br />
f (k) (a)<br />
(x − a) k <br />
k!<br />
n a f n <br />
a n f <br />
f x <br />
f <br />
f (x) <br />
∞∑<br />
k0<br />
f (k) (a)<br />
(x − a) k <br />
k!<br />
<br />
P n (x) n <br />
f x a M f (n+1) (x) <br />
[a, c] <br />
<br />
Pn (c) − f (c) |c − a|n+1<br />
≤ M<br />
(n + 1)! <br />
<br />
n cos(3x) x <br />
n 2 P 2 (x) <br />
n 4 P 4 (x) <br />
n 6 P 6 (x) <br />
(7) −1, ′ (7) −1<br />
′′ (7) 1 ′′′ (7) −4<br />
7
P 2 (x) <br />
7<br />
P 3 (x) <br />
(7.1)<br />
P 2 (7.1) ≈<br />
P 3 (7.1) ≈<br />
5 a −2<br />
x<br />
<br />
5<br />
x <br />
<br />
cos(x) a <br />
−π/4 <br />
cos(x) <br />
<br />
f (x) ln(x) <br />
a 3 <br />
ln(x) <br />
<br />
<br />
a 0 f (x) x 3 − 2x 2 + 3x − 1<br />
<br />
<br />
a 0 f (x) x 3 − 2x 2 + 3x − 1 <br />
<br />
f (x) m n <br />
a 0 n<br />
<br />
a <br />
<br />
<br />
f (x) sin(x) f <br />
x π 2 f (x) x π 2<br />
<br />
<br />
f (x) ln(x) f <br />
x 1 f (x) x 1
ln(2)<br />
<br />
<br />
<br />
sin(x 2 ) <br />
sin(x 2 ) a 0<br />
<br />
<br />
f (x) sin(x) <br />
∞∑<br />
(−1) k x 2k+1<br />
(2k + 1)! <br />
k0<br />
x 2 x <br />
<br />
sin(x 2 ) <br />
sin(x 2 ) <br />
<br />
<br />
f f (x) <br />
⎧⎪<br />
f (x) ⎨<br />
e −1/x2 x 0,<br />
⎪<br />
⎩0 x 0.<br />
f ′ (0) 0<br />
f (n) (0) 0 n ≥ 2 <br />
f <br />
f <br />
x <br />
f (x)
x<br />
e x 1 + x + x2<br />
2! + x3 xn<br />
+ ···+<br />
3! n! + ···<br />
<br />
e x <br />
1 + 2x + 3x 2 + 4x 3 + ···<br />
x <br />
f <br />
<br />
f (x) a 0 + a 1 x + a 2 x 2 + ···+ a k x k + ···<br />
a 0 a 1 ... <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y ′ ky<br />
<br />
k <br />
<br />
<br />
k 1
y f (x) f (x) <br />
<br />
y f (x) <br />
∞∑<br />
a k x k <br />
a k <br />
k0<br />
<br />
f (x) x <br />
<br />
∞∑<br />
ka k x k−1 <br />
k1<br />
∞∑<br />
a k x k<br />
k0<br />
f (x) ∑ ∞<br />
k0 a kx k <br />
<br />
a 1 a 0 <br />
a 2 a 1 a 2 a 0 <br />
a 3 a 2 a 3 a 0 <br />
a 4 a 3 a 4 a 0 <br />
a k <br />
a 0 <br />
y a 0 <br />
<br />
<br />
<br />
<br />
<br />
x
x a <br />
∞∑<br />
c k (x − a) k<br />
k0<br />
<br />
{c k } x <br />
x <br />
f x <br />
<br />
∞∑<br />
f (x) c k (x − a) k <br />
k0<br />
<br />
<br />
<br />
<br />
∞∑ x<br />
f (x) k<br />
<br />
2 k<br />
f (1) f 3<br />
2 f (x) <br />
<br />
k0<br />
<br />
f x 1 <br />
∞∑ 1<br />
2 k<br />
k0<br />
1 2 <br />
f (1) 1<br />
1 − 1 2<br />
2<br />
<br />
f (3/2) <br />
∞∑ ( ) 3 k<br />
1<br />
4 1 − 3 4<br />
k0<br />
4<br />
f (x) x 2 <br />
f (x) <br />
∞∑ ( ) x k<br />
1<br />
2 1 − x 2<br />
k0<br />
2<br />
2 − x
−1 < x 2<br />
< 1 <br />
f −2 < x < 2<br />
<br />
x <br />
x <br />
<br />
<br />
f (x) ∑ ∞<br />
<br />
k1<br />
x k<br />
k 2 <br />
<br />
<br />
n∑ x<br />
S n (x) k<br />
<br />
k 2<br />
n ≥ 1 S 10 (x) S 25 (x) S 50 (x) <br />
<br />
k1<br />
10<br />
y<br />
5<br />
x<br />
-2 -1 1 2<br />
-5<br />
-10<br />
∑ ∞ x k<br />
k1 k 2<br />
S 50 <br />
(−1, 1) <br />
−1 < x < 1 <br />
x <br />
<br />
<br />
|a k+1 |<br />
lim<br />
k→∞ |a k |<br />
a k xk<br />
k 2 <br />
|x| k+1<br />
(k+1) 2<br />
k→∞ |x| k<br />
k 2<br />
lim
(<br />
k<br />
lim |x|<br />
k→∞ k + 1<br />
|x| lim<br />
|x|<br />
k→∞<br />
(<br />
k<br />
k + 1<br />
f (x) <br />
|x| < 1 |x| > 1 |x| 1 <br />
x 1 x −1 <br />
x 1 <br />
f (1) <br />
p p > 1 x −1<br />
∞∑<br />
k1<br />
1<br />
k 2 <br />
) 2<br />
) 2<br />
f (−1) <br />
∞∑ (−1) k<br />
<br />
k1<br />
k 2<br />
{ 1<br />
n 2 }<br />
<br />
x −1 <br />
−1 ≤ x ≤ 1<br />
<br />
<br />
<br />
<br />
∑ ∞<br />
k1<br />
(x−1) k<br />
3k<br />
∑ ∞<br />
k1 kxk<br />
∑ ∞ k 2 (x+1) k<br />
k1<br />
4 k ∑ ∞<br />
k1<br />
x k<br />
(2k)!<br />
∑ ∞<br />
k1 k!xk<br />
<br />
sin(x)<br />
e x
f (x) 1<br />
1+x 2<br />
x 0<br />
f (x) 1 <br />
1+x 2<br />
f (x) <br />
f (x) <br />
(x) 1<br />
<br />
1−x<br />
<br />
(−x 2 ) f (x) −x 2 x <br />
(x) (−x 2 ) f (x)<br />
f (x)<br />
x f (x) <br />
<br />
<br />
<br />
sin(x) <br />
cos(x) <br />
∞∑<br />
k0<br />
(−1) k x 2k+1<br />
(2k + 1)!<br />
∞∑ (−1) k x 2k<br />
k0<br />
(2k)!<br />
−∞< x < ∞<br />
−∞< x < ∞<br />
∞∑<br />
e x x k<br />
<br />
k!<br />
k0<br />
−∞< x < ∞<br />
1<br />
∞∑<br />
1 − x x k − 1 < x < 1<br />
k0<br />
<br />
sin(x 2 ) e 5x3 cos(x 5 ) <br />
<br />
f <br />
f (x) <br />
∞∑<br />
(−1) k x2k<br />
(2k)! <br />
k0
f (x) 1 − x2<br />
2! + x4<br />
4! − x6<br />
x2k<br />
+ ···+ (−1)k<br />
6! (2k)! + ···<br />
f ′ (x)<br />
f (x) f ′ (x) <br />
<br />
f ′′ (x) f ′′ (x)<br />
<br />
f<br />
f ′ <br />
f ∫ f (x) dx <br />
<br />
<br />
<br />
<br />
f (x) <br />
f (x) <br />
∞∑<br />
c k x k<br />
k0<br />
f (x) −r < x < r <br />
∑ ∞<br />
k1 kc kx k−1 f (x) <br />
f ′ (x) −r < x < r <br />
f ′ (x) <br />
∞∑<br />
kc k x k−1 , |x| < r<br />
k1<br />
∑ ∞<br />
k0 c k xk+1<br />
k+1<br />
f (x) <br />
−r < x < r <br />
∫<br />
∞∑ x<br />
f (x) k+1<br />
dx c k + C, |x| < r<br />
k + 1<br />
k0
x 0 arctan(x) <br />
<br />
arctan(x) <br />
x 0 arctan(x) <br />
<br />
<br />
1<br />
∞∑<br />
1 + x 2 (−1) k x 2k<br />
−1 < x < 1 <br />
<br />
∫<br />
k0<br />
d<br />
dx [arctan(x)] 1<br />
1 + x 2 <br />
1<br />
dx arctan(x) + C<br />
1 + x2 1<br />
<br />
1+x 2<br />
arctan(x) <br />
∫ ∞∑<br />
arctan(x) (−1) k x 2k dx<br />
k0 <br />
∞∑ (∫<br />
)<br />
(−1) k x 2k dx<br />
k0<br />
∞∑<br />
(−1) k x2k+1 + C<br />
2k + 1<br />
k0<br />
<br />
<br />
−1 < x < 1<br />
C arctan(0) 0 <br />
0 arctan(0) ∞∑<br />
(−1) k 02k+1 + C C<br />
2k + 1<br />
k0<br />
<br />
C 0 <br />
−1 < x < 1<br />
arctan(x) <br />
∞∑<br />
(−1) k x2k+1<br />
2k + 1<br />
<br />
∞∑<br />
(−1) k x2k+1<br />
2k + 1<br />
k0<br />
k0
x −1 x 1 <br />
1<br />
<br />
2k+1<br />
<br />
arctan(x) −1 ≤ x ≤ 1<br />
ln(1 + x) x 0 <br />
<br />
<br />
<br />
∞∑<br />
a k x k <br />
k0<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
4<br />
∞∑<br />
<br />
(1 − 10x) f (x) c n x n<br />
c 0 <br />
c 1 <br />
c 2 <br />
c 3 <br />
c 4 <br />
R <br />
n0<br />
<br />
f (x) 4x arctan(6x) <br />
∞∑<br />
f (x) c n x n .<br />
n0<br />
<br />
c 0
c 1 <br />
c 2 <br />
c 3 <br />
c 4 <br />
R <br />
R <br />
<br />
<br />
∫ 1<br />
0 sin(x2 ) ds<br />
sin(x) sin(x 2 ) <br />
sin(x 2 ) <br />
sin(x 2 ) <br />
∫ sin(x 2 ) dx<br />
∫ 1<br />
0 sin(x2 ) dx <br />
∫ 1<br />
0 sin(x2 ) dx <br />
f <br />
<br />
f (x) <br />
∞∑<br />
a k x k <br />
k0<br />
f a k <br />
f (n) (0) n!a n n<br />
<br />
<br />
<br />
{ f n } <br />
f 0 0, f 1 1, f 2 1, f 3 2, f 4 3, f 5 5, f 6 8, f 7 13, ···<br />
<br />
f 0 0 f 1 1 n ≥ 2 <br />
<br />
f n f n−1 + f n−2 <br />
F(x) <br />
∞∑<br />
f k x k <br />
<br />
k0
xF(x) ∑ ∞<br />
k1 f k−1x k<br />
x 2 F(x) ∑ ∞<br />
k2 f k−2x k<br />
<br />
F(x) − xF(x) − x 2 F(x) x<br />
<br />
F(x) − xF(x) − x 2 F(x) x<br />
F(x) <br />
<br />
<br />
x<br />
1−x−x 2<br />
<br />
y ′′ − xy 0<br />
<br />
x y <br />
x <br />
<br />
y f (x) <br />
y <br />
∞∑<br />
a k x k <br />
k0<br />
<br />
y y ′ <br />
y ′′ <br />
<br />
<br />
∞∑<br />
∞∑<br />
(k − 1)ka k x k−2 − a k x k+1 0<br />
k2<br />
k0<br />
<br />
<br />
x k−2 x k+1 <br />
<br />
∞∑<br />
∞∑<br />
(k − 1)ka k x k−2 (k + 1)(k + 2)a k+2 x k <br />
<br />
k2<br />
k0<br />
∞∑<br />
a k x k+1 <br />
k0<br />
∞∑<br />
a k−1 x k <br />
k1<br />
<br />
y ′′ ± k 2 xy 0
∞∑<br />
∞∑<br />
(n + 1)(n + 2)a n+2 x n − a n−1 x n 0<br />
n0<br />
n1<br />
<br />
x <br />
∞∑<br />
2a 2 + [(k + 1)(k + 2)a k+2 − a k−1 ] x k 0<br />
k1<br />
<br />
<br />
a 3k+2 0 k<br />
1<br />
a 3k <br />
(2)(3)(5)(6)···(3k−1)(3k) a 0 k ≥ 1<br />
1<br />
a 3k+1 <br />
(3)(4)(6)(7)···(3k)(3k+1) a 1 k ≥ 1<br />
<br />
<br />
y a 0<br />
1 +<br />
<br />
∞∑<br />
k1<br />
+ a 1<br />
x +<br />
<br />
k1<br />
x 3k<br />
(2)(3)(5)(6) ···(3k − 1)(3k) <br />
∞∑ x 3k+1<br />
(3)(4)(6)(7) ···(3k)(3k + 1) <br />
<br />
a 0 a 1
∫ du<br />
1 a 2 +u 2 a arctan u a + C<br />
∫<br />
√<br />
du<br />
ln |u + √ u 2 ± a 2 | + C<br />
u 2 ±a 2<br />
∫ √ √<br />
u 2 ± a 2 du u 2 u 2 ± a 2 ± a2<br />
2 ln |u + √ u 2 ± a 2 | + C<br />
∫<br />
√<br />
u 2 du<br />
√ u<br />
u 2 ±a 2 2 u 2 ± a 2 ∓ a2<br />
2 ln |u + √ u 2 ± a 2 | + C<br />
∫ du<br />
u √ − 1 u 2 +a 2 a ln a+ √ u<br />
2 +a 2<br />
u<br />
+ C<br />
∫ du<br />
u √ 1 u 2 −a 2 a sec−1 u a + C<br />
∫<br />
√<br />
du<br />
arcsin u<br />
a 2 −u 2<br />
a + C<br />
∫ √ √<br />
a 2 − u 2 du u 2 a 2 − u 2 + a2<br />
2 arcsin u a + C<br />
∫ √<br />
u 2<br />
du − √ u<br />
a 2 −u 2 2 a 2 − u 2 + a2<br />
2 arcsin u a + C<br />
∫ du<br />
u √ − 1 a 2 −u 2 a ln a+ √ a<br />
2 −u 2<br />
u<br />
+ C<br />
∫<br />
√<br />
du<br />
u 2√ − a 2 −u 2<br />
+ C<br />
a 2 −u 2 a 2 u
u <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x a