Active Calculus 2.0, 2017a

s t
s(t) 64 − 16(t − 1) 2
y s(t) 0 ≤ t ≤ 3
0

t < 1 1 < t < 3 t 1
s(1) − s(0.5)
AV [0.5,1] .
1 − 0.5
AV [0.5,1]
AV [0.5,1]
s(t)
s(t)
t s
AV [0.5,1]
[0.5, 1]
t s(t)
t a t b AV [a,b]
AV [a,b]
s(b) − s(a)
.
b − a
AV [a,b] s t
s(t)
64 − 16(t − 1) 2
[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))
B (0.8, s(0.8))
s(t) 64 − 16(t − 1) 2
t
0.8
(0.8, s(0.8))
t
0.8
0.4 0.8 1.2
64
56
48
feet
A
B
s
sec
s(t) 64 −
16(t − 1) 2
t a
t a

s(t) 64−16(t −1) 2
[1.5, 2]
[0, 0.5]
t 1.5 t 2
t 1
t a
s [a, b]
b
b a + h h
[a, a + h]
s(a + h) − s(a)
AV [a,a+h] ,
h
h (a + h) − a h h
h
t a t a h < 0 AV [a,a+h]
[a + h, a]
t a
h
s(t) 16 − 16t 2 s t
[0.5, 0.5+
h] −0.5 < h < 0.5 h 0

[0.5, 0.75] [0.4, 0.5]
t 0.5
−0.5 < h < 0.5 h 0 h
0 ≤ t ≤ 1
AV [0.5,0.5+h]
s(0.5 + h) − s(0.5)
(0.5 + h) − 0.5 .
s(0.5 + h)
s s(t) 16 − 16t 2
s(0.5 + h) 16 − 16(0.5 + h) 2
16 − 16(0.25 + h + h 2 )
16 − 4 − 16h − 16h 2
12 − 16h − 16h 2 .
s(0.5 + h) − s(0.5)
AV [0.5,0.5+h]
(0.5 + h) − 0.5
(12 − 16h − 16h2 ) − (16 − 16(0.5) 2 )
0.5 + h − 0.5
12 − 16h − 16h2 − 12
h
−16h − 16h2
.
h
h h
h
h
AV [0.5,0.5+h] −16 − 16h.
h
[0.5, 0.75] h 0.25
−16−16(0.25) −20 [0.4, 0.5] h −0.1
−16 − 16(−0.1) −14.4
AV [0.5,0.5+h] h h −16h
t 0.5 −16

s(t) 64 − 16(t − 1) 2
[2, 2 + h] [1.5, 2]
t 2
[a, b]
(a, s(a)) (b, s(b)) y s(t)
s
(a,s(a))
m = s(b)−s(a)
b−a
(b,s(b))
t
t s
[a, b] AV [a,b] s(b)−s(a)
b−a
[a, b]
[a, a + h]
AV [a,a+h] s(a+h)−s(a)
h
s (a, s(a))
(b, s(b)) m s(b)−s(a)
b−a
s
[a, b]
s
t
s
0 ≤ t ≤ 0.2
t 0.2

f (x) 36 − x 2
f (4)− f (0)
4−0
f (6)− f (4)
6−4
f (6)− f (0)
6−0
f (4)− f (0)
4−0
f (6)− f (4)
6−4
f (6)− f (0)
6−0

f (x) (x)
f (x) (x)
0 ≤ x ≤ 2.2
5.2 ≤ x ≤ 6.1
5.2 ≤ x ≤ 8
2.2 ≤ x ≤ 4
2.2 ≤ x ≤ 6.1
t 0 h
t
s(t) 100 cos(0.75t) · e −0.2t + 100
t 0 t 15

[0, 15]
[0, 2] [1, 6] [8, 10]
200
150
100
50
s
5 10 15 20
t
t 5
t 0
t 1.1 t 2.45
t 7
t 13
s(t)
t −3.5
s(t)
s
v
2 4 6 8 10 12
t
2 4 6 8 10 12
t
s(t)
v(t)

t 2.45
t 7 [2.45, 7]
v(t) t
s(t)
v(t)
v(t)
f f [a, b]
f (b) − f (a)
.
b − a
[a, b]
f (b)− f (a)
b−a
y f (x)
P(t) t t
P
t 0 t 20
P(t) 181843(1.04) t/10 .
[5, 10] [5, 9] [5, 8] [5, 7] [5, 6]

lim x→a f (x) L
x
y f (x) x 2
(−2, 4) (−1, 1) (0, 0) (1, 1) (2, 4)
s(t) 64 − 16t 2 [1, x]
AV [1,x]
s(x) − s(1)
x − 1
(64 − 16x2 ) − (64 − 16)
x − 1
16 − 16x2
.
x − 1
x AV [1,x] x
x 1
(x) 16−16x2
x−1
t 1 t x
t 1 (x) x
1 (1) 0/0
x
1 (x)

(−2) (−1)
(0) (1) (2)
a −1 a 0
a 2
x
a (x)
x
a 1
(x)
g
3
2
1
-2 -1 1 2 3
-1
y (x)
f L x a
x
(x)
(0) 1 4
L a

f x a L
f L x a
lim f (x) L
x→a
f (x) L x
a f (x) x
a f x a
lim (x) 3, lim (x) 4, lim (x) 1,
x→−1 x→0 x→2
x → 1
x 1 x 1
x 1
x 1
f f
x a
f (x)
a
a
a
f (x) 4−x2
x+2 a −1 a −2 (x) sin
π
x a 3 a 0
a −2
f a −1

x f(x)
−0.9 2.9
−0.99 2.99
−0.999 2.999
−0.9999 2.9999
−1.1 3.1
−1.01 3.01
−1.001 3.001
−1.0001 3.0001
x f(x)
−1.9 3.9
−1.99 3.99
−1.999 3.999
−1.9999 3.9999
−2.1 4.1
−2.01 4.01
−2.001 4.001
−2.0001 4.0001
f
5
3
1
-3 -1 1
f x −1
f x −2
f (x) [−4, 2]
f x
−1 lim x→−1 f (x) 3
f
f f (x) 4−x2
x+2
x → −1 (4 − x 2 ) → (4 − (−1) 2 ) 3
(x + 2) → (−1 + 2) 1 x →−1 f
lim x→−1 f (x) 3 1 3
x →−2 f (−2)
x →−2 (4−x 2 ) → (4−(−2) 2 ) 0 (x+2) → (−2+2) 0
x →−2 f 0/0
f 4 x −2
f (x)
4 − x
lim f (x) 2
lim
x→−2 x→−2 x + 2
(2 − x)(2 + x)
lim
.
x→−2 x + 2
x →−2
x −2 x
−2 2+x
x+2
x

lim f (x) lim 2 − x.
x→−2 x→−2
2 − x
4 lim x→−2 f (x) 4.
x (x)
2.9 0.84864
2.99 0.86428
2.999 0.86585
2.9999 0.86601
3.1 0.88351
3.01 0.86777
3.001 0.86620
3.0001 0.86604
x (x)
−0.1 0
−0.01 0
−0.001 0
−0.0001 0
0.1 0
0.01 0
0.001 0
0.0001 0
2
g
-3 -1 1 3
-2
x 3
x 0
(x) [−4, 4]
x → 3
0.866025 π x → π 3
(x) sin( π x ) → sin( π 3 ) x → 3 sin( π √
3 ) 3
2
√
3
lim (x)
x→3 2 .
x → 0 π x
x
π x
x π x
x 0
(x) sin
π
x x 0
x 0
x 0 0
{0.1, 0.01, 0.001,...}
(10 −k )(10 −k ) ( )
π
sin
10 sin(10 k π) 0
−k
k {0.3, 0.03, 0.003,...}
(
(3 · 10 −k ) sin
)
π
3 · 10 −k
( ) √
10 k π 3
sin
3
2 ≈ 0.866025.

2
x → 0
√
3
lim x→1
x 2 −1
x−1
lim x→0
(2+x) 3 −8
x
lim x→0
√
x+1−1
x
t s
[a, b] AV [a,b] s(b)−s(a)
b−a
.
a b a
b → a
a
IV ta
t a
s(b) − s(a)
IV ta lim AV [a,b] lim .
b→a b→a b − a
b b a + h h
s(a + h) − s(a)
IV ta lim AV [a,a+h] lim
.
h→0 h→0 h

s(t) t 2
s t
[3, 3 + h] h > 0
[3, 3.2]
t 3
s t
s t
[0.5, 1]
[1.5, 2.5] [0, 5]
t 2
t 2
[1.5, 2.5]
5
3
1
1 3 5
s
t
y s(t)

lim x→a f (x) L f x
a L f (x) L
x a
lim x→a f (x) a f
f
x a
f x → a
f (x)
lim f (x)
x→−2
lim f (x)
x→0
lim
x→2
f (x)
lim
x→4
f (x)

lim
θ→0
sin(5θ)
.
θ
θ
sin(5θ)
lim
θ→0
θ
x ⎧⎪
2 − 4, 0 ≤ x < 1
f (x) ⎨2, x 1
⎪
⎩2x − 5, 1 < x
lim
x→1 +
lim
x→1 −
f (x)
f (x)
lim f (x)
x→1
f (x)
x 2 − 36
lim
x→−6 x + 6
f (x) 16−x4
x 2 −4
f
x a 2 lim x→2 f (x),
16−x4
x 2 −4
lim x→2 f (x)
f (2) −8
16−x4
x 2 −4
−4− x2 .
f
y f (x)
[1, 3]
16−x
lim 4
x→2 x 2 −4

(x) − |x+3|
x+3
a −3 lim x→−3 (x),
|x+3|
x+3 lim x→−3 (x)
|a| a a ≥ 0
|a| −a a < 0
(−3) −1
− |x+3|
x+3
−1.
y (x)
[−4, −2]
lim x→−3 (x)
3
3
-3 3
-3 3
-3
-3
y f (x) y (x)
y f (x)
f (−2) 2 lim x→−2 f (x) 1
f (−1) 3 lim x→−1 f (x) 3
f (1) lim x→1 f (x) 0

f (2) 1 lim x→2 f (x)
y (x)
(−2) 3 (−1) −1 (1) −2 (2) 3
x −2, −1, 1 2
(0) lim x→0 (x)
t 0 s t
s(t) 100 cos(0.75t) · e −0.2t + 100
[1, 1+
h]
h → 0

s t a
t a + h
AV [a,a+h]
s(a + h) − s(a)
.
h
y f (x).
f f [a, a + h]
f (a + h) − f (a)
AV [a,a+h] .
h

f [a, b]
AV [a,b]
f (b) − f (a)
.
b − a
f
f
a a + h x
(a, f (a))
(a + h, f (a + h))
(a, f (a)) (a + h, f (a + h))
y
f
(a, f (a))
(a + h, f (a + h))
a
a + h
x
y f (x)
f
f a f a
f ′ (a)
f x a

f x x a f ′ (a)
f ′ f (a + h) − f (a)
(a) lim
,
h→0 h
f ′ (a) f a f
x a
f x a
f [a, a + h] h → 0
x a x a
y s(t) s ′ (a)
t a
f (a+h)− f (a)
h
f x
s t
s ′ (a)
f (a+h)− f (a)
h
(a, f (a))
(a + h, f (a + h))
(a, f (a)) (a + h, f (a + h)) (a, f (a)) h → 0

(a, f (a)) (a + h, f (a + h)) h
y f (x)
f (a+h)− f (a)
m
h
h
h → 0
(a, f (a))
y f (x) (a, f (a)) (a, f (a))
m f ′ (a)
y
y
y
y
f
f
f
f
x
x
x
x
a
a
a
a
f (a, f (a))
x a
f (a, f (a))
(a, f (a))
y f (x) (a, f (a))
x a
y
f
a
x
f (a, f (a))
(a, f (a)) f
(a, f (a)) f ′ (a)

f x x a f ′ (a)
y f (x) (a, f (a))
f (x)
x − x 2 f ′ (2)
f ′ f (2 + h) − f (2)
(2) lim
.
h→0 h
f f (2) 2−2 2 −2 f (2+h) (2+h)−(2+h) 2 .
f ′ (2 + h) − (2 + h) 2 − (−2)
(2) lim
.
h→0 h
h h → 0
h
f ′ 2 + h − 4 − 4h − h
(2) 2 + 2
lim
.
h→0 h
f ′ −3h − h
(2) 2
lim .
h→0 h
h
f ′ (2) lim
h→0
(−3 − h).
h → 0 f ′ (2) −3

f ′ (2)
y x − x 2 (2, −2)
f ′ (2) f
(2, −2)
(2, −2) m f ′ (2) −3
-2
m = f ′ (2)
1 2
-4
y = x − x 2
y x − x 2 (2, −2)
f f (x) 3 − 2x
f f
x
f [1, 4] [3, 7] [5, 5+ h]
f x a 1 f ′ (1)
f ′ (2) f ′ (π)
f ′ (− √ 2)
t s(t) −16t 2 +
16t + 32

s
s [1, 2]
s
t a 1
s
[1, 2]
s a 1
32
16
y
a
s ′ (a)
1 2
t
y s(t)
P t t
P(t) 25000e t/5
P t 0 t 5
P
P t a 2

P a 2
h
P ′ (2)
P
[2, 4]
P a 2
y
P ′ (a) a
P
t
y P(t)
f (b)− f (a)
f [a, b]
b−a
f x
(a, f (a)) (b, f (b)) y f (x) [a, a+h]
[a, b]
f (a+h)− f (a)
h
x f x a
f ′ (a) f a f a
f ′ f (a + h) − f (a)
(a) lim
,
h→0 h
x a [a, a + h] h → 0
f ′ (a)

f x x a
y f (x) (a, f (a)) f ′ (a)
(a, f (a))
y f (x)
x
x
x
x
x
x 1.75
x 1.75 x
(x) B (2.6, 8)
A (2.55, 8.06)
B

( )
′ (
)
f (x)
f (9) f (12)
f (9) − f (6) f (6) − f (3)
f (6)− f (3)
6−3
f ′ (3) f ′ (12)
f (9)− f (3)
9−3
(t) t 2 + 6t t 3
′ (3)
f ′ (3) f (x) 3 x
f ′ (3) ≈

y f (x)
y f (x)
y f (x)
[−3, −1]
y f (x)
[0, 2]
y f (x)
x −3
y f (x) x 0
f
[−3, −1] [0, 2]
f
x −3 x 0
4
-4 4
-4
y
f
x
y f (x)
3
3
-3 3
-3 3
-3
-3
y f (x) y (x)

y f (x)
f [−3, 0] −2 f
[1, 3]
f x −1 −1
f x 2
y (x)
(3)−(−2)
5
0 (1)−(−1)
2
−1
′ (2) 1 ′ (−1) 0
P
P(t) 1.15(1.014) t t
y P(t)
t
f ′ (a)
h y f (x) (a, f (a))
f ′ (a)
f (x) x 2 − 3x a 2
f (x) 1 x a 1
f (x) √ x a 1
f (x) 2 − |x − 1| a 1
f (x) sin(x) a π 2

f
f ′
f ′ (a) f ′ (x)
f ′ (x) f (x)
f f ′
y f (x)
x a y f (x)
x a f ′ (a)
a a 1 a 3
a
f ′ (1) f ′ (3) f ′ (a) f ′ (x)
y f (x)
f (x) 4x − x 2
f ′ (0) f ′ (1) f ′ (2)
f ′ (3)
f ′ (a) a
f ′ (4) f ′ (5)
f ′ (a) a
f (x) f (x) 4x − x 2
f ′ (a) a
f (x) 4x − x 2
f ′ (0) 4 f ′ (1) 2 f ′ (2) 0 f ′ (3) −2
f ′ (4) −4 f ′ (5) −6.
0 1 2 3

f ′ (a) a
f ′ f (a + h) − f (a)
(a) lim
h→0 h
4a + 4h − a 2 − 2ha − h 2 − 4a + a 2
lim
h→0 h
lim
h→0
h(4 − 2a − h)
h
lim
h→0
4(a + h) − (a + h) 2 − (4a − a 2 )
h
lim
h→0
(4 − 2a − h).
lim
h→0
4h − 2ha − h 2
h
4 2a hh → 0 (4 − 2a − h) →
(4 − 2a) f ′ (a) 4 − 2a
f ′ (3) 4−2(3)
−2 a a
a x
x
y f (x)
f (x) 4x − x 2 f ′ (x) 4 − 2x.
4
3
2
m = 0
m = 2
m = −2
4
3
2
(0,4)
(1,2)
1
m = 4
m = −4
1
(2,0)
-1
-2
-3
1 2 3 4
y = f (x)
-1
-2
-3
1 2 3 4
(3,−2)
y = f ′ (x)
-4
-4
(4,−4)
f (x) 4x − x 2 f ′ (x) 4 − 2x
f f ′

f (x) 4x−x 2
f ′ (x) 4−2x
x
x
x
f y f ′ (x)
f x
y f (x)
f (x)
f ′ (x)
a f ′ (a) a
x
f x
f x x f ′ (x) f ′ (x)
f (x+h)− f (x)
lim h→0 h
,
y f (x)
y f ′ (x) y f (x)
y f ′ (x)
y f (x)
y f ′ (x)
f 1 × 1 f ′
f
f ′

f
g
x
x
f ′
g ′
x
x
p
q
x
x
p ′
q ′
x
x

s
x
x
r ′
s ′
x
x
w
z
x
x
w ′
z ′
x
x

f ′
f
y f (x) 4x −
x 2 f ′ (a) 4 − 2a
f ′ (x) 4−2x f f ′
f ′ (x) f f ′
f ′ (x)
y f (x)
f ′ (x)
f x p(z) p ′ (z)
q(s) s 3
f (x) 1
p(z) z 2 G(y) √ y
(t) t
F(t) 1 t
f ′ f (x+h)− f (x)
(x) lim h→0 h
x
y f ′ (x) f f ′
f f ′
f ′ (a)
f ′ (x)
y f (x)

y f ′ (x)
f
f (x)
x
x
(x) 3x 2 −4
′ (x) lim [(
)/h]
h→0
f (x)

f (x)
f
x
f (x) x
f (x) x
f ′ (x) x
f ′ (x) x
f (x) 1
x − 3

f ′ (2)
f ′ (4)
f ′ (5)
f ′ (7)
f f x
f f (−2) 1 f ′ (−2) −2 f ′ (−1) −1 f ′ (0) 0
f ′ (1) 1 f ′ (2) 2
y f (x)
y f ′ (x)
y f ′ (x)
y f (x)
y f ′ (x)
3
3
-3 3
-3 3
-3
-3
y f (x) y f ′ (x)
(x) x 2 − x + 3
′ (x)
y (x) y ′ (x)
′ (x)
p ′ (x) p(x) 5x 2 −
4x + 12.

′ (x) p ′ (x)
y ′ (x)
2
2
-2 2
-2 2
-2
-2
y (x) y ′ (x)
x 0 < x < 2 ′ (x)
y (x)
0 < x < 2 y (x)
x ′ (x)
y (x)
(0) 1
y (x)
y f (x)
y f ′ (x)
f 1×1
f ′ f
f ′

f
f
x
x
f ′
f ′
x
x
f
f
x
x
f ′
f ′
x
x
y f (x) y f ′ (x)

f ′
f
y s(t)
t v(t) s ′ (t)
s(t) t v(t) s ′ (t)
t 0
t
[57, 68]
[68, 104]
(57, 63.8) (104, 106.8)

[68, 104]
t 80
100
80
60
40
20
s
(104,106.8)
(57,63.8)
(68,63.8)
t
20 40 60 80 100
y s(t)
t
f x
f ′ f (x + h) − f (x)
(x) lim
,
h→0 h
x a f ′ (a)
(a, f (a))
df dy
dx
dx f ′ (x)
f x
f (x+h)− f (x)
h
f x
f ′ (x) f ′ (x) f
x f
y P(t) P
t t 0
P ′ (2) 21.37 P
t

f ′ f (x+h)− f (x)
(x) x
h
h h
y f (x) f (1)
2.5 f (2) 3.25 f (3) 3.625 f ′ (2)
f ′ f (2+h)− f (2)
(2) lim h→0 h
y f (x)
h
h −1 h 1
(2, 3.25)
f ′ (2) ≈
f (1) − f (2)
1 − 2
2.5 − 3.25
−1
0.75.
f ′ (2) ≈
f (3) − f (2)
3 − 2
3.625 − 3.25
1
0.375.
(2, 3.25)
(2, 3.25)
f ′ (2) ≈
0.75 + 0.375
2
0.5625.
f ′ (2) f

3
3
2
2
1
1
1 2 3
1 2 3
y f (x) (1, 2.5)
(2, 3.25) (2, 3.25) (3, 3.625)
(1, 2.5) (3, 3.625)
y f (x)
f (1)− f (2)
1−2
0.75
f (3)− f (2)
3−2
0.375.
(2, f (2)) f ′ (2)
f (3) − f (1)
3 − 1
3.625 − 2.5
2
1.125
2
0.5625.
f ′ f (a + h) − f (a − h)
(a) ≈ ,
2h
y f (x) (a −
h, f (a − h)) (a + h, f (a + h))

F
t
t 0 15 30 45 60 75 90
F(t) 70 180.5 251 296 324.5 342.8 354.5
t 30
t 60
F ′ (75) F ′ (90)
F(64) 330.28 F ′ (64) 1.341
t 65 t 66
[0, 90]
r
C(r)
C(2000) 800
C ′ (r)
C(2000) 800 C ′ (2000) 0.35 C(2100)
C ′ (2000) C ′ (3000)
C ′ (5000) −0.1

C s
C f (s) s
f (80) 0.015 f (90) 0.02
f (100) 0.027
s 90
f (80) 0.015
f ′ (90)
f
f ′
f ′
P(t) 400 − 330e −0.03t
t
P P ′
400
◦ F
16
◦ F/min
300
y = P(t)
12
200
8
y = P ′ (t)
100
min
4
min
20 40 60 80
20 40 60 80
P(t) 400 − 330e −0.03t P ′ (t)
P ◦ P ′ ◦

y f (x)
y f ′ (x) f x
f
f ′ f (a + h) − f (a − h)
(a) ≈ ,
2h
y f (x)
(a − h, f (a − h)) (a + h, f (a + h))
f ′ (7) 2
x 7 f
f (8)
f (7)
x
H
H f (t) t
f ′ (t)
f ′ (35)
| f ′ (35)| 1.2 f (35) 52
C C f ()
f (300) 350

f ′ (300) 1.2
W W f (c)
c
f (1500) 150
f ′ (2000) 0
f −1 (157) 2300
f ′ (c) dW/dc
f ′
f ′(
)
s t 2 − 5t + 12,
t

t 4
F t
F(t) 75 + 110e −0.05t
h 0.01 F ′ (10)
F ′ (10)
F ′ (10)
F ′ (10) F ′ (20)
y F ′ (t)
0 ≤ t ≤ 30
T
q T f (q)
f (50) 0.75
s s(q) f ′ (q)
f ′ (50) −0.02
v(t) 16 − 32t
v t
t 0 t 2
v ′ (1)
v ′ (1)
t 1
v ′ (t)
V m
V h(m)
h(40000) 15500 h(55000) 13200
h [40000, 55000]

h(70000) 11100
h ′ (55000)
h ′ (30000) h ′ (80000)
V h(m)
h ′ (m)

f
y f (x) y f ′ (x)
x a y f (x)
(a, f (a))
f
f ′ (x)
f f ′ (x)
f
f ′ (x)
f
A
B
f
f
A
f ′ (x)
B

f B
y f ′ (x)
t
y s(t)
(2, 4)
14
10
y
s
6
2
2 6 10
t
y s(t)
t
[0, 1] [1, 2] [2, 3] [3, 4] [4, 5]
[0, 12]
y s ′ (t)

y s ′ (t)
s ′ (t) s ′ (t)
s ′ (t)
y v(t)
v v ... v
...
y v ′ (t)
v ′ (t)
y v(t)
y
y
2 6 10
t
2 6 10
t
y v(t) s ′ (t) y v ′ (t)
f (x) (a, b) f
(a, b) x y (a, b) x < y f (x) < f (y)
f (a, b) x y (a, b)
x < y f (x) > f (y)

f (a, b) f
(a, b) f ′ (x) > 0 x a < x < b f
(a, b) f ′ (x) < 0 f ′ (a) 0 f
x a
−2 < x < 0 x ±2
x 0 f
f ′ (x) 0
2
A
-2 2
y = f (x)
-2
B
−3 < x < −2 0 < x < 2
−2 < x < 0 2 < x <
3
f
f ′ f (x + h) − f (x)
(x) lim
.
h→0 h
f ′
y [ f ′ (x)] ′
y f (x) y f ′′ (x)
f ′ f
f

f ′′ f ′ (x + h) − f ′ (x)
(x) lim
.
h→0 h
y f ′′ (x)
y f ′ (x) y f ′ (x)
f
x

−100
−2
x < y x y
−100 −2 −100
−2
y x 2 y e x
y −x 2 y −e x

f
f ′
f
f ′
f (a, b) f
(a, b) f ′ (a, b) f (a, b)
f ′ (a, b)
t
y s(t)
y v(t) s ′ (t) y v ′ (t)

y s(t)
14
y
s
y v(t) s ′ (t)
10
a(t) a(t)
v(t) a(t)
s(t)
6
2
2 6 10
t
s ′′ s ′
y s(t)
t
s
v a
v s
v s
v s
a v
a v
a v
a s
a s
a s
s v a f f ′

f ′′ f ′
f f ′ f ′′
f ′ f f ′
f ′′
F
t F ′ (30)
F ′ (60)
t F(t)
0 70
15 180.5
30 251
45 296
60 324.5
75 342.8
90 354.5
t F ′ (t)
0
15 6.03
30 3.85
45 2.45
60 1.56
75 1.00
90
F(t)
F ′ (t)
F ′ (t)
F ′′ (30)
F ′′ (30)
F(30) F ′ (30) F ′′ (30)
f ′ f
f
f ′ f ′′
f f ′ f ′′

f ′
f f ′′
f 1 × 1 f ′
f
f ′ f ′′
f
f
x
x
f ′
f ′
x
x
f ′′
f ′′
x
x
f f ′ f ′′

f
f
f ′′
f
y x 2 y e x
y −x 2 y −e x
f (x)
f (4)
f ′ (4)
f ′′ (4)

f
f ′ f ′′
f f ′ f ′′
f
f ′
f ′′
t
v(t)
P(t) t
P(t)

P(t)
P(t)
P(t)
P(t)
f ′ f
f ′ f
x
f (x) x
f (x) x
f ′ (x) x
f ′ (x) x
f ′′ (x) x
f ′′ (x) x
y f (x)
f (2) −3 f ′ (2) 1.5 f ′′ (2) −0.25
f x 2 f x 2
f (2.1) −3 −3 −3
f ′ (2.1) 1.5 1.5 1.5
y f (x) (2, f (2))
y (x)

y (x)
(2, (2))
(x) 0
′
−3 < x < 3
′′ (2)
y = g ′ (x)
4
2
-3 -1 1 3
y ′ (x)
h t
t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
h(t) 200 184.2 159.9 131.9 104.7 81.8 65.5 56.8 55.5 60.4 69.8
t 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
h(t) 81.6 93.7 104.4 112.6 117.7 119.4 118.2 114.8 110.0 104.7
h ′ (4.5) h ′ (5) h ′ (5.5)
h ′′ (5)
h ′′ (5)
y h(t)
−3 <
x < 3
y f (x) f −3 < x < 3 −3 < x < 0
0 < x < 3
y (x) −3 < x < 3 −3 < x < 0
0 < x < 3
y h(x) h −3 < x < 3 −3 < x < −1
−1 < x < 1 1 < x < 3

y p(x) p −3 < x < 0
0 < x < 3

f L x → a
x a x a
f x a
f x a
x a
L x a
f (a) lim x→a f (x)
f ′ (a)
f −4 < x < 4
x 2 f
sin( π x
)
a −3, −2, −1, 0, 1, 2, 3 lim x→a f (x)
L
lim x→a f (x) L
a f
f (a) a f (a)
lim x→a f (x)
a −3, −2, −1, 0, 1, 2, 3 f ′ (a)

f (a, f (a)) a
f ′ (a)
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
y f (x)
f L x
a lim x→a f (x) L f (x)
L x a
f a 1
x f
x 1 f f
x f
a 1
f a 1
f L 1 x a
lim
x→a − f (x) L 1
f (x) L 1 x
a x < a L 1 f x
a L 2 f x a
lim
x→a + f (x) L 2

f (x) L 2 x
a x > a f
lim f (x) 2 lim f (x) 3
x→1− x→1 +
f x → 1
3
f
3
g
2
2
1
1
1
f a 1
a 1
a 1
x
a 1
x → 1 + lim x→1 (x) .
f L x → a
lim f (x) L lim f (x).
x→a− x→a +
x a
x a
f
a −2 −1
x 2 lim x→2 + f (x) a −1
a 0

3(x + 2) + 2 −3 < x < −2
2
3
(x + 2) + 1 −2 ≤ x < −1
⎧⎪
f (x) ⎨ 2(x + 2) + 1 −1 < x < 1
2 x 1
⎪
⎩4 − x x > 1
3
3
2
1
-2 -1 1 2
-1
y f (x)
a −2, −1, 0, 1, 2 f (a)
a −2, −1, 0, 1, 2 lim x→a − f (x) lim x→a + f (x)
a −2, −1, 0, 1, 2 lim x→a f (x)
a
a
lim f (x) f (a)
x→a
y f (x)
◦ •

a 1
3
f
3
g
3 h
2
2
2
1
1
1
f h a 1
f (1) f a 1
f a 1
lim x→1 (x) 3 (1) 2
a 1 h
a 1
lim h(x) 3 h(1).
x→1
h a 1 h
f x a
f x → a
f x a
lim x→a f (x) f (a).
x a x → a
x a [a, b]
[a, b]

p(x) x 2 − 2x + 3
lim x→2 p(x)
lim
x→2 (x2 − 2x + 3) 2 2 − 2 · 2 + 3 3.
f
a lim x→a f (x)
a f (a)
a f lim x→a f (x) f (a)
a f
x a
f
x a f
x a
f x a f
x a
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
y f (x)
f x a f ′ (a)
f ′ (a) y f (x)
(a, f (a)) f ′ (a) f
(a, f (a)) f x a f

x a f (a) f (a) lim x→a f (x)
f x a
x a f x a
f
f x 1
x 1
f
f a 1 lim x→1 f (x) 1 f (1).
f
1
(1,1)
1
f a 1 a 1
(1, 1)
f a 1 f ′ (1)
f ′ (x) −1 x
f ′ (x) +1 x +1
−1 x 1
f ′ (1)
f
(1, 1) f
x a
(a, f (a))

x a
f x a f x a f
x a f x a
f x a
(a, f (a))
f x a f x a
(x) |x|
f
x
x 0
′ (0)
|h|
lim h→0 h .
′ (0)
h
a f
x a
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
y f (x)
p x b lim x→b p(x)

f L x → a f x a
x a
x a
x a
f x a f (a) f x → a
f x a
f x a f ′ (a) f
(a, f (a)) f x a
(a, f (a))
(a, f (a))
x a
x a x a
f
x a f x a f x a
f x a
f (x)
lim f (x)
x→−2
lim f (x)
x→0
lim
x→2
f (x)
lim
x→3
f (x)

2x − 5, 0 ≤ x < 3
⎧⎪
f (x) ⎨4, x 3
⎪
⎩x 2 − 6x + 10, 3 < x
lim
x→3 +
lim
x→3 −
f (x)
f (x)
lim f (x)
x→3
f (x)
x x
x x
x x
k
k
⎧⎪
f (x) ⎨
kx 0 ≤ x < 2
⎪
⎩9x 2 2 ≤ x.
y p(x)
p

p
3
3
-3 3
-3 3
-3
-3
y p(x) y
p ′ (x)
a lim x→a p(x)
a p a
a p x a
y p ′ (x)
f a 2 a 2
a 3 a 3
h a −2 a −2
a −2
p
p(−1) 3 lim x→−1 p(x) 2
p(0) 1 p ′ (0) 0
lim x→1 p(x) p(1) p ′ (1)
h(x) y h ′ (x)
y h ′ (x)
y h(x)

x y h ′ (x)
y h(x)
y h(x) h
y h(x) y h ′ (x)
y h(x)
3
3
2
1
y = h ′ (x)
-3 3
-3 -2 -1 1 2 3
-1
-2
-3
-3
y h(x) y h ′ (x)
(x) √ |x|
x 0
′ (0) lim
h→0
√
|h|
h .
′ (0)
√
|−0.01|
h
0.01
h
′ (0)
y ′ (x)

y f (x) (a, f (a))
f (a, f (a))
y f (x) (a, f (a))
y f (x)
f L
f f
f x a f ′ (a)
y f (x) (a, f (a))
y (x) −x 2 + 3x + 2
y ′ (x)
y (x) x 2
(2)
y (x) (2, (2))
y (x)
(2, (2))

y (x) (2, (2))
m (x 0 , y 0 ) y − y 0 m(x − x 0 )
f x a
y f (x) (a, f (a)) f ′ (a)
(a, f (a)) m f ′ (a)
y − f (a) f ′ (a)(x − a),
y f ′ (a)(x − a) + f (a)
f (a) f (x)
a
f ′ (a) f ′ (x)
a
f
(a, f (a)) f

y
y = f (x)
y = f (x)
(a, f (a))
(a, f (a))
y = f ′ (a)(x − a)+ f (a)
y = L(x)
a
x
y f (x) (a, f (a))
y L(x)
x a f (x) ≈ L(x)
y f (x) (a, f (a)) f x a
y
y f ′ (a)(x − a) + f (a)
x y
L(x)
L(x) f ′ (a)(x − a) + f (a)
f (a, f (a))
L(x) x
a f (x) ≈ L(x)
y f (x)
L(x) 3 − 2(x − 1) (1, 3)
f f (x) x f (1.2)
f (1.2) ≈ L(1.2)
f (1.2) ≈ L(1.2) 3 − 2(1.2 − 1) 3 − 2(0.2) 2.6.
y L(x)

L(x) f (a) + f ′ (a)(x − a) L(x) ≈ f (x) x a
f (x) ≈ f (a) + f ′ (a)(x − a) x a.
y L(x) f a
y (x)
a −1 L(x) −2 + 3(x + 1)
L(−1) L ′ (−1)
(−1) ′ (−1)
(−1.03)
(−1)
(−1.03)
′′ (−1) 2.
y (x) a −1
x −1 y L(x)
y (x)
y L(x) y (x)
y L(x)
y f (x)
L(x) f (a) + f ′ (a)(x − a) L(a) f (a)
L L ′ (x) f ′ (a)
x L ′ (a) f ′ (a) L
f (a, f (a))

y L(x)
f
f ′′ (a) < 0 f ′′ (a) 0 f ′′ (a) > 0 f ′′ (a) > 0 f
f ′′ (a) < 0
f
f ′′ (a) 0 f ′′ x a
f f (x) L(x) x
f
f ′′ (a) > 0
L(x) ≤ f (x) x a
f (x)
f x
f (2) −1
y f ′ (x)
f ′′ (a) 0 f ′′ x a

2
2
y = f ′ (x)
2
2
x
2
x
2
x
y f ′ (x) y f (x)
y f ′′ (x)
f
a 2
L(x) f (2, −1)
f (2.07)
y f ′′ (x)
y f (x)
x 2
y f (x) x 2
y L(x)
y f (x)
f (2.07)
sin(x)
lim
x→0 x
sin(x)
x
f (x) sin(x) (0, 0) L(x) x x sin(x) ≈ x
x
sin(x)
≈ x x x 1,

sin(x)
lim
x→0 x
1.
y f (x) (a, f (a))
y − f (a) f ′ (a)(x − a).
y f (x)
y L(x) L(x) f (a) + f ′ (a)(x − a)
f (x) ≈ L(x) x x a
L(x) f (a) + f ′ (a)(x − a)
L(a) f (a) L ′ (a) f ′ (a)
y f (x) x a
f ′′ (a)
y f (x) f
√ 25.3
f (x) √ x f (x) x 25
y mx + b m b
m
b
√ 25.3
f (x) x a y 2x − 3

a
a
f (a)
f (a)
f (3.1)
f (3.1)
f (x) f (125) 76 f ′ (125) 8 f (128.5)
f (128.5)
H
H f (t) t
f ′ (t)
f ′ (25)
| f ′ (25)| 0.6 f (25) 65
y p(x) a 3 L(x) −2x + 5
p(3) p ′ (3)
p(2.79)
p ′′ (3) 0 p ′′ (x) < 0 x < 3
p ′′ (x) > 0 x > 3
y p(x) x 3 y L(x)

F
t
t F(t)
0 70
15 180.5
30 251
45 296
60 324.5
75 342.8
90 354.5
F ′ (60)
y L(t) y F(t) a 60
F(63)
y s(t)
s(t) t
t 9 s(9) 4
t 9 −1.2
0.08
t 9.34
t 9
f f ′ (x) (x − 1)e −x2
y f (x)
x f ′ (x) 0
f ′
x f ′′ (x) > 0
f
f (2) −3 f (1.88)
f x 2
f (1.88)

f (x)
f ′ (x)
f (x) x n
f (x) a x
y f (x) y kf(x)
k
y f (x) y (x) y
f (x) + (x)
f ′ f f
x y f (x) x
f ′ f (x + h) − f (x)
(x) lim
,
h→0 h
f ′ (x)
f (x)
f (x) x f ′ (x) 1
f (x)

x
(x) 4x 7 − sin(x) + 3e x
′ (x)
f (x) x n n 1, 2, 3,...
x
f ′ (x) f (x) x 2
f ′ (x) f (x) x 3
f ′ (x) f (x) x 4 (a+b) 4
a 4 +4a 3 b +6a 2 b 2 +4ab 3 + b 4 (x + h) 4
f (x) x 5 f (x) x 13
f (x) x n
n f (x) x n n
f ′ (x)
f ′
f (x) x 2 f ′ (x)
f ′ (x) 2x y
x y x
dy
dx
Δy
Δy
Δx
Δx
y x dy
dx
y x 2 dy
dx 2x.
dy
dx
d
dx [□]
□ x
d
dx [x2 ] 2x.

x
f (z) z 2 f ′ (z) 2z y t 2 dy
dt
2t
d
dq [q2 ] 2q
[
f ′′ (z) d df
]
d
dz dz 2 f
dz 2
f (x) c
d
dx
[c] 0
c f (x) c f ′ (x) 0
f (x) 7 f ′ d
(x) 0
dx [√ 3] 0.
n f (x) x n f ′ (x) nx n−1
n
f (x) x n f ′ (x) nx n−1
(z)
z −3 ′ (z) −3z −4 h(t) t 7/5 dh
dt
7 5 t2/5 d
dq [qπ ] πq π−1
a f (x) a x f ′ (x) a x ln(a)
f (x) 2 x f ′ (x) 2 x ln(2) p(t) 10 t
p ′ (t) 10 t ln(10) a e e
d
dx [e x ] e x ln(e) e x

ln(e) 1 e x
x 2
2 x
h(z)
h ′ (z) dh
dz
f (t) π
p(x) 3 1/2
d
(z) 7 z
r(t) ( √ dq [q−1 ]
2) t m(t) 1
h(w) w 3/4 t 3
p(t) 3t 5 −7t 4 +t 2 −9
t
y f (x)
y kf(x)
k |k|
y 0 k < 0
y kf(x) k y f (x)
k
k
k f (x) f ′ (x)
d
dx [kf(x)] kf′ (x)

(t) 3 · 5 t ′ (t) 3 · 5 t ln(5)
d
dz [5z−2 ] 5(−2z −3 )
y f (x)
y (x) y ( f + )(x)
( f + )(x) f (x) + (x)
f (x) (x) f ′ (x) ′ (x)
d
dx [ f (x) + (x)] f ′ (x) + ′ (x)
y ( f − )(x)
f (x) − (x) y f (x) + (−1 · (x))
d
dx [ f (x) + (−1 · (x))] f ′ (x) − ′ (x)
d
dw (2w + w 2 ) 2 w ln(2) + 2w h(q)
3q 6 − 4q −3 h ′ (q) 3(6q 5 ) − 4(−3q −4 ) 18q 5 + 12q −4
f ′ (x)
h ′ (z) dr/dt
f (x) x 5/3 − x 4 + 2 x
s(y) (y 2 + 1)(y 2 − 1)
(x) 14e x + 3x 5 − x
h(z) √ q(x) x3 −x+2
x
z + 1 + 5 z
z 4 p(a) 3a 4 − 2a 3 + 7a 2 − a + 12
r(t) √ 53 t 7 − πe t + e 4
f

f
f
f
p(t) 7t 5 − 4t 3 + 8t p
p ′ (t) 35t 4 − 12t 2 + 8
(z) 7 · 2 z ′ (z) 7 · 2 z ln(2)
h(z) √ z + 1 z
z 4
P(t) 2(1.37) t + 32 t
p(a) 3a 4 − 2a 3 + 7a 2 − a + 12
a −1
y f (x) f
f ′ (x) df
dx dy
dx d
dx [ f (x)]

f f (x) x n
f ′ (x) nx n−1 n
a f (x) a x f ′ (x)
a x ln(a)
f (x) (x) f ′ (x) ′ (x) a b
d af(x) + b(x) af ′ (x) + b ′ (x).
dx
y x 15/16
dy
dx
f (x) 1
x 17
f ′ (x)
y 9 √ x
dy
dx
f (t) 2t 2 − 2t + 15
f ′ (t)
y 4t 12 − 7 √ t + 4 t
dy
dt
y √ x(x 2 + 4)
dy
dx
y x5 + 2
x
dy
dx

f (2, 27) f
f (x) 4x 3 − 4x 2 + 11
y
f (x) x 3 + 3x 2 − 189x + 1 x f ′ (x) 0
x
f
f (2) 5 (2) −3 f ′ (2) −1/2 ′ (2) 2
h h(x) 3 f (x) − 4(x) h(2)
h ′ (2)
y h(x) (2, h(2))
p p(x) −2 f (x) + 1 2
(x) p
a 2
p(2.03) p (2, p(2))
p q
p
3
2
1
-3 -2 -1 1 2 3
-1
q
-2
-3
p q
x p x q
r(x) p(x) + 2q(x) x r
r ′ (−2) r ′ (0)

y r(x) (2, r(2))
r(t) t t s(t) arccos(t)
r ′ (t) t t (ln(t)+1) s ′ (t) − √ 1
1−t 2
0 < t < 1
w(t) 3t t − 2 arccos(t) w ′ (t)
y w(t) ( 1 2 , w( 1 2 ))
v(t) t t + arccos(t) v t 1 2
f (x) a x a
f (x) d
dx [ax ]
f ′ a
(x) x · a h − a x
lim
.
h→0 h
f ′ (x) a x a h − 1 · lim .
h→0 h
h
L lim
h→0
a h − 1
h
a 2 a 3
L 1 f
a x
a 2
3 a
a h − 1
L lim 1.
h→0 h
ln(2) ln(3)
d
dx [2x ] d
dx [3x ]
f (x) e x

d
dx [ax ] a x ln(a)
y sin(x) y cos(x)
sin(x) cos(x)
f ′ (x) f (x)
f (x) x n f ′ (x) nx n−1
n
a f (x) a x
f ′ (x) a x ln(a)
(x) 2 x
x −2, −1, 0, 1, 2
y (x)
′ (0)
h
y (x) x 0
y ′ (x)
y (x)
′ (x)
c(x) c ln(2)
′ (x) 2 x ln(2)

7
6
5
4
3
2
1
-2 -1 1 2
7
6
5
4
3
2
1
-2 -1 1 2
y (x) 2 x y ′ (x)
d
d
dx
[sin(x)]
dx
[cos(x)]
f (x) sin(x)
1 × 1
1.57 × 1 π/2
−2π −π π
-1
1
2π
1
−2π −π π 2π
-1
y f (x) sin(x)
x −2π, − 3π 2 , −π, − π 2 , 0, π 2 ,π,3π 2
, 2π,

y f (x)
f ′ (0)
h
y f (x) x 0
f ′ (2π) f ′ (−2π)
y f ′ (x)
y f (x)
f (x) sin(x)
(x) cos(x)
1 × 1
1.57 × 1 π/2
1
1
−2π −π π
-1
2π −2π −π π
-1
2π
y (x) cos(x)
x −2π, − 3π 2 , −π, − π 2 , 0, π 2 ,π, 3π 2
, 2π,
y (x)
′ ( π 2
)
h
y (x) x π 2
′ (− 3π 2
)
y ′ (x)
y (x)
(x) cos(x)
sin 2 (x) + cos 2 (x) 1 cos(x − π 2
) sin(x)

x
d
[sin(x)] cos(x)
d
dx
[cos(x)] − sin(x).
dx
f (x) sin(x)
cos(x)
h(t) 3 cos(t) − 4 sin(t)
y f (x) 2x + sin(x)
2
x π 6
y (x) x 2 + 2 cos(x)
x π 2
p(z) z 4 + 4 z + 4 cos(z) − sin( π 2 )
P(t) 24 + 8 sin(t)
P t
P
f (x) a x (a > 1)
f ′ (x)
a x f (x) 2 x
d
dx [2x ] 2 x ln(2)
y sin(x) y cos(x)
d
dx
[sin(x)] cos(x)
sin(x + h) cos(x + h)

d
dx
[cos(x)] − sin(x)
V(t) 24 · 1.07 t + 6 sin(t)
t t 0
V ′′ (2)
0 ≤ t ≤ 20 V(t) 24 · 1.07 t + 6 sin(t)
A(t) 24 · 1.07 t V(t) 24 · 1.07 t + 6 sin(t)
A ′ (t) V ′ (t) 6 sin(t)
V(t)
f (x) 3 cos(x) − 2 sin(x) + 6
y f (x) a π 4
y f (x) a π
a π 2
f
a 3π 2
y f (x)
d
dx
[sin(x)] cos(x) f (x) sin(x)
f ′ sin(x + h) − sin(x)
(x) lim
.
h→0 h
α β sin(α + β)
sin(α) cos(β) + cos(α) sin(β)
f ′ sin(x)(cos(h) − 1) + cos(x) sin(h)
(x) lim
.
h→0 h
h x
f ′ cos(h) − 1
sin(h)
(x) sin(x) · lim + cos(x) · lim .
h→0 h
h→0 h

h
cos(h) − 1
lim
h→0 h
sin(h)
lim .
h→0 h
f ′ (x)
d
dx
[cos(x)] − sin(x).

x n
a x sin(x) cos(x)
f (x) 7x 11 − 4 · 9 x + π sin(x) − √ 3 cos(x),
f
f ′ (x) 77x 10 − 4 · 9 x ln(9) + π cos(x) + √ 3 sin(x)
p(z) z 3 cos(z),
q(t) sin(t)
2 t .
f f (t) 2t 2 (t)
t 3 + 4t
f ′ (t) ′ (t)

p(t) 2t 2 (t 3 + 4t) p(t) f (t) · (t)
p 2t 2 p ′ (t)
p ′ (t) f ′ (t) · ′ (t)
q(t) t3 +4t
2t 2
q(t) (t)
f (t)
q
q
t q ′ (t)
q ′ (t) ′ (t)
f ′ (t)
p(x) f (x) · (x)
f f
N(t)
t t 0
S(t)
t S(t)
t
V(t) N(t) · S(t)
V(t) N(t) · S(t) ,
N ′ (t)
S ′ (t)
N S V
N(100) 520 S(100) 27.50
N ′ (100) 12 S ′ (100) 0.75

S(100) · N ′ (100) 27.50
· 12
330
.
V
N(100) · S ′
(100) 520 · 0.75
390
.
V ′ (100) S(100) · N ′ (100) + N(100) · S ′ (100) 330 + 390 720
.
p f P(x)
f (x) · (x) P ′ (x) f (x) ′ (x) + (x) f ′ (x).
f P(x) f (x) · (x)
P ′ (x) f (x) ′ (x) + (x) f ′ (x).
P f
f
V(101) N(101) · S(101) 532 · 28.25 15029
V(100) N(100) · S(100) 520 · 27.50 14300
V(101) V(100) + 720 15020.
N ′ (100) 12

P f
P
P(z) z 3 · cos(z) P
z 3 cos(z) P ′
z 3 − sin(z) cos(z)
3z 2
P ′ (z) z 3 (− sin(z)) + cos(z)3z 2 −z 3 sin(z) + 3z 2 cos(z).
m(w) 3w 17 4 w m ′ (w)
h(t) (sin(t) + cos(t))t 4 h ′ (t)
y f (x)
a 1 f f (x) e x sin(x)
L(x) y (x)
a −1 (x) (x 2 + x)2 x
Q(x) Q(x)
f (x)/(x) f Q ′
f f ′ ′ Q (x) 0
Q f /
f (x) Q(x) · (x).
f
f ′ (x) Q(x) ′ (x) + (x)Q ′ (x).
Q ′
Q ′ (x)
Q ′ (x)(x) f ′ (x) − Q(x) ′ (x).

(x)
Q ′ (x) f ′ (x) − Q(x) ′ (x)
.
(x)
Q(x) f (x)
(x)
.
Q ′ (x)
f ′ (x) − f (x)
(x) ′ (x)
(x)
f ′ (x) − f (x)
(x) ′ (x)
(x)
· (x)
(x)
(x) f ′ (x) − f (x) ′ (x)
(x) 2 .
f Q(x) f (x)
(x)
x ′ (x) 0
Q ′ (x) (x) f ′ (x) − f (x) ′ (x)
(x) 2 .
Q
f Q ′
Q(t) sin(t)/2 t
sin(t) 2 t Q ′
2 t cos(t) sin(t)
2 t ln(2) (2 t ) 2
Q ′ (t) 2t cos(t) − sin(t)2 t ln(2)
(2 t ) 2 .
Q ′ (t) 2 t
Q ′ (t)
cos(t) − sin(t) ln(2)
2 t .

f (x) f ′ (x)
r(z)
v(t)
3z
z 4 +1 r′ (z)
sin(t)
cos(t)+t 2 v ′ (t)
R(x) x2 − 2x − 8
x 2
− 9
x 0
I
I(t) 100t
e t ,
I t I ′ (0.5)
I ′ (2) I ′ (5)
x
f (x) x sin(x) 2
+
cos(x) + 2 .
f
f ′ (x) d
dx
[
x 2
x sin(x) +
cos(x) + 2
[
d
d
[x sin(x)] +
dx dx
]
x 2
cos(x) + 2
f ′ (x) (x cos(x) + sin(x)) + (cos(x) + 2)2x − x2 (− sin(x))
(cos(x) + 2) 2
d
dx []
]

x cos(x) + sin(x) + 2x cos(x) + 4x2 + x 2 sin(x)
(cos(x) + 2) 2 .
s(y) y · 7y
y 2 + 1 .
s
d
dy
(y 2 + 1) ·
d
s ′ dy y · 7
y
− y · 7 y d ·
dy y 2 + 1
(y)
(y 2 + 1) 2 .
d
dy y · 7
y
d
dy y 2 + 1
s ′ (y) (y2 + 1)[y · 7 y ln(7) + 7 y · 1] − y · 7 y [2y]
(y 2 + 1) 2 .
s ′ (y)
s s
f (r) (5r 3 + sin(r))(4 r − 2 cos(r)) f ′ (r)
p(t) cos(t)
t 6 · 6 t p′ (t)
(z) 3z 7 e z − 2z 2 sin(z) +
z
z 2 +1 . ′ (z)
t
s(t) 3 cos(t)−sin(t) t 1
e t
f (x) (x) f (3)

−2 f ′ (3) 7 (3) 4 ′ (3) −1 p(x) f (x) · (x) q(x) f (x)
(x)
p ′ (3) q ′ (3)
f f ′ (a)
f x x a y f (x)
(a, f (a))
P f
P(x) f (x)(x)
P ′ (x) f (x) ′ (x) + (x) f ′ (x).
Q f
Q(x) f (x)
(x)
Q ′ (x) (x) f ′ (x) − f (x) ′ (x)
(x) 2 .
F
2a(x) − 5b(x)
F(x) ,
c(x) · d(x)
F
F
f (x)
f (x) x · 12 x
f ′ (x)

f (x)
f (x) (x 8 − 7 √
x)8
x
f ′ (x)
z
dz
dt
z 9t + 4
4t + 5
h(r)
h ′ (r)
h(r)
r 2
15r + 11
s(q) 4 cos q sin q
s ′ (q)
f (x) x 4 cos x
f ′ (x)
h(t) t sin t + cos t
h ′ (t)
h(x) f (x) · (x) k(x) f (x)/(x)
f (x)
(x)

h ′ (0)
k ′ (−2)
F(4) 2, F ′ (4) 4, H(4) 3, H ′ (4) 1
G(z) F(z) · H(z) G ′ (4)
G(w) F(w)/H(w) G ′ (4)
f
f (2) 5 (2) −3 f ′ (2) −1/2 ′ (2) 2
h h(x) (x) · f (x) h(2)
h ′ (2)
y h(x) (2, h(2)) h
r r(x) (x)
f (x)
r
a 2
r(2.06) r
r (2, r(2))
r(t) t t s(t) arccos(t)
r ′ (t) t t (ln(t)+1) s ′ (t) − √ 1
1−t 2
0 < t < 1
w(t) t t arccos(t) w ′ (t)
y w(t) ( 1 2 , w( 1 2 ))
v(t)
t t
arccos(t) v t 1 2
p q

(x) p(x)·q(x) r ′ (−2)
r ′ (0)
x r ′ (x)
y r(x) (2, r(2))
z(x) q(x)
p(x) z′ (0)
z ′ (2)
x z ′ (x)
p
3
2
1
-3 -2 -1 1 2 3
-1
q
-2
-3
p
q
t 0 A(t)
t Y(t) t
C(t)
C(t) A(t) Y(t)
C(0)
C ′ (t) A(t) A ′ (t) Y(t) Y ′ (t)
C ′ (0)
C(1)
f (v) v
f (v)
v f (80) 0.05 f ′ (80)
0.0004
(v) v
f (v) (v) (80) ′ (80)

h(v) v
h(v)
v h(v) f (v) h(80)
h ′ (80)

tan(x) cot(x) sec(x) csc(x)
(x,y)
1
sin(θ)
θ
cos(θ)
θ (x, y) x
y θ
cos(θ) sin(θ) cos(θ) x
θ sin(θ) y
sin 2 (θ) + cos 2 (θ) 1,

tan(θ) sin(θ)
cos(θ)
cos(θ)
cot(θ)
sin(θ)
sec(θ)
cos(θ) 1
csc(θ)
sin(θ) 1
y tan(x)
f (x) tan(x) tan(x)
sin(x)
cos(x)
f
f ′ (x)
f ′ (x)
cos(x) cos(x) + sin(x) sin(x)
cos 2 .
(x)
f ′ (x)
sec(x)
cos(x) 1 f ′ (x)
x f ′ (x)
f
(x) cot(x) ′ (x) (x) cos(x)
sin(x)
′ sin(x)(− sin(x)) − cos(x) cos(x)
(x)
sin 2 (x)
− sin2 (x) + cos 2 (x)
sin 2 (x)
′ (x) − 1
sin 2 (x)

csc(x) 1
sin(x) ′
′ (x) − csc 2 (x).
′ sin(x) 0
π
x x kπ k 0, ±1, ±2,...
d
dx [cot(x)] − csc2 (x).
x x (2k+1)π
2
k ±1, ±2,...
d
dx [tan(x)] sec2 (x).
h(x) sec(x) sec(x)
cos(x) 1
h
h ′ (x)
sin(x) cos(x)
h ′ (x)
tan(x) sec(x)
h ′ h
p(x) csc(x) csc(x)
sin(x) 1
p
p ′ (x)
sin(x) cos(x)
p ′ (x)
cot(x) csc(x)

p ′ p
tan(x) cot(x) sec(x) csc(x)
f (x) 5 sec(x) − 2 csc(x) f
x π 3
p(z) z 2 sec(z) − z cot(z) p
z π 4
h(t) tan(t)
t 2 + 1 − 2e t cos(t) h ′ (t)
(r) r sec(r)
5 r ′ (r)
s(t)
15 sin(t)
e t .
s t
t ≥ 0
ds/dt
s ′ (2)

d
dx [tan(x)] sec2 (x),
d
dx [cot(x)] − csc2 (x),
d
d
[sec(x)] sec(x) tan(x), [csc(x)] − csc(x) cot(x).
dx dx
x x kπ 2
k ±1, ±2,...
h(t) t tan t + sin t
h ′ (t)
f (x) 5 tan(x)
x
f ′ (x)
f ′ (4)
f (x) tan(x) − 5
sec(x)
f ′ (x)
f ′ (1)
f (x) 2x2 tan(x)
sec(x)
f ′ (x)
f ′ (3)
y 2 tan x (π/4, 2)
y mx+b m
b

t
h(t) 3 + 2 cos(t)
1.2 t
t 2
t 2
t 2
f (x) sin(x) cot(x)
f ′ (x)
x f (x) cos(x)
f
f ′
p(z)
p(z)
z tan(z)
z 2 sec(z) + 1 + 3e z + 1.
p ′ (z)
p z 0
z 0p

C(x) f ((x))
f C ′ (x) f f ′ ′
f (x) sin(x)
(x) x 2
f
s(x) 3x 2 − 5 sin(x),
p(x) x 2 sin(x),
q(x) sin(x)
x 2 .
s ′ p ′
q ′
s(x) 3(x) − 5 f (x) p(x) (x) · f (x) q(x) f (x)
(x) .
C(x) sin(x 2 ),
x
C(x) x
x −→ x 2 −→ sin(x 2 ).
f x
f
C(x) f ((x)) sin(x 2 )
C f
x f
C(x) f ((x)) f C ′ (x)

f f ′ ′
p(x) f (x) · (x) f
C(x) f ((x))
f
C ′ f
C(x) sin(x 2 ).
f
f ((x))
h(x) tan(2 x )
p(x) 2 x tan(x)
r(x) (tan(x)) 2
m(x) e tan(x)
w(x) √ x + tan(x)
z(x) √ tan(x)
C(x) sin(x 2 )
f (x) −4x +7 (x) 3x −5 C(x) f ((x))
C ′ (x) C ′ f
f
C(x) f ((x))
f (3x − 5)
− 4(3x − 5) + 7
− 12x + 20 + 7
− 12x + 27.
C ′ (x) −12 f ′ (x) −4 ′ (x) 3 C ′
f ′ ′

f
C(x) sin(2x). C
C ′ C ′
C(x) sin(2x) 2 sin(x) cos(x).
C ′ (x) 2 sin(x)(− sin(x)) + cos(x)(2 cos(x)) 2(cos 2 (x) − sin 2 (x)).
cos(2x) cos 2 (x) − sin 2 (x).
C ′ (x)
C ′ (x) 2 cos(2x).
C(x) sin(2x) C ′ (x) 2 cos(2x) (x)
2x f (x) sin(x) C(x) f ((x)) ′ (x) 2 f ′ (x)
cos(x) C ′ (x)
C ′ (x) 2 cos(2x) ′ (x) f ′ ((x)).
C(x) f ((x)) C ′
f f ′ (x) x
C(x) f ((x)) C
x x x
f (x)

x f (x) C
C(x) f ((x)) x
C ′ (x) f ′ ((x)) ′ (x).
C f
C ′
f
r(x) (tan(x)) 2 .
r (x) tan(x) f (x) x 2
f
f (x) x 2
f ′ (x) 2x
f ′ ((x)) 2 tan(x)
(x) tan(x)
′ (x) sec 2 (x)
r ′ (x) f ′ ((x)) ′ (x) r(x)
(tan(x)) 2
r ′ (x) 2 tan(x) sec 2 (x).
r(x) r(x) tan 2 (x)
cos 4 (x) sin 5 (x) sec 2 (x)
f f ((x)) f ′ (x) ′ (x) f ′ ((x))
h(x) cos(x 4 )
p(x) √ tan(x)
s(x) 2 sin(x)
z(x) cot 5 (x)
m(x) (sec(x) + e x ) 9

h(t) 3 t2 +2t sec 4 (t)
h
h(t) a(t) · b(t) a(t) 3 t2 +2t b(t) sec 4 (t)
h a
b
a ′ (t)
b ′ (t)
a(t) f ((t)) 3 t2 +2t f
f (t) 3 t
(t) t 2 + 2t
f ′ (t) 3 t ln(3) ′ (t) 2t + 2
f ′ ((t)) 3 t2 +2t ln(3)
a ′ (t) f ′ ((t)) ′ (t) 3 t2 +2t ln(3)(2t + 2).
b b(t) r(s(t)) sec 4 (t) r
r(t) t 4
r ′ (t) 4t 3
r ′ (s(t)) 4 sec 3 (t)
s(t) sec(t)
s ′ (t) sec(t) tan(t)
b ′ (t) r ′ (s(t))s ′ (t) 4 sec 3 (t) sec(t) tan(t) 4 sec 4 (t) tan(t).
h
h(t) 3 t2 +2t sec 4 (t)
h ′ (t) 3 t2 +2t d dt [sec4 (t)] + sec 4 (t) d dt [3t2 +2t ].
a b 3 t2 +2t sec 4 (t)
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
m(v) sin(v 2 ) cos(v 3 )
h(y) cos(10y)
e 4y +1
s(z) 2 z2 sec(z)
c(x) sin(e x2 )
y √ e x + 3
x 0
1
s(t)
(t 2
+ 1) 3
t s t
t 1
h
P P
30e −0.0000323h dP/dh
dP/dh
1000
f (x) (x)
x f(x) f ′ (x) (x) ′ (x)
−1 2 −5 −3 4
2 −3 4 −1 2
f
C(x) f ((x)) C ′ (2)
D(x) f ( f (x)) D ′ (−1)

d
dx
[sin(x)] cos(x)
d
dx [sin(u(x))],
u x
d
dx [sin(u(x))] cos(u(x)) · u′ (x).
d
dx [ax ] a x ln(a)
d
dx [au(x) ] a u(x) ln(a) · u ′ (x).
u(x)
d (5x + 7)
10 10(5x + 7) 9 · 5,
dx
d
dx [tan(17x)] 17 sec2 (17x),
d e
−3x −3e −3x .
dx
x
h(x) 2 sin(x) x −→ sin(x) −→ 2 sin(x) .
C(x) f ((x))
f C ′ (x) f f ′ ′
C ′ (x) f ′ ((x)) ′ (x).

f (x) e 4x (x 2 + 3 x )
f ′ (x)
v(t) t 5 e −ct
c
v ′ (t)
y √ e −5t2 + 5
dy
dt
f (x) axe −bx+10
a b
f ′ (x)
f (x)
x 2 (x)
h(x) f ((x))

h ′ (1)
h ′ (2)
h ′ (3)
F(2) 3, F ′ (2) 4, F(4) 1, F ′ (4) 5 G(1) 3, G ′ (1) 4, G(4) 2, G ′ (4) 7
H(4) H(x) F(G(x))
H ′ (4) H(x) F(G(x))
H(4) H(x) G(F(x))
H ′ (4) H(x) G(F(x))
H ′ (4) H(x) F(x)/G(x)
f (x) 9x sin(2x)
f ′ (x)
f (x) x 3 (x) sin(x)
h(x) f ((x)) h
x π 4 .
x 0.25 h(x) f ((x)) r(x) ( f (x))
h(x) f ((x)) r(x) ( f (x))
u(x)
u u ′
p(x) e u(x)
q(x) u(e x )
r(x) cot(u(x))
s(x) u(cot(x))
a(x) u(x 4 )
b(x) u 4 (x)

p q
p
3
2
1
-3 -2 -1 1 2 3
-1
q
-2
-3
p q
C(x) p(q(x)) C ′ (0) C ′ (3)
x C ′ (x)
Y(x) q(q(x)) Z(x) q(p(x)) Y ′ (−2) Z ′ (0)
h
V π 3 h2 (12 − h).
h 1
t
h(t) sin(πt) + 1 t h
t 2
t 2

arcsin(x) arctan(x)
f ′ f f ′
f (x) e x (x) ln(x)
y 5 9
(x − 32) x
y
y 5 9
(x − 32) x x
y
C(x) 5 9
(x − 32)
F(y)
y
F(y)
F(y)
p(x) F(C(x))
F C p(x)
r(y) C(F(y)) F C
r(y)

C ′ (x) F ′ (y)
f : A → B A
B A f B f
: B → A ( f (a)) a a A
f ((b)) b b B f
f −1 f f
f y f (x)
f −1 (y) f −1 ( f (x)) x,
y f (x) x f −1 (y)
x y
f : A → B
f f
f −1 f −1 f f −1
f
f −1 ( f (x)) x x f f ( f −1 (y)) y y
f
y f (x) x f −1 (y)
f f −1
y f (x) (x, y) f
x f −1 (y) (y, x) f −1
f f −1 y x
y x
y f (x) 2 x
(−1, 1 2 ) ( 1 2
, −1)
y x
f
f

y = f (x)
2
(−1, 1 2 )
-2 2
y = x
-2
( 1 2 ,−1)
y = f −1 (x)
y f (x) y f −1 (x)
y f (x) e x
x f −1 (y) ln(y) y e x x ln(y) ln(e x ) x
x e ln(y) y y.
(x) ln(x)
(x) ln(x)
e (x) x
d
dx
[
e
(x) ] d
dx [x].
1
e (x) ′ (x) 1.
′ (x) ′ (x)
′ (x) 1
e (x) .
(x) ln(x) e (x) e ln(x) x
′ (x) 1 x .

x
d
dx [ln(x)] 1 x
d
dx [ln(x)] 1 x
x
f (t) ln(t 2 + 1) f ′ (t) 1
t 2 +1 · 2t
f (x) e x
f −1 (x) ln(x)
y = e x
8
y e x
B
A (ln(0.5), 0.5)
m A 0.5
4
B (ln(5), 5)
m B 5
B ′
A ′ B ′
A
-4 4 8
y x
A ′
x
y = ln(x)
d
dx [ln(x)] 1 x A′
-4
(0.5, ln(0.5)) m A ′ 1
0.5
2
B ′ (5, ln(5)) m B ′ 1 5
y
e x y ln(x)
x
m A ′ 1
m A
m B ′ 1
m B
y f (x)
y x
x y

h(x) x 2 ln(x)
p(t) ln(t)
e t +1
s(y) ln(cos(y) + 2)
z(x) tan(ln(x))
m(z) ln(ln(z))
[− π 2 , π 2 ]
f (x) sin(x) [− π 2 , π 2 ]
[−1, 1] f −1
f −1 : [−1, 1] → [− π 2 , π 2 ].
f −1
f −1 (y) arcsin(y).
y sin(x) x arcsin(y)
y
y π 6
− π
1 2
2
arcsin( 1 2 ) π 6
sin( π 6 ) 1 2 .
π
2
− π 2
(1, π 2 )
f −1 f
( π 2 ,1)
π
2
f (x) sin(x)
[− π 2 , π 2 ]
f −1 (x) arcsin(x)

h(x) arcsin(x)
h ′ (x) h(x) x
sin(h(x)) x.
d
dx [sin(h(x))] d
dx [x],
1
cos(h(x))h ′ (x) 1.
h ′ (x)
h ′ (x)
1
cos(h(x)) .
h(x) arcsin(x) h ′ (x) cos(arcsin(x))
x
θ arcsin(x)θ
x
θ
1
x
√ 1 − x 2
θ arcsin(x) sin(θ) x
cos(arcsin(x))
cos(arcsin(x)) cos(θ) √ 1 − x 2 .
1
θ
√
1 − x 2
x
θ arcsin(x)
h(x) arcsin(x)
h ′ 1
(x)
cos(arcsin(x))
h ′ (x)
1
√
1 − x 2 .

x −1 < x < 1
d
dx [arcsin(x)] 1
√ .
1 − x 2
r(x) arctan(x)
r ′ (x) r ′ (x)
r(x)
r(x) arctan(x) r ′ (x)
x
θ θ arctan(x)
x 1 cos(arctan(x))
1 x
r ′ (x)
f (x) x 3 arctan(x) + e x ln(x)
p(t) 2 t arcsin(t)
h(z) (arcsin(5z) + arctan(4 − z)) 27
s(y) cot(arctan(y))
m(v) ln(sin 2 (v) + 1)
( ) ln(w)
(w) arctan
1 + w 2

y x (ln(2), 2)
f (x) e x f ′ (ln(2)) 2
(2, ln(2)) f −1 (x) ln(x)
( f −1 ) ′ (2) 1 2 f ′ (ln(2))
f
y f (x) x (y) f ((x)) x
x f −1
x
d
dx [ f ((x))] d
dx [x],
f ′ ((x)) ′ (x) 1.
′ (x) ′ 1
(x)
f ′ ((x))
.
(x, (x))
f ((x), f ((x))) ((x), x)
m = f ′ (a)
y = f (x)
(a,b)
(b,a)
m = g ′ (b)
y = g(x)
y f (x) y (x) f −1 (x)
y f (x)
y (x) (a, b) f
(b, a) f (a) b (b) a
′ (x) 1/ f ′ ((x)) x b
′ (b)
1
f ′ ((b)) 1
f ′ (a) ,

(b, a)
f (a, b)
y x
f (a, b)
f f ′ (a) 0
′ (b) 1
f ′ (a) .
x ′ ′ (x) 1/ f ′ ((x)).
ln(x) arcsin(x) arctan(x)
(x) ln(x)
f (x) e x
′ (x)
1
f ′ ((x)) 1
e ln(x) 1 x .
d
x
dx [ln(x)] 1 x
d
x −1 < x < 1
dx [arcsin(x)] √ 1
1−x 2
d
x
dx [arctan(x)] 1
1+x 2
f x
′ ′ 1
(x)
f ′ ((x)) .
f (t)
f (t) ln(t 2 + 7)
f ′ (t)
(t)
(t) cos(ln(t))
′ (t)
h(w)
h(w) 5w arcsin w

h ′ (w)
x > 0 f (x) arctan x + arctan(1/x)
f ′ (x)
f
(x 0 , y 0 ) (2, 4) (x 1 , y 1 ) (2.5, 4.6) f
h(x) ( f (x)) 5
h ′ (2)
(x) f −1 (x)
′ (4)
f (x) 6 sin −1 x 4
f ′ (x)
arcsin(x) sin −1 (x) sin(x)
f (x) 7x 4 arctan(7x 2 ) f ′ (x).
f ′ (x)
f (x) 8 sin(x) sin −1 (x) f ′ (x)
f ′ (x)

f (x) ln(2 arctan(x) + 3 arcsin(x) + 5)
r(z) arctan(ln(arcsin(z)))
q(t) arctan 2 (3t) arcsin 4 (7t)
(v) ( )
arctan(v)
ln
arcsin(v)+v 2
y f (x)
f ′ (1)
y
f −1 (x)
f
( f −1 ) ′ (−1)
y = f (x)
y f (x)
f (x) 1 4 x3 + 4.
y f (x) f
f
f ′ (x) ′ (x) f ′ (2) ′ (6) f ′ (2)
′ (6)
h(x) x + sin(x)
y h(x) h
h −1
( π 2 , π 2
+ 1) y h(x)
(h −1 ) ′ ( π 2 + 1)

x
x
dy
dx
y
x
dy
dx
x
x y
y x
x 2 + y 2 = 16
4 x 3 − y 3 = 6xy
A
-4 4
x
B
-4
x 2 + y 2 16
x 2 + y 2 16
x 3 − y 3 6xy
x
y A (−3, √ 7)
B (−3, − √ 7)
y f (x)
x
dy
dx
y x

dy
dx
(x, y)
y y x
y x
d
dx [e u(x) ] e u(x) u ′ (x)
f x
d
dx [ f (x)] f ′ (x)
x
f c
d
dx x 2 + f (x)
d
d
dx x 2 f (x)
c + x + f (x)
2
d
dx
dx
d
dx
f (x 2 )
xf(x) + f (cx) + cf(x)
x
x 2 + y 2 16.
dy
dx
y x y
f (x) y
x dy
dx
x 2 + y 2 16 y x
x
d x 2 + y 2 d
dx
dx [16] .
d x
2 + d y
2 0.
dx dx
x
x

x y x
d
dx x
2
2x. y y x
d
dx [y2 ]
d
dx [ f (x)2 ]
d
dx [y2 ] 2y 1 dy
dx
.
x 2 + y 2 16
2x + 2y dy
dx 0.
dy
dy
dx
dx
2x 2y
dy
dx − 2x
2y − x y .
dy
dx − x y
x y
x −4 4
(a, b)
m r b a
(a, b)
m t − a b
(0, 4) (0, −4)
(−4, 0) (4, 0)
dy
dx − x y
m t = − a b
(a,b)
m r = b a
x 2 +y 2 16
(a, b)
m r m t

x 3 + y 2 − 2xy 2
(−1, 1)
3
y
-3 3
x
-3
x 3 + y 2 − 2xy 2
x
d x 3 + y 2 − 2xy d
dx
dx [2] ,
d
dx [x3 ] + d
dx [y2 ] − d [2xy] 0.
dx
y x
y x
3x 2 + 2y dy
dx
− [2x
dy
dx
+ 2y] 0.
dy
dx
dy
dx
dy
dx
3x 2 − 2y
2y dy
dx
− 2x
dy
dx 2y − 3x2 .

dy
dx
dy
dx (2y − 2x) 2y − 3x2 .
(2y − 2x)
dy
dx
2y − 3x2
2y − 2x .
dy
dx
x y
(−1, 1) dy
dx
dy
dx
(−1,1)
2(1) − 3(−1)2
2(1) − 2(−1) −1 4 .
dy
dx
dy
dx
dy
dx
dy
dy
dx
dx
dy
dx
x y
dy
dx
(a,b)
dy
dx (a, b) f ′ (a) f ′
d
dx
d
dx [x2 + y 2 ]
dy
dx
x x 2 + y 2
y x
dy
dx (x2 + y 2 )
y x x 2 + y 2

x y 5 − 5y 3 + 4y
y x
dy/dx
3
y
x y 5 − 5y 3 + 4y
(0, 1)
x y 5 − 5y 3 + 4y
-3 3
-3
x
x y 5 − 5y 3 + 4y
dy
dx
x y
dy
dx
p(x, y)
q(x, y) .
q(x, y) 0 p(x, y) 0
x y p(x, y) 0
y(y 2 −1)(y −2) x(x −1)(x −
2)

dy
dx (x − 1)(x − 2) + x(x − 2) + x(x − 1)
(y 2 − 1)(y − 2) + 2y 2 (y − 2) + y(y 2 − 1) .
(x, y)
(x, y)
x 1
2
1
-1
y
1 2 3
x
y(y 2 −1)(y−2) x(x−1)(x−
2)
dy/dx
x 3 − y 3 6xy (−3, 3)
sin(y) + y x 3 + x (0, 0)
3xe −xy y 2 (0.619061, 1)
x y y
x
x y x
x y
x dy
dx
dy
dx

dy
dy
dx
x y
dx
dy
dy
dx
0
dx
dy/dx x y x 5 y − x − 9y − 8 0
dy
dx
dy
dx x y x ln y + y2 6lnx
dy
dx
dy/dx x y arcsin(x 3 y) xy 3
dy
dx
x 3 + 2xy + y 2 64 (1, 7)
xy 3 +4xy
40 (8, 1)
(8, 1)
2y 3 + y 2 − y 5 x 4 − 2x 3 + x 2
sin(x + y) + cos(x − y) 1
( π 2 , π 2 )
d
dx [ax ] a x ln(a) d
dx [ln(x)] 1 x
y a x x
y a
x
y x
dy
dy
dx y
dx
x

0 0
lim x→∞ f (x) L lim x→a f (x) ∞
∞ ∞
0 0 f f ′ (x)
f ′ f (x + h) − f (x)
(x) lim
,
h→0 h
h → 0 ( f (x + h) − f (x)) → 0
f
f ′ (x) 0 0
0 0
f ′ (x) x
f ′ (x) x 2
h h(x) x5 +x−2
x 2 −1
h
x 5 + x − 2
lim
x→1 x 2
− 1
h x 1 f (x) x 5 + x − 2 (x) x 2 − 1
f a 1 L f (x) L (x)
h(x) ≈ L f (x)
L (x)
x a 1

L f (x)
lim
x→1 L (x) .
lim x→1 h(x)
h(x) x 1
lim x→1 h(x)
0 0
f
L f
L f ≈ f
a
L g
a
L g ≈ g
g
f a
L f L x a a
0 0
h(x)
h(x) f (x)
(x)
f x a
f (a) (a) 0
lim x→a h(x).
f f x
x a
L f L x a
f (x)
(x)
x → a

f (x) (x) x → a
x → a
L f (x) f ′ (a)(x − a) + f (a) L (x) ′ (a)(x − a) + (a)
x
a f L f L
lim
x→a
f (x)
(x) lim
L f (x)
x→a L (x)
f ′ (a)(x − a) + f (a)
lim
x→a ′ (a)(x − a) + (a) .
f (a) 0 (a) 0
f (a)
(a)
lim
x→a
f (x)
(x) lim
x→a
f ′ (a)(x − a)
′ (a)(x − a)
lim
x→a
f ′ (a)
′ (a) ,
x a x−a
x−a
f ′ (a)
′ (a)
x
lim
x→a
f (x)
(x) f ′ (a)
′ (a) .
1
′ (a) 0
f ′ (a)
′ (a)
f x a f (a) (a) 0
′ f (x)
(a) 0 lim x→a (x) f ′ (a)
′ (a) .
′ x a
lim
x→a
f (x)
(x) lim
x→a
f ′ (x)
′ (x) ,
f (x)
(x) 0 0
x → a

f ′ (x)
′ (x)
.
x 5 + x − 2
lim
x→1 x 2 − 1 ,
x 5 + x − 2 5x 4 + 1
lim
x→1 x 2 lim 6 − 1 x→1 2x 2 3.
lim x→0
ln(1+x)
x
lim x→π
cos(x)
x
lim x→1
2ln(x)
1−e x−1
lim x→0
sin(x)−x
cos(2x)−1
f
m = f ′ (a)
m = f ′ (a)
a
m = g ′ (a)
a
m = g ′ (a)
g
f
f x a f (x)
(x)

x → a
f ′ (a) 2 ′ (a) −1
lim
x→a
f (x)
(x) f ′ (a)
′ (a) 2
−1 −2.
f
f (a) (a) 0
f (x)
(x)
x → a f x a
2
g
f
2
1
2 p 1
q
1
r
1 2 3 4
1 2 3 4
1 2 3 4
-1
-1
-1
-2
-2
-2
s
f (2) f ′ (2) (2) ′ (2)
f (x)
lim
x→2
(x)
p(2) p ′ (2) q(2) q ′ (2)
p(x)
lim
x→2
q(x)
r(2) r ′ (2) s(2) s ′ (2)
r(x)
lim
x→2
s(x)
r(x)
lim
x→2
s(x)

∞
∞
∞
∞
f (x) 1 x
x 0
f
x → 0 x → 0 +
f (x)
f (x)
x
x > 0
1
1
f (x)= 1 x
f (x) 1 x
lim x→a f (x) L f (x) L
x a
L a ∞ f (x) 1 x
1
lim
x→0 + x ∞,
1 x
x
1
lim
x→∞ x 0,
1 x
x x
lim x→a f (x) ∞ f (x)
x a
lim x→∞ f (x) L f (x) L x
−∞ f (x) 1 x
1
lim
x→0 − x
−∞ lim
1
x→−∞ x 0.

lim f (x) ∞
x→∞
f (x) x
lim
x→∞ x2 ∞.
lim x→a f (x) ∞ x a f lim x→∞ f (x)
L y L f −∞
x → a − x → a +
x →∞ x →−∞
∞
8
y = e x
64
y = f (x)
1
y = sin(x)
4
-2 2
10
-4 4 8
-4
y = ln(x)
-64
y = g(x)
x →±∞
f (x) x 3 − 16x (x) x 4 − 16x 2 − 8
e x lim x→∞ e x ∞ lim x→−∞ e x 0,
e −x
lim x→∞ e −x 0 lim x→−∞ e −x ∞.
lim x→0 + ln(x) −∞ lim x→∞ ln(x) ∞. e x ln(x)
x →∞
p(x) a n x n + a n−1 x n−1 + ···a 1 x + a 0
a n n n
a n lim x→∞ p(x) ∞ lim x→−∞ p(x) ∞
a n lim x→∞ p(x) −∞ lim x→−∞ p(x) −∞
n lim x→∞ p(x) ∞ lim x→−∞ p(x) −∞
a n f lim x→∞ p(x) −∞
lim x→−∞ p(x) ∞ a n

x →∞
−1
1 x →∞lim x→∞ sin(x)
x → ∞
q(x) 3x2 − 4x + 5
7x 2 + 9x − 10 .
(3x 2 − 4x + 5) →∞ x →∞ (7x 2 + 9x − 10) →∞ x →∞
lim x→∞ q(x) ∞ ∞
0 0
1
x 2
lim q(x) lim
x→∞ x→∞
3x 2 − 4x + 5
7x 2 + 9x − 10 ·
1
x 2
1
x 2
lim
x→∞
3 − 4 1 x + 5 1 x 2
7 + 9 1 x − 10 1 x 2 3 7
1 → 0 1 x 2
x
→ 0 x →∞ q
y 3 7
x →∞
x 2
lim
x→∞ e x ?
x 2 →∞ e x →∞
∞
f ±∞ x → a
a ∞
lim
x→a
f (x)
(x) lim
x→a
f ′ (x)
′ (x) .
′ (x) 0
a a ∞
x
lim 2
x→∞ e
x 2 →∞
x
e x →∞
x 2
lim
x→∞ e x lim 2x
x→∞ e x .

∞ ∞ 2x
x 2
x 2
lim
x→∞ e x lim 2x
x→∞ e x lim 2
x→∞ e x .
2 e x →∞ x →∞ 2
e x
x 2
lim
x→∞ e x 0.
→ 0 x →∞
lim x→∞
x
ln(x)
lim x→∞
e x +x
2e x +x 2
lim x→0 + ln(x)
1
x
lim x→
π
2
− tan(x)
x− π 2
lim x→∞ xe −x
f (x)
(x)
∞ ∞
f x →∞
lim
x→∞
f (x)
(x) 0,
f (x)
f lim x→∞ (x) ∞ lim x→∞ f (x)
(x)
f
x
lim 2
x→∞ e
0, e x x 2 3x
lim 2 −4x+5
x
x→∞ 7x 2 +9x−10
3 7
f (x) 3x 2 − 4x + 5 (x) 7x 2 + 9x − 10
0 0
f (a) (a) 0
f a
lim
x→a
f (x)
(x) lim
x→a
f ′ (x)
′ (x) .
x →∞ x ∞
lim x→∞ f (x) L f (x)

L x lim x→a f (x)
∞ f (x) x
a
∞ ∞ f
±∞ x → a a ∞
lim
x→a
f (x)
(x) lim
x→a
f ′ (x)
′ (x) .
lim
x→a
(x)
f (x)
f (x)
(x)
f (x)
lim
x→a (x)
f (x)
lim
x→a (x)
ln(x/4)
lim
x→4 x 2 − 16
∞
−∞
1 − cos(7x)
lim
x→0 1 − cos(6x)
7 x − 6 x − 1
lim
x→1 x 2
− 1
13x 2
lim
x→∞ e 8x
f
f (3) (3) 0 f ′ (3) ′ (3) 0 f ′′ (3) −2 ′′ (3) 1 h
h(x) f (x)
(x)
f x 3
lim h(x).
x→3

R(x)
3(x − a)(x − b)
5(x − a)(x − c) ,
a b c x
R R a b c
(x) x 2x x > 0 lim x→0 + (x)
0 0 0 k 0 k > 0
b 0 1 b 0 0 0
h(x) ln((x)) h(x) 2x ln(x).
h(x) 2ln(x)
1
x
lim x→0 + h(x)
lim x→0 + h(x) lim x→0 + (x).
f lim x→∞ f (x) ∞
f (x)
lim x→∞ (x) ∞ lim x→∞ (x) 0
ln(x) √ x
ln(x) n√ xn
e x
x n ln(x) n

f
s(t) −16t 2 + 32t + 48
f f (c) f (c) ≥ f (x) x
f f (c) f (c) ≤ f (x)
x f
(c)
(c, (c))
(a, (a)) (b, (b))

40
V
y = s(t)
(c,g(c))
30
20
10
(a,g(a))
(b,g(b))
1 2
y = g(x)
s(t) −16t 2 + 24t + 32 ( 3 4
, 41)
f f (c)
f (c) ≥ f (x) x c f (c) f (c) ≤
f (x) x c f
(b) (b, (b))
(a) (a, (a)) (c)
h
c h(c) h
c h(c) h
h [−3, 3]
h [−3, 3]
c h ′ (c) 0

c
h ′ (c)
h
h ′ (c)
h ′ (c) h
y = h(x)
2
1
-2 -1 1 2
-1
-2
h
[−3, 3]
(c, f (c))
c
c
c c
c
c
f ′ (c) 0 f ′ (c)

c
f x c c
f f ′ (c) 0 f ′ (c)
f
c (c, f (c))
f (c)
p f p
x p f p f ′
p f p f ′
p
f f ′ (x) e −2x (3−
x)(x + 1) 2 f
f ′ (x)
f f ′ (x) x
f ′ (x) 0
e −2x (3 − x)(x + 1) 2 0
x 3 x −1 f
x e −2x 0
f ′ (x)
f ′
f ′ x < −1
f ′
f

f ′ (x)=e −2x (3 − x)(x + 1) 2
sign( f ′ )
behav( f )
+++ +++
+
+
INC −1
INC 3
+ − +
−
DEC
f
f ′ (x) e −2x (3 − x)(x + 1) 2
f ′ (x)
x < −1 x −2 e −2x
(3 − x) (x + 1) 2 x −2 e −2x (x + 1) 2
x (3 − x) x −2
f ′ + ++
+
x −1 f ′ f ′
f
f f ′
f −1 < x < 3 f ′ f x > 3
f
f ′ f ′ x 3 f ′
f x 3 f
x −1 f x −1 f
x −1
(x) x 2
′ (x) (x+4)(x−1)2
x−2
x 2
lim x→∞ ′ (x)

y (x)
f ′′
f ′ f
y f (x)
f ′ (p) 0
f
p
f
f p
f ′ (p) 0 f ′′ (p) p p
f ′′ f
f
p f
f ′ (p) 0 f ′′ (p) < 0 f p
f
f ′′ (p)
p f f ′ (p) 0 f ′′ (p)
0 f p f ′′ (p) < 0 f
p f ′′ (p) > 0
f ′′ (p) 0
f ′′ (p) 0 f
f (x) x 4 (x) −x 4 h(x) x 3 p 0

f (x) f ′ (x) 3x 4 − 9x 2
f f
f
f ′ (x) 3x 4 −9x 2 f
3x 4 − 9x 2 0
0 3x 2 (x 2 − 3) 3x 2 (x + √ 3)(x − √ 3),
x 0, ± √ 3 f
f
f ′ (x)=3x 2 (x + √ 3)(x − √ 3)
sign( f ′ )
behav( f )
+ −− ++−
INC
√
− 3
+
−
DEC 0
++−
−
DEC
+++
+
√
3
INC
f f ′ (x) 3x 4 − 9x 2 3x 2 (x 2 − 3)
f (−∞, − √ 3) ( √ 3, ∞) f
(− √ 3, 0) (0, √ 3)
f x − √ 3 x √ 3 f
x 0 f ′
x 0
f ′ (x) 3x 4 − 9x 2
f ′′ (x) 12x 3 − 18x f ′′
f ′′ (x) 0
0 12x 3 − 18x 12x
(
x 2 − 3 )
√
12x 3
+
2
2
x
x −
√
3 ,
2
√
3
x 0, ±
2 f ′′
f ′

( √
f ′′ (x)=12x x +
3
2
)( √
x −
3
2
)
sign( f ′′ )
behav( f )
−−−
−
CCD
√
3
−
2
− + −
+
CCU 0
++−
+++
−
+
CCD √ CCU
3
2
f f ′′ (x) 12x 3 − 18x
√ )
12x
(x 2 2 −
3
2
√ √
3
f (−∞, −
2 ) (0, 3
2
)
√ √
3
(−
2 , 0) ( 3
2 , ∞)
f
A
B
C
f
D
E
− √ 3 − √ 1.5
√
1.5
√
3
f
A (− √ 3, f (− √ 3)) f A
E ( √ 3, f ( √ 3) f
A B
B D

C f
f C C
f f
f
y
f x − √ 3
f (− √ 3) f
f
B C D f
p
f f (p, f (p)) f
f f ′ (p) 0
f ′ (p) f ′′ (p) 0 f ′′ (p)
f ′ f
f ′′ f
′′
x
′ (−1.67857351) 0
g ′′
2
1
1 2
y ′′ (x)
′′ (x) ′′

(x)
f (x) a(x −
h) 2 + k a h k
h(x) x 2 + cos(kx) k
h k
k 1, 3, 5, 10
h(x) x 2 + cos(3x)
k
h
h
k ≤ √ 2
k > √ 2
12
8
4
-2 2
k h
y h(x)
f p f ′ (p) 0
f ′ (p)
f f ′ f
f ′ f
f f

f f
f x p
f ′′ (p) f ′′ f
f x p f ′ (p) 0
f ′′ (p) < 0 f p f
f ′ (p) 0 f ′′ (p) > 0 f p f ′ (p) 0 f ′′ (p) 0
f p
f (x) 3e −9x2
f (x)
x
x
x
x
x
x
f (x) 2x 4 +27x 3 −21x 2 +15
f f ′ f ′′
f f ′ f ′′
f x
f (0) −1/2

y f ′ (x)
2
1
x
1
f ′ x
1
x
y f ′ (x) y f (x)
y f ′′ (x)
f f
f f
y f ′′ (x)
f f
y f (x)
′ (2) 0
1 < x < 2 2 < x < 3 ′ (x)
x 2
′′ (x) x 1 < x < 3
′′ (x) x 2
x 2
h

h(x) 0
h(x) 0
lim x→∞ h ′ (x) 3
y h(x)
x →∞
y h(x)
h ′
y h ′ (x)
p p ′′ (x) (x + 1)(x − 2)e −x
p p
√
5−1
x
2
p p
x
√
5−1
2
(2, 12
e 2 ) y p(x) p ′ (2) − 5 e 2
y p(x) x 2

d + a
d
f (t)=asin(b(t − c)) + d
c
c + 2π b
f (t) a sin(b(t − c)) + d a b c d
f (t) a sin(b(t −
c)) + d a 2π b
c d

a b c
d
y mx + b m y
(0, b) y mx + b
x a
m
b
a h k a 0 f
f (x) a(x − h) 2 + k
f f
a h k
a h k f ′ (x)
f a h k
a < 0 f
f
(x) axe −bx ,
a b
a b

′ (x)
′ (x) ax d
dx
[
e
−bx ] + e −bx d
dx [ax],
′ (x) axe −bx (−b) + e −bx (a).
′ (x) 0
′ (x)
0 ae −bx (−bx + 1).
a 0 e −bx 0 x
−bx + 1 0. x x 1 b
′ (x) ae −bx (1 − bx)
x 1 b
g ′ (x)=ae −bx (1 − bx)
sign(g ′ )
behav(g)
++ +−
+
−
INC
1
b
DEC
(x) axe −bx
′ (x) ae −bx (1 − bx) ae −bx
(1 − bx) x 1 b
−b (1 − bx) x < 1 b x > 1 b
x < 1 b x > 1 b
( 1 b , ( 1 b
))
′ (x) −abxe −bx +ae −bx
′′ (x) −abxe −bx (−b) + e −bx (−ab) + ae −bx (−b).
′′ (x) ab 2 xe −bx − 2abe −bx abe −bx (bx − 2).

g ′′ (x)=abe −bx (bx − 2)
sign(g ′′ )
behav(g)
+− ++
−
+
CCD CCU
2
b
(x) axe −bx
abe −bx
′′ (bx − 2) x 2 b
(bx − 2)
(b) (bx − 2) x < 2 b
x > 2 b ′′
x < 2 b x > 2 b
ax
lim (x) lim
x→∞ x→∞ axe−bx lim
x→∞ e . bx
∞ ∞
lim x→∞ (x) 0
lim (x) lim
x→−∞ x→−∞ axe−bx −∞,
ax →−∞ e −bx →∞ x →−∞
(x) → 0
a b

a
b e−1
global max
1
b
2
b
inflection pt
g(x)=axe −bx
(x) axe −bx
b
a a
( 1 b , ( 1 b )) ( 1 b , a b e−1 )
a
x → ∞
p(x) x 3 − ax a 0
p ′ (x) p
p
p
p a a
p
p ′′ (x) p
p a
p
p(x)

a > 0 a < 0
p(x) x 3 − ax a
p
a
h(x) a(1 −
e −bx ), a b
h
x h
x
lim x→∞ a(1 − e −bx ) lim x→−∞ a(1 − e −bx )
b
h
a b
A
L(t) A c k
1+ce −kt
L(t) A(1+ce −kt ) −1 L ′ (t)
L L
L ′′ (t) Ack 2 e −kt ce −kt − 1
(1 + ce −kt ) , 3
t L ′′ (t) 0
t
A
A
lim t→∞ lim
1+ce −kt t→−∞
1+ce −kt
L(x)
L
A c k
L(t)
A

C T D
T C
2 − D
3 D 2 .
D
dT/dD
D
(x) −2 ≤ x ≤ 2
′ (x) (x)
(x)
(x)
x
[−2, 2]
x
x
(−2) −8 (0)
(0)
−1 < (0) ≤ 3 ∞
(2) (0)

(2)
(0)
< >
p(x) x 3 − ax 2 a > 0
x 0 x a
p
p ′′ p ′′ (x) 0
p
p
a a
q(x) e−x
x−c
c > 0
q x c
lim x→∞ q(x) lim x→−∞ q(x)
q ′ (x) q
q
q
E(x) e − (x−m)2
2s 2 m s
E ′ (x) E
E
E ′′ (x)
( )
E ′′ (x) e − (x−m)2 (x − m) 2 − s 2
2s 2 s 4 .
x E ′′ (x) 0
lim x→∞ E(x) lim x→−∞ E(x)
E
y E(x) m s

f x p f (p) ≥ f (x) x p
p f (p) ≥ f (x) x f
f
x c
x a f (c) f (x)
x f (a)
f (x) x a
f
f x b
global max
relative max
relative min
f
a b c
f

3
f (x) 2 +
1+(x+1) 2
f
f
f
f
lim x→∞ f (x) lim x→−∞ f (x)
f (x) > 2 x
f
(−∞, ∞)
3
f (x) 2 +
1+(x+1) 2
x x 0 ≤ x ≤ 4
x [0, 4]
f
(x) 1 3 x3 − 2x + 2.
−2 ≤ x ≤ 3
−2 ≤ x ≤ 3
x
[−2, 3]
−2 ≤ x ≤ 2
−2 ≤ x ≤ 1
[a, b]
a b

f [a, b] f
[a, b] x m a ≤
x m ≤ b f (x m ) ≤ f (x) x [a, b] x M
[a, b] f (x M ) ≥ f (x) x [a, b] m f (x m ) M f (x M )
m ≤ f (x) ≤ M x [a, b]
[a, b]
f [a, b]
f
f
f
h(x) xe −x [0, 3]
p(t) sin(t) + cos(t) [− π 2 , π 2 ]
q(x) x2
x−2 [3, 7]
f (x) 4 − e −(x−2)2 (−∞, ∞)
h(x) xe −ax [0, 2 a
] a > 0
f (x) b − e −(x−a)2 (−∞, ∞) a, b > 0

2
g
2
g
2
g
-2 3
-2 2
-2 1
[−2, 3]
[−2, 2] [−2, 1]
x 20 − x

x
x 3
20−x
20−x
4
x
3
x
20 − x
20−x
4
x 0 ≤ x ≤ 20
x 0
5 × 5
A Δ 1 2 bh 1 2 · x
3 · x√ 3
6
√ 3
A □ s 2
20−x 2
4
A(x)
√
3x
2 ( ) 20 − x 2
36 + .
4
[0, 20]
A(x)
A ′ (x)
√ ( )(
3x 20 − x
18 + 2 − 1 ) √
3
4 4 18 x + 1 8 x − 5 2 .
A ′ (x) 0 x 180
4 √ ≈ 11.3007 A
3+9
[0, 20]
A
( )
) 2
A
≈ 10.8741
180
4 √ 3+9
A(0) 25
A(20)
√
3
36
√
3(
180
4 √ 3+9 )2
4
+
(
20−
180
4 √ 3+9
4
√
100
(400)
9 3 ≈ 19.2450

x ≈ 11.3007
10.8741
y A(x)
[0, 20]
25
20
15
10
5
y = A(x)
5 10 15 20
10 × 15
V
V x
V
V

f f ′′ (x) < 0 x
a b a < b ′ (x) < 0 x < a
′ (x) < 0 a < x < b ′ (x) > 0 x > b
h a b a < b h ′ (x) < 0 x < a
h ′ (x) > 0 a < x < b h ′ (x) < 0 x > b lim x→∞ h(x) 0
lim x→−∞ h(x) 0
p x a p ′′ (x) > 0 x < a
p ′′ (x) < 0 x > a
p(x) x 3 − a 2 x [0, a] a > 0
r(x) axe −bx [ 1
2b
, b] a, b > 0

w(x) a(1 − e −bx ) [b, 3b] a, b > 0
s(x) sin(kx) [ π 3k , 5π
6k ]
[a, b]
[a, b]
[a, b]
f ′ (x) ≤ 0 x [a, b]
c a < c < b ′ (x) > 0 x < c ′ (x) < 0
x > c
h(a) h(b) h ′′ (x) < 0 x [a, b]
p(a) > 0 p(b) < 0 c a < c < b p ′ (x) < 0 x < c
p ′ (x) > 0 x > c
s(t) 3 sin(2(t − π 6
)) + 5. s
s
[ π 6 , 7π 6 ]
[0, π 2 ] [0, 2π]
[ π 3 , 5π 6 ]

x y
x y

x y
x y
x y
x y

V x 2 y 4x + y 108
x y
y 108 − 4x y V(x) x 2 (108 − 4x)
x
0.027

P
Q P
Q
Z P
Q
P Z
P
3
Z
cabin
Q
2
P Z
xy x
f (x) 25 − x 2 x
y 25 − x 2
a b
f (x) b − ax 2
4 × 24

θ
1 1
2
θ

printed
material
x
y 12 − x 2

y x
y x
dy
dx
y x x
t
d 12
d 16
r V 4 3 πr3
V r t
V r t dV
dt
dr
dt V 4 3 πr3 t
dV
dt
r dr
dt
V r

dr dV
dt
dt
dV
dt
t
dV
dt
4πr 2 dr
dt
dr
dt
d 12
d 16
d 12 d 16
t
h
r

V 1 3 πr2 h.
V r h t
t
d
dt [V] d dt
[ 1
h]
3 πr2 .
d
dV
dt
[V]
dt
r
h t
dV
dt d [ 1
h]
dt 3 πr2
1 3 πr2 d dt [h] + 1 3 πh d dt [r2 ]
1 dh
πr2
3 dt + 1 dr
πh2r
3 dt
d
y x
dx [y2 ] 2y dy
dx
d dt [r2 ] 2r dr
dt
dV
dt 1 dh
πr2
3 dt + 2 3 πrhdr dt ,
V h r
t
h 1 dh
2
r t
dt
1 dr
2 dt
dV
dt
r h r
dr
dh
dt
h
dt
dV
dt 1 3 πr2 · 1 dr
2 dt + 2 3 πr · 1
2 r · dr
dt .
dV
dt

dV
dt
10 3
r 4 r 4 dV
dt
10
10 1 3 π42 · 1
2
dr
dt
r4
+ 2 3 π4 · 1
2 4 · dr
dt
r4
8 3 π dr
dt
r4
+ 16
3 π dr
dt
dr
dr
dt
r4 dt
10
r4
8π dr
dt
r4
dr
dt
10
r4
8π ≈ 0.39789
dh
dh
dt
r 4
dt
1 dr
2 dt
t
dh
dt
5 ≈ 0.19894 .
8π
r4
dr dr
dt
dt
r4
r t t
r t r 4
t
r
h r h 1 2
r t
r h
h 1 2
r
V 1 3 πr2 h V r
V 1 ( ) 1
3 πr2 2 h 1 6 πr3 .
r4
.
t
dV
dt 1 dr
πr2
2 dt ,
dr
dt
dh
dt

t
dr
dt
r4
r h t
t
h

t
h 3 h 4 h 5
r h
h
dV
dt
dh
dt

h
dh
dt
t
h
dV
dt 1 dh
πh2
16 dt ,
dh
dt 16
πh 2 dV
dt .
dV
dh
dt
h
dt
x
s
x s
dx
dt
ds
dt

90 ′
x y z
x 2 + y 2 z 2
t
2x dx
dt
+ 2y
dy
dt
2z
dz
dt ,
x y z
10
15
h r
V 1 3 πr2 h
15

8600.0 3 /
12.0
4.0 24.0 /
5.0
3 /
25
60
3
15
θ
θ

t
s t s(t) 64 − 16(t − 1) 2
[a, b]
s(b)−s(a)
b−a
t
s ′ s(t + h) − s(t)
(t) lim
.
h→0 h
f

f f ′
f ′ f
v(t) 3
8
mph
8
miles
4
4
hrs
hrs
1 2
1 2
y v(t) y s(t)
y v(t)
s(t) t
s(0) 0

y s(t)
t s
v
1
A 2
y = v(t)
3 mph v(t)=2
1
hrs
3 mph hrs
A 1
1 2 3
1 2 3
[1, 1.5] A 1 y v(t) [1, 1.5]
A 1 2
· 1 1 .
2
v(t) [a, b]
[a, b] A
A v(a)(b − a) v(a)Δt,
Δt t
v(t) [a, b] v(a)
v(t)

[1, 1.5] v(1) 2.5 v(1.5) ≈ 2.1
A 2 v(1)Δt 2.5
· 1 1.25 .
2
v [1, 1.5] A 2 1.25
[0, 3]
3 mph y = v(t)
1
hrs
1 2 3
y v(t) [0, 3]
t 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
v(t) 1.500 1.789 1.938 1.992 2.000 2.008 2.063 2.211 2.500

t 0 t 2
Δt 0.5
[0, 2]
v
v(t) 0.5t 3 − 1.5t 2 +
1.5t + 1.5 v
s s
s ′ v
s(2) − s(0)
3 mph y = v(t)
2
1
hrs
1 2
y v(t)
v
v

y v(t)
t v
s s v
t
v(t) 3
8
mph
8
miles
s(t)=3t
4
v(t)=3
4
A = 3 · 1.25 = 3.75
1 2
hrs
s(0.25)=0.75
s(1.5)=4.5
1 2
hrs
v(t) 3 s(t) 3t
s(t) 3t s ′ (t) 3 s(t) 3t
v(t) s(1.5) 4.5 s(0.25) 0.75
t 0.25 t 1.5
s(1.5) − s(0.25) 4.5 − 0.75 3.75 .
[0.25, 1.5]
[0.25, 1.5]
v s s ′ v
s(0) 0
s(0)

s v f ′ f
f G G ′
G (x) 3x 2 + 2x
G(x) x 3 + x 2 G ′ (x) (x)
H(x) x 3 + x 2 + 5
H
v(t) 32 − 32t t v
0 ≤ t ≤ 2
t
sv s(0) 0
s(1) − s( 1 2
)
y v(t)
t 1 2
t 1
[0, 1]
s(2) − s(0)
y v(t)
24
12
-12
-24
ft/sec
v(t)=32 − 32t
sec
1 2
y v(t)

s(0) 0
s 1
s −1
t 0 3
4
3
t 0 t 3
t 0 t 1.5
D [0,1.5] 3 · 1.5 4.5 .
t 1.5 t 2
D [1.5,2] 4 · 0.5 2 .
D [2,3] 3 · 1 3 ,
D D [0,1.5] + D [1.5,2] + D [2,3] 4.5 + 2 + 3 9.5 .
1.5 < t < 2 v −4
4.5 − 2 + 3 5.5 .

4.5
3.0
mph
y = v(t)
(3,5.5)
(1.5,4.5)
1.5
A 1 = 4.5
A 3 = 3
hrs
4.5 miles y = s(t)
3.0
1.5
(2,2.5)
hrs
-1.5
-3.0
1 3
A 2 = 2
-1.5
-3.0
1 3
-4.5
-4.5
A 1
4.5 3 · 1.5 A 2 A 3
A 2
D A 1 + A 2 + A 3 4.5 + 2 + 3 9.5 .
t t
s(3) − s(0) (+4.5) + (−2) + (+3) 5.5 ,
−2 [1.5, 2]
y s(t) −4
1.5 < t < 2 −4
s

v t
4 m/sec y = v(t)
2
sec
8
4
2 4 6 8
2 4 6 8
-2
-4
-4
-8
t 0 s(0) 1
0 ≤ t ≤ 2 t 2
t 8
t 0 t 8
t 1, 2, 3,...,8
y s(t)

y v(t)
s
t
f F f F
f F ′ f
v v s
s ′ v v [a, b]
s(b) − s(a) [a, b]
f (t) 9t f (t)
t 0 t 10

x f (t) 12 − 4t
f (t) t 0 t 4
v
t
v(t)
v(t) t 2 − 6t + 8
[−2, 5]
s(t)

0 ≤ t ≤ 4 t s(0) 1
10
mph
y = v(t)
10
miles
6
6
2
hrs
2
hrs
-2
1 2 3 4 5
-2
1 2 3 4 5
-6
-6
-10
-10
y v(t)
y s(t)
0 ≤ t ≤ 2 t 2
0 ≤ t ≤ 4
y s(t)
0 ≤ t ≤ 4 s
t v(t) 500 − 32t
a t a
y v(t) t
0 ≤ t ≤ a
s v s s ′ (t) v(t)
s(a) − s(0)
s(5) − s(1)
t
v v

y v(t)
1
y = v(t)
-1
1 2 3 4 5 6 7
y v(t)
0 ≤ t ≤ 2
s(5) − s(2) y s(t)
[0, 2] [2, 4][5, 7]
s
s
t
t 0 6 12 18 24 30
p(t) 7 8 10 13 18 35
y p(t) t
[0, 30]

v
v(t) y v(t) t
v(t)
t
A 1 A 2 A 3
D
[a, b]
y = v(t)
D A 1 + A 2 + A 3 ,
[a, b]
A 1
A 2
A 3
s(b) − s(a) A 1 − A 2 + A 3 .
a
b
v(t) < 0
A 2
D s(b) − s(a)
A 1 A 2 A 3

t v(t) 0.25t 3 − 1.5t 2 + 3t + 0.25
0 ≤ t ≤ 2
3 mph hrs
3 mph hrs
3 mph hrs
y = v(t)
y = v(t)
y = v(t)
2
2
2
1
A 4 B 4
A 3 B 3
A 2 B 2 2
B 1 C 1
1
1
C
C 3
C 4
A 1
1 2
1 2
1 2
y v(t)
[0, 2]
y v(t)
[0, 2]
S A 1 + A 2 + A 3 + A 4
y v(t)
A 3 v(1) · 1
2 2 · 1
2 1.

T B 1 + B 2 + B 3 + B 4 .
U C 1 + C 2 + C 3 + C 4 .
S T U D
[0, 2]
Σ S
1 + 2 + 3 + ···+ 100,
∑100
k 1 + 2 + 3 + ···+ 100.
k1
∑ 100
k1
k k k k
10∑
k1
(k 2 + 2k) (1 2 + 2 · 1) + (2 2 + 2 · 2) + (3 2 + 2 · 3) + ···+ (10 2 + 2 · 10),
n∑
f (k) f (1) + f (2) + ···+ f (n).
k1
n

∑ 5
k1 (k2 + 2)
3 + 7 + 11 + 15 + ···+ 27
∑ 6
i3 (2i − 1) 4 + 8 + 16 + 32 + ···+ 256
∑ 6
i1 1 2 i
y v(t) [a, b]
[a, b]
y f (x) [a, b]
f
f (x) ≥ 0
[a, b] f
[a, b]
a
b
x 0
x 1 x 2
··· x i x i+1
△x
··· x n−1
x n
[a, b] n Δx
n n
[a, b] Δx b−a
n
x 1 x 0 + Δx x 2 x 0 + 2Δx x i a + iΔx,
[x i , x i+1 ]
y f (x)
f [a, b]

[x i , x i+1 ]
A i+1 f (x i ) · Δx,
y = f (x)
A 1 A 2 ··· A i+1 ···
A n
x 0 x 1 x 2 x i x i+1 x n−1
x n
[a, b] n Δx
y f (x) [a, b]
L n
L n A 1 + A 2 + ···+ A i+1 + ···+ A n
f (x 0 ) · Δx + f (x 1 ) · Δx + ···+ f (x i ) · Δx + ···+ f (x n−1 ) · Δx.
L n
∑n−1
i0
f (x i )Δx.
0 n − 1
n L n f
[a, b]

y = f (x)
y = f (x)
B 1 B 2 ··· B i+1 ···
C 1 C 2 ··· C i+1 ···
C n
B n
x 0 x 1 x 2 x i x i+1 x n−1
x n
x 0 x 1 x 2 x i x i+1 x n−1
x n
[x i , x i+1 ] B i+1 f (x i+1 ) · Δx,
R n B 1 + B 2 + ···+ B i+1 + ···+ B n
f (x 1 ) · Δx + f (x 2 ) · Δx + ···+ f (x i+1 ) · Δx + ···+ f (x n ) · Δx
n∑
f (x i )Δx.
i1

R n f [a, b]
x i+1 x i + x i+1
2
x i+1 [x i , x i+1 ]
C 1
C 1 f (x 1 ) · Δx.
M n C 1 + C 2 + ···+ C i+1 + ···+ C n
f (x 1 ) · Δx + f (x 2 ) · Δx + ···+ f (x i+1 ) · Δx + ···+ f (x n ) · Δx
n∑
f (x i )Δx,
i1
M n f [a, b]
f (x) ≥ 0 [a, b] L n R n M n
y f (x) [a, b]
f
y v(t)
[a, b]
v
x ∗ i+1 [x i, x i+1 ] x i ≤ x ∗ i+1 ≤ x i+1
f (x ∗ 1 ) · Δx + f (x∗ 2 ) · Δx + ···+ f (x∗ i+1 ) · Δx + ···+ f (x∗ n) · Δx
n∑
f (x ∗ i )Δx.
L n R n M n
i1
t v(t) 2 9 (t − 3)2 + 2
2 ≤ t ≤ 5

[2, 5] L 4 R 4 M 4
L 4 R 4 M 4
L 4 R 4
f
v f [a, b] L n
f [a, b] R n
f [a, b]
L n
∑n−1
i0
f (x i )Δx,
f
f [a, b] L n
f
y = f (x)
y = f (x)
y = f (x)
A 1
A 2
A 3
a b c d
a b c d
a b c d
f
f [a, d]
[a, d]
b ≤ x ≤ c [b, c]
f (x i )Δx
y f (x) x
[b, c]

f [a, b]
[a, b] f
[a, b] f
L 24 ≈ A 1 − A 2 + A 3 ,
L 24
A 1 A 3 f
A 2 f A 1 − A 2 + A 3
f [a, d]
f
v t
v(t) 1 2 t2 − 3t + 7 2 .
M 5 v [1, 5]
Δt t 0 t 1 ··· t 5
[1, 5]
[1, 5]
M 10 M 20
n
a b

f (x ∗ )Δx
i
L n R n M n
L n f (x 0 )Δx + f (x 1 )Δx + ···+ f (x n−1 )Δx
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx
M n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx
∑n−1
i0
f (x i )Δx,
n∑
f (x i )Δx,
i1
n∑
f (x i )Δx,
i1
x 0 a x i a + iΔx x n b Δx b−a
n
x i (x i−1 + x i )/2
f (x) −x2
4 +
2x [2, 6]
y f (x)

y −x2
4
+ 2x [2, 6]
f (x) −x2
4 +
2x [2, 6]
y f (x)
y −x2
4
+ 2x [2, 6]
0 ≤ t ≤ 8

y e x n 2 ∫ 3
2 e x dx
n 2 ∫ 3
2 e x dx
f (x) 3x + 4
M 4 y f (x) [2, 5]
Δx x 0 , x 1 ,...,x 4
y f (x) x [2, 5]
n 4 M n L 4 R 4
M n
n
x [a, b]
S
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.
S f [a, b]
S f x [a, b]
S
S S
R
S

t 0 0.3 0.6 0.9 1.2 1.5 1.8
v(t) 100 88 74 59 40 19 0
[0, 1.8] L 6 R 6 1 2 (L 6 + R 6 )
v(t) [0, 1.8]
r
M 4 [0, 4]
M 4
r(t) 0.5e 0.5t r L 5 [0, 4]
4
3
tons/week
y = r(t)
2
1
weeks
1 2 3 4
r(t)

f [a, b]
y = f (x)
y = f (x)
y = f (x)
A 1
A 2
A 3
a b c d
a b c d
a b c d
f
f [a, d]
n

f (x) sin(2x)−
x 2
10 + 3 [1, 7] R 10 4.90595
x
n 5
f (x) 2x + 1
f (x) 2x + 1 [1, 4]
L n M n R n n 5 n 25 n 100
f (x) 2x + 1 x [1, 4]
f (x) 2x + 1

x [1, 4]
f (x) x 2 + 1 [1, 4]
L n M n R n n 5 n 25 n 100
f (x) x 2 + 1 x [1, 4]
f (x) x 2 +1
x [1, 4]
f (x) 2x + 1
[1, 4] L 5 16.2
L n M n R n
[a, b]
x ∗ i+1 [x i, x i+1 ]
n →∞

lim
n→∞ L n lim
n→∞
R n lim
n→∞
M n lim
n→∞
n∑
f (x ∗ i )Δx.
f
f [a, b]
∫ b
a
f (x) dx
∫ b
n∑
f (x) dx lim f (x ∗
n→∞
i )Δx,
a
Δx b−a
n x i a + iΔx i 0,...,n x ∗ i
x i−1 ≤ x ∗ i
≤ x i i
1,...,n
∫ a b
f ∫ b
a
f (x) dx
∫ b
a
f (x) dx y f (x) x
[a, b]
i1
i1
f
A 1 A 2 A 3
f x
[a, b] [b, c] [c, d]
∫ b
∫ c
f (x) dx A 1 , f (x) dx −A 2 ,
a
∫ d
c
∫ d
a
b
f (x) dx A 3 ,
f (x) dx A 1 − A 2 + A 3 .
y = f (x)
A 1
A 2
A 3
a b c d
f
[a, d]

v [a, b]
s(b) − s(a)
s(b) − s(a)
∫ b
a
v(t) dt.
[a, b] ∫ b
a
v(t) dt
[a, b],
v
x
∫ 4
1 (x2 + 1) dx
∫ 4
1
(2x + 1) dx
x
A 1 2 (3 +
9) · 3 18
∫ 4
1
(2x + 1) dx 18.
9
3
∫ 4
1
f (x)=2x + 1
(2x + 1)dx
1 4
f (x)
2x + 1 x [1, 4]

∫ 1
0 3xdx
∫ 4
−1
(2 − 2x) dx
∫ 1
√
−1 1 − x 2 dx
y = g(x)
1
-3 -2 -1 1 2 3 4
-1
∫ 4
−3
(x) dx
f [a, b]
f x
∫ a
a
f (x) dx a
f a ∫ a
a
f (x) dx 0.

∫ b
∫ c
f (x) dx A 1 , f (x) dx A 2 ,
a
∫ c
a
b
f (x) dx A 1 + A 2 ,
y = f (x)
A 1 A 2
a b c
f a b c
y f (x)
[a, c]
∫ c
a
f (x) dx
∫ b
a
f (x) dx +
∫ c
b
f (x) dx.
a < b < c
a b c
f a b
∫ a
b
∫ b
f (x) dx −
a
f (x) dx.
a b
Δx b−a
a−b
n
b a Δx
n
− b−a
n
f
k
d
dx [kf(x)] kf′ (x),

f
d
dx [ f (x) + (x)] f ′ (x) + ′ (x).
B = 2 f (x i )△x
y = 2 f (x)
a
A = f (x i )△x
A
x i x i+1
y = f (x)
b
a
x i
B
x i+1
b
y f (x) y 2 f (x) [a, b]
x x
f (x i ) 2 f (x i )
A B B 2A
n
y 2 f (x)
y f (x) 2
k

f k
∫ b
a
k · f (x) dx k
∫ b
a
f (x) dx.
f
C =(f (x i )+g(x i ))△x
A = f (x i )△x
A
f
B = g(x i )△x
B
g
C
f + g
a
x i
x i+1
b
a
x i
x i+1
b
a
x i
x i+1
b
y f (x) y (x) [a, b]
y f (x) + (x)
f
f + ( f + )(x i ) f (x i ) + (x i )
f
A B C C A + B
f
∫ b
a
[ f (x) + (x)] dx
∫ b
a
f (x) dx +
∫ b
a
(x) dx.
f c k
∫ b
∫ b
∫ b
[cf(x) ± k(x)] dx c f (x) dx ± k (x) dx.
a
a
a

f
x 2 x 3
∫ 2
0 f (x) dx −3 ∫ 5
2
f (x) dx 2
∫ 2
0 (x) dx 4 ∫ 5
2
(x) dx −1
∫ 2
0 x2 dx 8 3 ∫ 5
2 x2 dx 117
3
∫ 2
0 x3 dx 4 ∫ 5
2 x3 dx 609
4
∫ 2
5
∫ 5
0
f (x) dx
(x) dx
∫ 5
0 ( f (x) + (x)) dx
∫ 5
2 (3x2 − 4x 3 ) dx
∫ 0
5 (2x3 − 7(x)) dx
n y 1 y 2 ... y n
AVG y 1 + y 2 + ···+ y n
.
n
n
R n f
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx ( f (x 1 ) + f (x 2 ) + ···+ f (x n ))Δx.
Δx b−a
n
R n ( f (x 1 ) + f (x 2 ) + ···+ f (x n )) · b − a
n
(b − a) f (x 1) + f (x 2 ) + ···+ f (x n )
.
n
n
(b − a) n
n

f
f [a, b]
f (x 1 ) + f (x 2 ) + ···+ f (x n )
f AVG[a,b] lim
.
n→∞ n
f [a, b]
lim n→∞ R n ∫ b
a
f (x) dx
n →∞
∫ b
a
f (x) dx (b − a) · f AVG[a,b] .
f AVG[a,b]
f [a, b] [a, b]
f AVG[a,b] 1 ∫ b
b − a · f (x) dx.
a
f a b (b − a)
f [a, b]
∫ b
a f (x)dx y = f (x)
f AVG[a,b]
(b − a) · f AVG[a,b]
y = f (x)
A 1
A 2
y = f (x)
a
b
a
b
a
b
y f (x) [a, b]
∫ b
a
f (x) dx
(b − a) f AVG[a,b]
y
f AVG[a,b] (b − a) · f AVG[a,b]
∫ b
a
f (x) dx

A 1 A 2
v(t) √ 4 − (t − 2) 2
0 ≤ t ≤ 4 t v
y v(t) y √ 4 − (t − 2) 2
∫ 4
0
v(t) dt
v(t)
∫ 4
0
v(t) dt
v(t) [0, 4]
t 0 t 4 t
∫ 4
0
v(t) dt
[0, 4]
f [a, b]
f [a, b] ∫ b
a
f (x) dx
f [a, b]
∫ b
a
f (x) dx lim
n→∞
n∑
f (x ∗ i )Δx,
i1
Δx b−a
n x i a + iΔx i 0,...,n x ∗ i
x i−1 ≤ x ∗ i
≤ x i
i 1,...,n

∫ b
a
f (x) dx f
[a, b]
[a, b] f [a,b] 1
b−a · ∫ b
a
f (x) dx.
v ∫ b
a
v(t) dt
[a, b] v
v(t) dt [a, b]
∫ b
a
∫ b
∫ b
∫ b
[cf(x) ± k(x)] dx c f (x) dx ± k (x) dx
a
a
a
f [a, b] c k
f (x)
∫ b
a
f (x)dx
∫ c
b
f (x)dx
∫ c
a
f (x)dx
∫ c
a | f (x)|dx
f (x)
∫ 0
−4 f (x)dx
A ∫ 6
−4 f (x)dx
A

f (x) 6x + 5 [2, 6]
f (x) (x)
f (x)
(x)
f (x) 0 ≤ x ≤ 2
(x) 0 ≤ x ≤ 2
f (x) · (x) 0 ≤ x ≤ 2
( f ) · ( f · )
y f (x)
∫ 3
−3
f (x) dx ≈
f
∫ 13
11.5
f (x)dx
∫ 14.5
10
f (x)dx 5,
∫ 11.5
10
f (x)dx 6,
∫ 14.5
13
f (x)dx 8

∫ 11.5
13
(5 f (x) − 6)dx
v
t 0 t 4
[0, 4]
v
[0, 4]
[0, 4]
[0, 4]
[0, 4]
[0, 1.5]
s(0) 0
2 ft/sec 1
sec
-1
-2
1 2 3 4
y = v(t)
v(t) t(t − 1)(t − 3)
0 ≤ t ≤ 4
[0, 4]
[0, 4]
[0, 4]
[0, 4]

[0, 4]
f
f
2
y = f (x)
2
y = g(x)
1
1
1 2 3 4
1 2 3 4
-1
-1
-2
-2
f
∫ 1
0
[ f (x) + (x)] dx
∫ 4
1
[2 f (x) − 3(x)] dx
h(x) (x) − f (x) [0, 4]
c
∫ 4
0
cdx
∫ 4
0
[ f (x) + (x)] dx
f (x) 3 − x 2 (x) 2x 2
[−1, 1] y f (x)
y f (x) [−1, 1]
[−1, 1] y (x)
y (x) [−1, 1]
y f (x) y (x) [−1, 1]
∫ 1
−1 [ f (x) −
(x)] dx

p(x) ≥ q(x) x [a, b] y p(x)
y q(x)
∫ b
a
[p(x) − q(x)] dx.

v(t)
[a, b] [a, b]
y v(t) t [a, b]
y v(t) [a, b]
t
∫ b
a
v(t) dt
D
[a, b]
D A 1 + A 2 + A 3 ,
[a, b]
s(b) − s(a) A 1 − A 2 + A 3 .
A 1 A 2 A 3
y = v(t)
a
A 1
A 2
A 3
b

v(t) −32t + 16 v t
s(t) t
s v v s s ′ (t) v(t)
s(t)
s s(0) 32
s( 1 2 ) − s(0) s(2) − s( 1 2
) s(2) − s(0)
y v(t) [0, 2]
y v(t) t [0, 2]
a
D = ∫ b
a v(t)dt
= s(b) − s(a)
y = v(t)
b
s(t)
v(t)
v(t)
[a, b]
D D
s(b) − s(a)
D ∫ b
a
v(t) dt
v

s(b) − s(a)
∫ b
a
v(t) dt.
s(b) − s(a) [a, b]
[a, b] ∫ b
a
v(t) dt
s s v
v s s v
v(t) 3t 2 + 40
[1, 5]
D
∫ 5
1
v(t) dt
∫ 5
1
(3t 2 + 40) dt s(5) − s(1),
s v t 3 3t 2
40t 40 s(t) t 3 +40t s
v(t) s ′ (t) 3t 2 + 40 v
D
∫ 5
1
3t 2 + 40 dt s(5) − s(1)
(5 3 + 40 · 5) − (1 3 + 40 · 1) 284 .
s
s
140
120
100
80
60
40
20
y = v(t)
D = ∫ 5
1 v(t)dt
= 284
1 3 5
v(t) 3t 2 + 40 [1, 5]

f f F
F f F ′ (x) f (x) x V
s V v
∫ b
V(b) − V(a) v(t) dt.
∫ b
a
f (x) dx
f f
x
v
f [a, b] F f ∫ b
a
f (x) dx
F(b) − F(a).
a
F(b) − F(a)
F(b) − F(a) F(x)| b a ,
F a b
∫ b
a
f (x) dx F(x)| b a .
F
f d
dx [ 1 3 x3 ] x 2
∫ 1
x 2 dx 1 1
3 x3
0
0
1 3 (1)3 − 1 3 (0)3
1 3 .

∫ 4
−1
(2 − 2x) dx
∫ π
2
sin(x) dx
0
∫ 1
0 e x dx
∫ 1
−1 x5 dx
∫ 2
0 (3x3 − 2x 2 − e x ) dx
f (x)
sin(x) F(x) − cos(x) (x) x 2 G(x) 1 3 x3
f
5 sin(x) − 4x 2 , x 2 sin(x),
sin(x)
x 2 , sin(x 2 ).
d
[− cos(x)] sin(x),
dx
F(x) − cos(x) f (x) sin(x)
f (x) sin(x) F(x) − cos(x) F f
f (x) 5 sin(x) − 4x 2
− cos(x) sin(x) 1 3 x3 x 2
F(x) −5 cos(x) − 4 3 x3

f
(x) x 2 G(x) 1 3 x3
G(x) 1 3 x3 + 7 G ′ (x) x 2
(x) x 2
G(x) 1 3 x3 + C,
C
+C G
C
∫ 1
0
x 2 dx.
(x) x 2 G(x)
1
3 x3 + C.
∫ 1
0
x 2 dx 1 3 x3 + C
1
0
( ( 1 1
3 (1)3 + C)
−
3 (0)3 + C)
1 3 + C − 0 − C
1 3 .
C
+C

F
f
f (x)
kk
x n n −1
1
x x > 0
sin(x)
cos(x)
sec(x) tan(x)
csc(x) cot(x)
sec 2 (x)
csc 2 (x)
e x
a x (a > 1)
1
1+x 2
1
√
1−x 2
F(x)
∫ 1
0
∫ π/3
0
∫ 1
0
x 3 − x − e x + 2 dx
(2 sin(t) − 4 cos(t) + sec 2 (t) − π) dt
( √ x − x 2 ) dx
v(t)
[a, b] ∫ b
a v(t) dt v(t) ≥ 0 [a, b] ∫ b
a
v(t) dt
[a, b]
f ∫ b
a
f (x) dx

y f (x) x [a, b]
x
f [a, b] f AVG[a,b]
f AVG[a,b] 1 ∫ b
b − a a
f (x) dx.
v V
V(b) − V(a)
∫ b
a
v(t) dt.
V s v s ′
s(b) − s(a)
∫ b
a
s ′ (t) dt.
f
f [a, b] f ′ f (b) −
f (a) ∫ b
a
f ′ (x) dx. [a, b]
[a, b]
f
f ′ f ′ f
f f ′

4
3
2
1
3
1
3 4
4
3
2
1
(2,4)
(3,3)
(1,3)
(0,0)
(4,0)
-1
-2
-3
1 2
1
y = f ′ (x)
3
-1
-2
-3
1 2 3 4
y = f (x)
-4
-4
f ′ (x) 4 − 2x f (x) 4x − x 2
f f ′
f (1) − f (0)
f (1) 3 f (0) 0
y f ′ (x) [0, 1] f (1) − f (0) ∫ 1
0 f ′ (x) dx
f ′ [0, 4]
0 ∫ 4
0 f ′ (x) dx 0 f (4) − f (0) 0 f (4) f (0)
r(t) t r(t)
r(t) 0.0069t 3 −0.125t 2 +11.079 y r(t)
∫ 10
4
r(t) dt
4 ≤ t ≤ 10

12
gal/day
10
8
y = r(t)
6
4
2
days
2 4 6 8 10 12
r(t)
r(t) ≥ 0 ∫ 10
4
r(t) dt
[4, 10]
· .
R(t)
t R ′ (t) r(t)
∫ 10
4
r(t) dt R(10) − R(4),
r(t) 0.0069t 3 − 0.125t 2 + 11.079
∫ 10
4
0.0069t 3 − 0.125t 2 + 11.079 dt 0.0069 · 1
4 t4 − 0.125 · 1
3 t3 + 11.079t
10
≈ 44.282.
4

4 ≤ t ≤ 10
r [4, 10]
∫
1 10
r AVG[4,10]
r(t) dt ≈ 44.282 7.380,
10 − 4 4
6
c y c(t)
0 ≤ t ≤ 10 c c(t) −0.05t 2 + t + 10
30 ≤ t ≤ 40 c(t) −0.05t 2 + 3t − 30
15
cal/min
y = c(t)
10
5
10 20 30
min
c(t)
C(t) c(t) C(40) − C(0)
0 ≤ t ≤ 40

f [a, b]
F f
∫ b
a
f (x) dx F(b) − F(a).
f
F [a, b]
∫ b
a
f ′ (x) dx f (b) − f (a),
f
f [a, b]
f [a, b]
f (x)
∫ b
a
f (x)dx
∫ c
b
f (x)dx
∫ c
a
f (x)dx
∫ c
a | f (x)|dx
f (x)

∫ 0
−3 f (x)dx
A ∫ 5
−3 f (x)dx
A
f (x) 4x + 4 [4, 9]
f (x) (x)
f (x)
(x)
f (x) 0 ≤ x ≤ 2
(x) 0 ≤ x ≤ 2
f (x) · (x) 0 ≤ x ≤ 2
( f ) · ( f · )
y f (x)

∫ 3
−3
f (x) dx ≈
f
∫ 6
∫4.5
4.5
6
f (x)dx
∫ 7.5
3
(4 f (x) − 1)dx
f (x)dx 4,
∫ 4.5
3
f (x)dx 1,
∫ 7.5
6
f (x)dx 1
v 0 ≤ t ≤ 4 v(t)
v(t) − 1 4 t3 + 3 2 t2 +1 v
0 ≤
t ≤ 4
[12, 24]
c
t c
[12, 24]
15
12
9
6
3
m/min
y = v(t)
min
4 8 12 16 20 24
f
f (x)
⎧⎪
⎨
⎪
⎩
−x 2 + 2x + 1, 0 ≤ x < 2
−x + 3, 2 ≤ x < 3
x 2 − 8x + 15, 3 ≤ x ≤ 5
f x
[2, 5]

f [0, 5]
5 ≤ x ≤ 7
f f (x) (x) 5 ≤ x ≤ 7 ∫ 7
0
f (x) dx 0.
c(h) h
h 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c 925 875 830 780 730 685 635 585 535 490 440
m(h)
h
m(h)
m(h)
M 5
s(t) [a, b]
s(b) − s(a)
AV [a,b] .
b − a
v(t) [a, b]
v AVG[a,b] 1 ∫ b
v(t) dt.
b − a a
[a, b]
[a, b]

f A(x) ∫ x
0
f (t) dt A
f
f
y f ′ (x) f
f
f ′ f
f f ′
f ′′ f
f ′ f ′′
f
f ′ f
f
[a, b] f ′ x

f
f f
f y f ′ (x)
f ′ x ≤ 0 x ≥ 6 f ′ (x) 0
f (0) 1
3
y = f ′ (x)
3
1
1
-1
1 3 5
-1
1 3 5
-3
-3
y f ′ (x) y f (x)
f f
f
f
f (1) − f (0)
∫ 1
0
f ′ (x) dx.
f (1)

f (2) f (3) f (4) f (5) f (6)
y f (x)
f x < 0 x > 6
f F
a F(a)
F(b) F(b) − F(a) ∫ b
f (x) dx
F(b) F(a) +
∫ b
a
f (x) dx.
a
f F
[a, b]
F(b) − F(a)
[a, b] ∫ b
a
f (x) dx
f (x) x 2 f
F(1) 2 a 1 b 2
F(2) F(1) +
∫ 2
1
2 + 1 2
3 x3
1
( 8
2 +
3 − 1 )
3
13
3 .
x 2 dx
f
f
F F

F
f f
y f (x)
f
F f F(0) −1
x ≤ 0 x ≥ 7 f (x) 0
1
y = f (x)
1 2 3 4 5 6 7
-1
y f (x)
F F
F
F
F(x) x 1, 2,...,7
F(−1) F(8)
y F(x)
F x < 0 x > 7
G
f G(0) 0
G
F
f f

f
3
f
G
1
2
F
-1
1 3 5
2 4
H
-2
-3
y f (x) f
F f F(0) 1
F(1) 1.5 F(2) 1.5 F(3) −0.5 F(4) −2
F(5) −0.5 F(6) 1 F ′ f F
F
G f
G(0) 3 G F f
G F
G(1) − G(0) ∫ 1
0
f (x) dx 0.5 F(1) − F(0) 0.5, G(0) 3
G(1) G(0) + 0.5 3.5 F(1) F(0) + 0.5 1.5 F(0) 1
H(0) −1
G H f G − H
G H f
G ′ f H ′ f
d
dx [G(x) − H(x)] G′ (x) − H ′ (x) f (x) − f (x) 0
G − H

f
f (x) x 2 F(x) 1 3 x3 +C +C
F f
f F
F(2) 3
x F(x) 1 3 x3 + C
3 1 3 (2)3 + C,
C 3 − 8 3
1
3 x3 + 1 3 .
1 3 F(x)
(x) |x| − 1 G(−1) 0
[−2, 2]
h(x) sin(x) H(0) 1
[0, 4π]
x ⎧⎪
, 0 < x ≤ 1
p(x) ⎨−(x − 2) 2 , 1 < x < 2 P(0) 1
⎪
⎩0
[−1, 3]
F b F(a)
a b f
F(b) F(a) +
∫ b
a
f (x) dx.
F(b)
(b, F(b)) F
b

f A
A(x)
∫ x
a
f (t) dt.
x
A x
a x
t x
A a x
y f (t)
f
a 0 A(x) ∫ x
0
f (t) dt
y = f (t)
3
1
π
2π
-1
x
1
A(x)
π
2π
x
f A(x)
f (t) dt
∫ x
0
A t 0 t x
y f (t)
y A(x) f A
f
A
a
a

A
A(x) ∫ x
1
(t) dt
3
g
3
1
1
-1
1 3 5
-1
1 3 5
-3
-3
y (t) y A(x) A
A(x) ∫ x
1
(t) dt
A A
A
A
A(0) A(1) A(2) A(3)
A(4) A(5) A(6)
y A(x)
A x < 0 x > 6
B A B B(x) ∫ x
0
(t) dt
f F
F F(a)

∫ b
a
f (x) dx a b
F(3) F(3) F(a) + ∫ 3
a
f (x) dx
F
F ′ f F
F
F f
G(x) F(x) + C f
f
f A(x) ∫ x
a
f (t) dt A
f [a, x] A
f
f (x)
∫ 0
−5 f (x)dx
A ∫ 7
−5 f (x)dx
A
f (x)
∫ 7
0 f (x)dx ≈
F f F(0) 40 F(7)
F(7) ≈
f ′ f f (5) 0

f
f (0)
f (7)
∫ 7
0 f ′ (x) dx
f
F ′ f F(0) 0 F(b) b
b
F(b)
v
A 1 7 6 A 2 8 3
s s(0) 0.5
s(1) s(3) s(5) s(6)
s s
s s
s
v(t) −2 + 1 2 (t − 3)2 s

3
v
3
s
A 1
A 2
t
1 6
1
-1
2 4 6
t
-3
-3
v s
c c
15
10
cal/min
c
5
10 20 30
min
10 20 30
c C
C c C

C(0) 0 C(t) t 5, 10, 15, 20, 25, 30
C
C 5 ≤ t ≤ 10
f A B
C A(x) ∫ x
−1 f (t) dt B(x) ∫ x
0 f (t) dt C(x) ∫ x
1
f (t) dt
x −1, 0, 1,...,6 A(x)
B(x) C(x)
A B C
A B C
A
f
3
3
1
f
1
-1
1 3 5
-1
1 3 5
-3
-3
f A B C

A(x) ∫ x
1
f (t) dt f
f [a, b] F f F ′ f
∫ b
f (x) dx F(b) − F(a).
a
f
f
[a, b] F f
∫ 4
1
x 2 dx,
F(x) 1 3 x3 f (x) x 2
∫ 4
1
x 2 dx 1 3 x3
4
1
1 3 (4)3 − 1 3 (1)3
21.
F(b)− F(a) F f
∫ b
a
f (x) dx
F

F
20
10
f (x)=x 2
∫ 4
1 x2 dx = 21
20
10
(4, 64 3 )
F(4) − F(1)=21
F(x)=
3 1x3
(1, 1 3 )
1 2 3 4
1 2 3 4
f (x) x 2 [1, 4]
F(x) 1 3 x3 [1, 4]
f AVG[a,b] 1
b−a
∫ b
a
f (x) dx
A(x) ∫ x
c
f (t) dt
A
A(x) ∫ x
c
f (t) dt f
c
A f
A
A(x)
∫ x
1
f (t) dt,

f (t) 4 − 2t
A(1) A(2)
A(x)
∫ x
1
(4 − 2t) dt
f A
A ′ (x)
f f (t) 4 − 2t f
f (x) 4 − 2x
A f
f (t) 4 − 2t
1 A
f (t) cos(t) − t A(x) ∫ x
2
f (t) dt A
A(x)
∫ x
2
(cos(t) − t) dt
sin(t) − 1 2 t2
x
sin(x) − 1 2 x2 − (sin(2) − 2) .
2
A(x) (sin(2) − 2)
A ′ (x) cos(x) − x,
A ′ (x) f (x) f A
f A(2) ∫ 2
2
f (t) dt 0 A
f A(2) 0
f A
A(x)
∫ x
c
f (t) dt,
c A f
A ′ (x) f (x)
A ′ A(x + h) − A(x)
(x) lim
h→0 h
∫ x+h
c
lim
h→0
f (t) dt − ∫ x
c
h
f (t) dt

f (t) dt
,
h
∫ x
c
f (t) dt+ ∫ x+h
x
f (t) dt
f (t) dt h
∫ x+h
c
∫ x+h
x
lim
h→0
∫ x+h
x
f (t) dt ≈ f (x) · h,
∫ x+h
A ′ x
(x) lim
h→0
f (t) dt
h
lim
h→0
f (x) · h
h
f (x).
A f A(c) ∫ c
c
f (t) dt 0.
f c f
A A(c) 0 A(x) ∫ x
c
f (t) dt
f f
1
y = f (x)
1 2 3 4 5 6 7
-1
y f (x) y A(x)
A A(x) ∫ x
2
f (t) dt
A f
A(1) A(3)
y A(x)
A A
A x 0, 1,...,7

A F
B(x) ∫ x
3 f (t) dt C(x) ∫ x
1
f (t) dt
G
G(x)
∫ x
c
(t) dt
G(c) 0
G ′ (x) (x) G
d
dx
[∫ x
]
(t) dt (x).
c
G
G ′ (x) (x)
∫ x
c
(t) dt
E(x)
∫ x
0
e −t2 dt.
E
e −t2
f (t) 4 − 2t F(t) 4t − t 2
e −t2
E E
x 0
erf(x) 2 √ π
∫ x
0 e−t2 dt 0 ≤ erf(x) < 1
x ≥ 0 lim x→∞ erf(x) 1

E E
E ′ (x) d [∫ x
]
e −t2 dt e −x2 ,
dx 0
E E(0) 0
E E(0) 0
E E ′ (x) e −x2
x e −x2 > 0 E ′ (x) > 0 x E
x →∞ E ′ (x) e −x2 → 0
E x →∞ x →−∞
E x
E ′′ (x) −2xe −x2 E ′′ (0) 0 E ′′ (x) < 0
x > 0 E ′′ (x) > 0 x < 0 E x < 0
x > 0 x 0
E x
E(2) E(2) ≈ 0.8822
E(3) E(3) ≈ 0.8862
x E 0.886
E
f (t) e −t2
1
1
E(x)= ∫ x
0 e−t2 dt
f (t)=e −t2 -1
-2 2
-2 2
-1
f (t) e −t2 E(x) ∫ x
0 e−t2 dt
f E(0) 0

E f (t) e −t2 E(0) 0
y E(x) f (t) e −t2
x E
x f (t) e −t2
x
f (t)
t F(x) ∫ x
1+t 2 0
f (t) dt
f (t)
t −10 ≤
1+t 2
t ≤ 10
F f
F
F
f ′ (t)
f ′ (t)
1−t2 .
(1+t 2 ) 2
F(5) F(10)
y F(x)
f F

F
f F F(x) ∫ x
c
f (t) dt
F(x) ∫ x
c
f (t) dt
∫
F ′ (x) d x
dx c
f (t) dt f (x)
F(x)
∫ x
π
sin(t 2 ) dt,
F ′ (x) sin(x 2 ).
f
[∫
d x
]
f (t) dt f (x).
dx
a
f f f t a
t x x
f (t)
t t a t x
∫ x
a
d f (t) dt?
dt
f (t) d dt
∫ x
a
d x
f (t) dt f (t)
dt
a
f (x) − f (a).
f (t)
.
f
a x f f (a)
f (a)

[∫ ]
d x
dx 4 e t2 dt
∫ [ ]
x d t
4
−2 dt 1+t dt 4
[∫
d 1
dx x cos(t3 ) ]
dt
∫ x
d
3 dt ln(1 + t 2 ) dt
[
d ∫ ]
x
3
dx 4
sin(t 2 ) dt
f A(x) ∫ x
1
f (t) dt
f
f c A(x)
f (t) dt f A(c) 0
∫ x
c
∫ x
d f (t) dt f (x) − f (c)
dt
c
d
dx
[∫ x
c
]
f (t) dt f (x).
(x) ∫ x
2
f (t) dt f (t)

(2)
′ (3)
x 0 ≤ x ≤ 8
x
∫
d a
cos(tan(t)) dt
dx x
F(9)
F ′ (x) e −x2 /5 F(0) 2
F(9) ≈
F F(x)
∫ x
2 (t) dt. A 1 4.29 A 2 12.75 A 3 0.36
A 4 1.79 A 2 x 0.5 x 2
6.06
F
F
6
15
4
2
y = g(t)
10
5
A 2
A 1
-2
1 2 3 4
A 4
5 6
A 3
-4
-1 1 2 3 4 5 6
-5
-10
F

( ) 4πt
R(t) 2 + 5 sin
25
S(t)
15t
1 + 3t
R(t) S(t) t
0 ≤ t ≤ 6 t 0
0 ≤ t ≤ 6
Y(x)
x
t 4
0 ≤ t ≤ 6 t
c(h) h
h 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c 925 875 830 780 730 685 635 585 535 490 440
m h y m(h)
h
m(h)
m
m(h)
M(h)
h
M(6000) M(10000)

u
F f
∫ b
a
f (x) dx F(b) − F(a).
∫ 1
x 3 − √ x + 5 x dx,
0
f (x) x 3 − √ x + 5 x .
F f F(x) 1 4 x4 − 2 3 x3/2 +
ln(5) 1 5x
∫ 1
x 3 − √ x + 5 x dx 1
0
4 x4 − 2 3 x3/2 + 1
1
ln(5) 5x
0
( 1
4 (1)4 − 2 3 (1)3/2 + 1 ) (
ln(5) 51 − 0 − 0 + 1 )
ln(5) 50
− 5 12 + 4
ln(5) .
f
∫ b
a
f (x) dx
G(x) ∫ x
a
f (t) dt f

u
x f u(x)
d f (u(x)) f ′ (u(x)) · u ′ (x).
dx
c(x) f (u(x)) f
u
(x)
′ (x)
(x) e 3x
h(x) sin(5x + 1)
p(x) arctan(2x)
q(x) (2 − 7x) 4
r(x) 3 4−11x
m M
m(x) e 3x
v(x) (2 − 7x) 3
n(x) cos(5x + 1)
w(x) 3 4−11x
s(x) 1
1+4x 2
a(x) cos(πx)
b(x) (4x + 7) 11
c(x) xe x2
+C

h(x) f (u(x)),
f u(x)
h(x) (5x − 3) 6 ,
f f (u) u 6 u(x) 5x − 3
f F(u) 1 7 u7 + C h
H(x) 1 7 (5x − 3)7 · 1
5 + C 1
35 (5x − 3)7 + C.
1 5
u ′ (x) 5 H ′ (x)
H ′ (x) 1
35 · 7(5x − 3)6 · 5 (5x − 3) 6 h(x),
H h
h(x) f (ax + b) F f
h
H(x) 1 F(ax + b) + C.
a
d f (x)
dx
f (x) x
∫
f (x) dx
f x
h(x) (5x − 3) 6 h
H
∫
(5x − 3) 6 dx 1 35 (5x − 6)7 + C.

d
dx
[□] x □
∫ □ dx x □
∫
sin(8 − 3x) dx
∫
sec 2 (4x) dx
∫ 1
11x−9 dx
∫
csc(2x + 1) cot(2x + 1) dx
∫ 1
√
1−16x 2 dx
∫ 5 −x dx
u
(x) xe x2 h(x) e x2 ?
+C
∫
d x
5 5x 4 ,
dx
5x 4 dx x 5 + C.
d f ((x)) f ′ ((x)) · ′ (x).
dx
∫
f ′ ((x)) ′ (x) dx f ((x)) + C.
f ′ ((x)) ′ (x) f
(x)

′ (x) f ′ ((x)) ′ (x)
u (x) u (x)
du
dx ′ (x) du ′ (x) dx
u
∫
∫
f ′ ((x)) ′ (x) dx f ′ (u) du.
f ′
u
f ′ ((x)) ′ (x) u u
∫
x 3 · sin(7x 4 + 3) dx
x 3·sin(7x 4 +
3) sin(7x 4 + 3)
x 3 (7x 4 + 3)
x 3 (7x 4 + 3)
f (u) sin(u)
u
u sin(7x 4 + 3) u
7x 4 + 3, du
dx 28x3 . du 28x 3 dx
x 3 dx 1
28
du
∫
sin(7x 4 + 3) · x 3 dx,
u x u 7x 4 + 3 1
28
du
x 3 dx
∫
∫
sin(7x 4 + 3) · x 3 dx
sin(u) ·
1
28 du.
u
u 7x 4 + 3
∫
∫
sin(7x 4 + 3) · x 3 1
dx sin(u) ·
28 du
du
dx ≈ Δu
du
Δx
dx ′ (x)
′ (x) ≈ Δu
Δx Δu Δu ≈ ′ (x)Δx
du ′ (x) dx

1 ∫
28
sin(u) du
1 (− cos(u)) + C
28
− 1 28 cos(7x4 + 3) + C.
[
d
− 1
]
dx 28 cos(7x4 + 3) − 1
28 · (−1) sin(7x4 + 3) · 28x 3 sin(7x 4 + 3) · x 3 ,
u
sin(7x 4 + 3) x 3 x 2
x 4
u u x 2 du 2xdx
∫ ∫
xe x2 dx e u · 1
2 du
1 ∫
e u du
2
1 2 e u + C
1 2 e x2 + C.
∫
e x2 dx,
x e x2 u u x 2
u
u du
u
u x

x
∫ x 2
5x 3 +1 dx
∫
e x sin(e x ) dx
∫ cos( √ x)
√ x
dx
u
u
f ((x)) ′ (x)
∫ 5
2
xe x2 dx.
∫ 5
2
xe x2 dx
∫ x5
x2
xe x2 dx.
u
u x 2
du 2xdx x 2 u 2 2 4 x 5 u 5 2 25.
u
∫ x5
x2
∫ u25
xe x2 dx e u · 1
u4 2 du
1 2 e u u25
u4
1 2 e25 − 1 2 e4 .
∫ xe x2 dx,
1 2 e x2 u
∫ 5
2
xe x2 dx 1 2 e x2
5
2
1 2 e25 − 1 2 e4 ,

u
∫ 2
1
x
1+4x 2 dx
∫ 1
0 e−x (2e −x + 3) 9 dx
∫ 4/π
2/π
cos( 1 x )
x 2
dx
∫ f (x) dx
f f F
F ′ f
∫
d
dx [F(x)] f (x) f (x) dx F(x) + C.
f F F f
u
∫
f ((x)) ′ (x) dx u (x) du ′ (x) dx
∫
∫
f ((x)) ′ (x) dx
f (u) du.
x u
(x)
′ (x)
′ (x)
∫
t 3 t 4 − 3 3
dt
F(x) f (x)
f (x) 2x 3 sin(x 4 )
F(x)

∫ ln 7 (z)
dz
z
∫
e 5x
dx
5 + e5x
∫ 7e
2 √ y
√ dy y
∫ 3π
5π/2
e sin(q) · cos q dq
sin(x) cos(x)
∫ tan(x) dx tan(x)
sin(x)
cos(x)
u
∫ tan(x) dx
∫ cot(x) dx
∫ sec 2 (x) + sec(x) tan(x)
dx.
sec(x) + tan(x)
u sec(x) + tan(x)
∫ sec(x) dx
∫ csc(x) dx
∫ x √ x − 1 dx.
x
√
x − 1 u x − 1 x dx u
x u
√ u u 1/2
u
∫ x 2√ x − 1 dx ∫ x √ x 2 − 1 dx

∫ sin 3 (x) dx
u sin(x)
sin 2 (x) + cos 2 (x) 1
sin 3 (x) sin(x)·sin 2 (x)
sin(x)
u cos(x)
∫ sin 3 (x) dx
∫ cos 3 (x) dx
r(t) 4 + sin(0.263t + 4.7) + cos(0.526t + 9.4).
t r
t
r(t)
∫ 24
0
r(t) dt

∫ x sin(x) dx ∫ xe x dx
u dv
u
∫ x 3 sin(x 4 ) dx
u
u
∫
x sin(x) dx.
f (x) x (x) sin(x)
∫ x sin(x) dx
f
x
d f (x) · (x) f (x) · ′ (x) + (x) · f ′ (x).
dx

(x)
′ (x)
(x) x sin(x)
h(x) xe x
p(x) x ln(x)
q(x) x 2 cos(x)
r(x) e x sin(x)
∫ xe x + e x dx
∫ e x (sin(x) + cos(x)) dx
∫ 2x cos(x) − x 2 sin(x) dx
∫ x cos(x) + sin(x) dx
∫ 1 + ln(x) dx
∫
x cos(x) dx.
d
[x sin(x)] x cos(x) + sin(x).
dx
∫ ( ) ∫
∫
d
dx [x sin(x)] dx x cos(x) dx + sin(x) dx.
∫
x cos(x) dx

d f (x)(x) f (x) ′ (x) + (x) f ′ (x).
dx
x
∫
∫
∫
d f (x)(x) dx f (x) ′ (x) dx + (x) f ′ (x) dx.
dx
∫
∫
f (x)(x) f (x) ′ (x) dx + (x) f ′ (x) dx.
f f (x) ′ (x) (x) f ′ (x)
∫
∫
f (x) ′ (x) dx f (x)(x) − (x) f ′ (x) dx.
u v u f (x)
v (x) du f ′ (x) dx dv ′ (x) dx
∫
∫
udv uv −
vdu.
u dv dv v ∫ vdu
∫ udv
∫
x cos(x) dx

u dv
u x dv cos(x) dx u cos(x) dv xdx
∫ u dv ∫
vdu udv
u x dv cos(x) dx
du 1 dx v sin(x)
∫
∫
x cos(x) dx x sin(x) − sin(x) · 1 dx.
∫ sin(x) · 1 dx.
∫
x cos(x) dx x sin(x) − (− cos(x)) + C x sin(x) + cos(x) + C.
∫ udv ∫ vdu
u v
∫ x cos(x) dx
∫ sin(x)·1 dx x
cos(x)
+C
∫
te −t dt
∫
4x sin(3x) dx
∫ z sec 2 (z) dz
∫ x ln(x) dx
u cos(x) dv xdx du − sin(x) dx
v 1 2 x2 ∫ x cos(x) dx 1 2 x2 cos(x) − ∫ 1
2
x 2 (− sin(x)) dx
x cos(x) 1 2 x2 sin(x)

∫
arctan(x) dx.
arctan(x) arctan(x)·1
arctan(x) 1
u arctan(x) dv 1 dx
∫ t 3 sin(t 2 ) dt
sin(t 2 )
t 3 t
sin(t 2 ) u
t 3 t · t 2
∫
t · t 2 · sin(t 2 ) dt,
z t 2 dz
2t dt tdt 1 2
dz z z
u z
∫
∫
t · t 2 · sin(t 2 ) dt z · sin(z) · 1
2 dz.
∫ arctan(x) dx u
arctan(x) dv 1 dx
∫ ln(z) dz
z t 2 ∫ t 3 sin(t 2 ) dt
z
∫ s 5 e s3 ds
∫ e 2t cos(e t ) dt e 2t e t · e t .

∫ udv ∫ vdu
∫ t 2 e t dt u t 2 dv e t dt
du 2t dt v e t
∫
∫
t 2 e t dt t 2 e t − 2te t dt.
u 2t dv e t dt du 2 dt
v e t
∫
( ∫ )
t 2 e t dt t 2 e t − 2te t − 2e t dt .
∫ 2te t dt
∫
t 2 e t dt t 2 e t − 2te t + 2e t + C.
t 3 e t
∫ e t cos(t) dt
u e t cos(t) u cos(t) dv e t dt
du − sin(t) dt v e t
∫
∫
e t cos(t) dt e t cos(t) − e t (− sin(t)) dt,
∫
∫
e t cos(t) dt e t cos(t) +
e t sin(t) dt
∫ e t sin(t) dt
∫
e t cos(t) dt

u sin(t) dv e t dt du cos(t) dt v e t
∫
( ∫
e t cos(t) dt e t cos(t) + e t sin(t) −
e t cos(t) dt
)
∫ e t cos(t) dt
∫
e t cos(t) dt
∫
2
e t cos(t) dt e t cos(t) + e t sin(t),
∫
e t cos(t) dt 1 e t cos(t) + e t sin(t) + C.
2
C
∫
x 2 sin(x) dx
∫ t 3 ln(t) dt
∫
e z sin(z) dz
∫
s 2 e 3s ds
∫ t arctan(t) dt
t
2
1 − 1 .
1+t 2 1+t 2
u
∫ π/2
0
t sin(t) dt.
∫ t sin(t) dt −t cos(t) +
sin(t) + C,

∫ π/2
0
t sin(t) dt (−t cos(t) + sin(t)) π/2
0
(− π 2 cos( π 2 ) + sin( π )
2 ) − (−0 cos(0) + sin(0))
1.
uv
u t dv sin(t) dt
du dt v − cos(t)
∫ π/2
0
t sin(t) dt − t cos(t) π/2
−
0
∫ π/2
0
(− cos(t)) dt
− t cos(t) π/2
+ sin(t) π/2
0
0
(− π 2 cos( π 2 ) + sin( π )
2 ) − (−0 cos(0) + sin(0))
1.
u
∫
∫
e x2 dx x tan(x) dx,
u
e x2 x tan(x)
F(x) ∫ x
0 e t2 dt
f (x) e x2 G(x) ∫ x
0
t tan(t) dt (x) x tan(x)
F
G u

∫ x sin(x) dx ∫ x ln(x) dx
∫ f (x) ′ (x) dx u f (x)
dv ′ (x) dx
∫
∫
udv uv − vdu
∫ f (x) ′ (x) dx ∫ vdu
∫
f ′ (x)(x) dx
u dv
∫ vdu
∫ udv
ln(x)
u dv
∫ x sin xdx
∫ x 2
dx
1+x 3
∫ x 2 e x3 dx
∫ x 2 cos(x 3 ) dx
∫ √
1
dx 3x+1
∫
3x cos(2x) dx +C
∫
(z + 1) e 4z dz

∫ 4
0
te −t dt
f (t) te −2t F(x) ∫ x
0
f (t) dt
F ′ (x)
F
F x > 0
∫ e 2x cos(e x ) dx
e 2x e x · e x z e x
z
∫ e 2x cos(e 2x ) dx
∫ e 2x sin(e x ) dx
∫ e 3x sin(e 3x ) dx
∫ xe x2 cos(e x2 ) sin(e x2 ) dx
u
u e 3x du 3e 3x dx
∫ x 2 cos(x 3 ) dx
∫ x 5 cos(x 3 ) dx x 5 x 2 · x 3
∫ x ln(x 2 ) dx
∫ sin(x 4 ) dx
∫ x 3 sin(x 4 ) dx
∫ x 7 sin(x 4 ) dx

∫ √ a 2 + u 2 du
u
∫ xe x2 dx u
∫
xe x dx ∫ e x2 dx
e x2
F(x) F ′ (x) e x2
u
u
∫
x 2 sin(x 3 ) dx ∫ ∫ ∫
x 2 sin(x) dx sin(x 3 ) dx x 5 sin(x 3 ) dx
∫ 1
1+x 2 dx
∫
x
1+x 2 dx
∫ 2x+3
1+x 2 dx
∫
e x
1+(e x ) 2 dx

∫ x ln(x) dx
∫ x √ 1 − x 2 dx
∫ ln(x)
x
∫
dx
1
√
1−x 2 dx
∫
ln(1 + x 2 ) dx ∫ x ln(1 + x 2 ) dx
∫
√
x
dx ∫
1−x 2
1
x √ 1−x 2 dx
R(x)
5x
x 2 −x−2
∫
5x
x 2 − x − 2 dx.
R
A
x−2 + B
x+1
.
5x
(x − 2)(x + 1) A
x − 2 +
B
x + 1 .
(x − 2)(x + 1)
5x A(x + 1) + B(x − 2).
x
x A B x −1
5(−1) A(0) + B(−3),
B 5 3 x 2
5(2) A(3) + B(0),
A 10 3
.
∫
5x
x 2 − x − 2 dx ∫ 10/3
x − 2 + 5/3
x + 1 dx.
∫
5x 10
x 2 dx
− x − 2 3 ln |x − 2| + 5 ln |x + 1| + C.
3
R(x) P(x)
Q(x)
P
Q
A
x − c ,
A
(x − c) n , Ax + B
x 2 + k
,
Ax + B
(x 2 + k) n
P Q R

A B c k
5x
x 2 − x − 2 10
3(x − 2) + 5
3(x + 1) .
∫
1
x 2 −2x−3 dx 1
x 2 −2x−3 1/4
x−3 − 1/4
x+1
∫ x 2 +1
∫
x 3 −x 2 dx x2 +1
x 3 −x 2 − 1 x − 1 x 2 + 2
x−1
x−2
x−2
dx
x 4 +x 2 x 4 +x 2
1 x − 2 x 2 + −x+2
1+x 2
√ a 2 ± w 2 √ w 2 − a 2 .
∫
∫
1
√
1 − x 2 dx,
x
√
1 − x 2 dx, ∫ √1
− x 2 dx.

arcsin(x) + C
u u 1 − x 2
u
(h)
∫ √a 2 − u 2 du u 2
√
a 2 − u 2 + a2
2 arcsin u a + C.
a 1 u x du dx
∫ √1
− x 2 dx x √
1 − x
2
2 − 1 arcsin x + C.
2
u
u x
∫ √9
+ 64x 2 dx.
a 3 u 8x
+ du 8dx dx 1 8 du
∫ √9
∫ √9
+ 64x 2 dx + u2 · 1
8 du 1 ∫ √9
+ u
8
2 du.
∫ √9
+ 64x 2 dx 1 ( u √
u
8 2
2 + 9 + 9 2 ln |u + √ u 2 + 9| + C)
1 8
( 8x
2
√
64x 2 + 9 + 9 2 ln |8x + √ 64x 2 + 9| + C)
.
1 8
u
∫ √
x 2 + 4 dx
∫
x
√
x 2 +4 dx
∫
2
√
16+25x 2 dx
∫ 1
x 2√ 49−36x 2 dx

ax 2 +bx+c
0 x a b c
x −b±√ b 2 −4ac
2a
∫
x 2 dx x3
3 + .
∫
1
16 − 5x 2 dx,
1
16 x − 5 3 x3 .

1
16−5x 2
∫ 1
dx,
16−5x 2
∫
√ √
1
5 5
16 − 5x 2 dx 20 arctanh( 4 x).
∫
1
16 − 5x 2 dx 1 (
8 √ √ √ √ √ )
log(4 5 + 5 x) − log(4 5 − 5 x) + .
5
∫
e −x2 dx,
∫
e −x2 dx
√ π
erf(x) + .
2
(x)
erf(x) √ 2 ∫ x
e −t2 dt.
π
∫
(1 + x)e x√ 1 + x 2 e 2x dx
0

u xe x du (1 +
x)e x dx
∫
(1 + x)e x√ 1 + x 2 e 2x dx
∫ √1
+ u 2 du,
Bx+C
x 2 +k
k
A
(x−c) n
n
∫ √ a 2 + u 2 du a
∫
2x
x 2 − 25 dx
∫
2x
x 2 −25 dx ∫ dx + ∫ dx
+C

+C
w x 2 − 25
∫
2x
x 2 −25 dx ∫ dw +C
+C
w
+C
∫
1
(x + 6)(x + 8) dx
∫
7x + 3
x 2 − 3x + 2 dx
∫ 6x 3 + 8x 2 + 2x + 6
x 4 + 1x 2 dx
a
b
c
a
x 2 + b x + cx + d
x 2 + 1
d
+C
25x − 10x 2 − 45
(x − 5)(x 2 + 9) A
x − 5 + Bx + C
x 2 + 9
A B C
∫ 25x − 10x 2 − 45
(x − 5)(x 2 + 9) dx

∫ x 3 +x+1
dx
x 4 −1
∫
x 5 +x 2 +3
x 3 −6x 2 +11x−6 dx
∫ x 2 −x−1
(x−3) 3
dx
u
∫ 1
x √ dx
9x 2 +25
∫ √
x 1 + x 4 dx
∫
√ e
x
4 + e 2x dx
∫
tan(x)
√9−cos 2 (x) dx
∫ √ x + √ 1 + x 2
x
dx.
u
u
u dv

∫ 1
0 e−x2 dx
n
f [a, b] L n
R n M n
L n f (x 0 )Δx + f (x 1 )Δx + ···+ f (x n−1 )Δx
R n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx
M n f (x 1 )Δx + f (x 2 )Δx + ···+ f (x n )Δx
∑n−1
i0
f (x i )Δx,
n∑
f (x i )Δx,
i1
n∑
f (x i )Δx,
i1
x 0 a x i a + iΔx x n b Δx b−a
n
x i
(x i−1 + x i )/2
n
f (x) 1
20 (x −4)3 +7 [1, 8]

y = f (x)
y = f (x)
y = f (x)
1 LEFT 8
1 RIGHT 8
1 MID 8
y f (x) [1, 8]
L n R n M n
f (x) x y
∫ 3
0 x2 dx
f (x) x 2
x 0 3 L 3
R 3 M 3
I
∫ 3
0 x2 dx

L 3 R 3 M 3
b 1 b 2 h
f (x) x 2 [0, 3]
∫ b
a
f (x) dx
n
L n R n M n
y f (x)
y = f (x)
D 1 D 2 D 3
x 0 x 1 x 2 x 3
∫ b
a
f (x) dx
a x 0 b x 3

D 1
y f (x) [x 0 , x 1 ]
f (x 0 ) f (x 1 )
x 1 − x 0 Δx b−a
3
D 1 1 2 ( f (x 0) + f (x 1 )) · Δx.
D 2 D 3 T 3
∫ b
a
f (x) dx
T 3 D 1 + D 2 + D 3
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.
T 3 1 2
Δx
f f
T 3 1 ( f (x0 ) + f (x 1 ) + f (x 2 )) Δx + ( f (x 1 ) + 1 f (x2 ) + f (x 3 )) Δx.
2
2
L 3
L 3 f (x 0 )Δx + f (x 1 )Δx + f (x 2 )Δx R 3 f (x 1 )Δx + f (x 2 )Δx +
f (x 3 )Δx L 3 R 3
T 3 1 2 [L 3 + R 3 ] .
T n ∫ b
a
f (x) dx n
[ 1
T n
2 ( f (x 0) + f (x 1 )) + 1 2 ( f (x 1) + f (x 2 )) + ···+ 1 2 ( f (x n−1) + f (x n ))]
Δx.
∑n−1
i0
1
2 ( f (x i) + f (x i+1 ))Δx.
T n 1 2 [L n + R n ] .

∫ 2 1
1
dx
x 2
∫ 2
1
1
x 2 dx
∫ 2 1
1
dx T
x 2 4 M 4 T 8 M 8
E T,4
E T,4
∫ 2
1
1
x 2 dx − T 4.
∫ 2
1
E M,8
1 x 2 dx − M 8.
E T,4 E T,8 E M,4 E M,8
f (x) 1 x 2
f
n L n R n M n
∫ b
a
f (x) dx
L n R n n
n →∞ Δx b−a
n
→ 0
n
n

T 1 M 1 M 1
∫ b
a
f (x) dx
T 1
M 1
M 1
M 1
∫ b
a
f (x) dx

n
f [a, b] T n ∫ b
a
f (x) dx M n
∫ b
a
f (x) dx f
M n T n M 1 T 1
f
f (x) 1− x 2 ∫ 1
0
f (x) dx
∫ 1
(1 − x 2 ) dx x − x3
1
3
2 0
3 .
0
M 4 M 8 T 4 T 8
E M,4 E M,8 E T,4 E T,8
∫ 2
1
∫ 1
1
x 2 dx
0 (1 − x2 ) dx 0.6
dx 0.5
x 2
T 4 0.65625 −0.0104166667 0.5089937642 0.0089937642
M 4 0.671875 0.0052083333 0.4955479365 −0.0044520635
T 8 0.6640625 −0.0026041667 0.5022708502 0.0022708502
M 8 0.66796875 0.0013020833 0.4988674899 −0.0011325101
T 4 M 4 T 8 M 8
∫ 1
0 (1 − x2 ) dx ∫ 2
1
dx
1
x 2
∫ 2
1
1
f [a, b] E T,4 ∫ b
a
f (x) dx − T 4
n 4 E T,4
T n M n n 4 n 8
T n M n
E M,4 ≈− 1 2 E T,4 E M,8 ≈− 1 2 E T,8

E M,n ≈− 1 2 E T,n.
T n 1 2 (L n + R n ).
[a, b]
L n R n ∫ b
a
f (x) dx
L n R n
L n R n
M n T n
M n T n
S 2n 2M n + T n
3
.
S 2n
S 2n S n n
n 2n
S 8 2M 4+T 4
3
S 16 2M 8+T 8
3

∫ 1
∫ 2
1
1
0 (1 − x2 ) dx 0.6
dx 0.5
x 2
T 4 0.65625 −0.0104166667 0.5089937642 0.0089937642
M 4 0.671875 0.0052083333 0.4955479365 −0.0044520635
S 8 0.6666666667 0 0.5000298792 0.0000298792
T 8 0.6640625 −0.0026041667 0.5022708502 0.0022708502
M 8 0.66796875 0.0013020833 0.4988674899 −0.0011325101
S 16 0.6666666667 0 0.5000019434 0.0000019434
S 8 S 16
S 16 ∫ 2 1
1
dx
x 2
E S,16 0.0000019434 L 8 0.5491458502
E L,8 −0.0491458502
L n R n M n
n T n S 2n
∫ 1
0 (1 − x2 ) dx
∫ 1
0 e−x2 dx
e −x2
v
t 0 0.3 0.6 0.9 1.2 1.5 1.8
v(t) 100 99 96 90 80 50 0
[0, 1.8]
[0, 1.8] L 3 R 3 T 3

[0, 1.8] M 3
v
[0, 1.8]
0.3 0.6 0.9 1.2 1.5 1.8
t
L n R n T n M n S 2n
f (x) 2 − x 2 (x) 2 − x 3 h(x) 2 − x 4
[0, 1]
[0, 1] L 1 R 1
M 1 ∫ 1
0 f (x) dx ∫ 1
0 (x) dx ∫ 1
0
h(x) dx
T 1 S 2
∫ 1
0 f (x) dx ∫ 1
0 (x) dx ∫ 1
0
h(x) dx
f h L 1 R 1 M 1 T 1
S 2

2
2
2
1
1
1
f [a, b] L n ∫ b
a
f (x) dx
f [a, b] M n
L n R n T n M n
f
f
∫ 5
2 (5x3 − 2x 2 + 7x − 4) dx S 2n
n
∫ 1
0 e−x2 dx
L n R n M n
n
∫ b
a
f (x) dx
T n 1 2 (L n + R n )
S 2n 2n + 1
M n n T n n + 1
S 2n
n

S n 2M n+T n
3
∫ b
a
f (x) dx
S n
∫ 3
0 e x dx
∫ 3
0 e x dx
n 4 n 2
n n 2 n 4
f (x)
∫ 3
0
f (x) dx

f
f (x) (x)
f
f
f
f
∫ b
a
f (x) dx
n 2

n 2
∫ 1
0
x tan(x) dx
u
L 4 R 4 M 4 T 4 S 4
∫ 1
0
x tan(x) dx
f (x)
f [3, 6]
f [3, 6]
f [3, 6]
∫ 6
3 f (x) dx L 4 7.23 R 4 6.75 M 4 7.05
f [3, 6]
f [3, 6]
∫ 6
3
f (x) dx
t 0 10 20 30 40 50 60
r(t) 2000 2100 2400 3000 3900 5100 6500
M n n
M n
S n
n
1
60 S n 2000+2100+2400+3000+3900+5100+6500
7

x y
v
a b
∫ b
a
v(t) dt
f [a, b] ∫ b
a
f (x) dx
x x a x b
f (x) 5 − (x − 1) 2 (x)
4 − x
f
f
y f (x) x f

y (x) x f
f
6
4
2
1 2 3
f
f (x) (x − 1) 2 + 1 (x) x + 2
6
6
6
4
g
4
g
4
g
2
2
2
f
f
1 2 3
1 2 3
1 2 3
f (x) (x − 1) 2 + 1 (x) x + 2
[0, 3]
(x) x + 2 (0, 2) (3, 5)

y (x − 1) 2 + 1 y x + 2 x + 2 y
x + 2 (x − 1) 2 + 1 x + 2 x 2 − 2x + 1 + 1
x 2 − 3x x(x − 3) 0,
x 0 x 3 y x + 2
y
[0, 3]
∫ 3
(x + 2) dx 21
0
2 ,
f
∫ 3
0
[(x − 1) 2 + 1] dx 6.
A
∫ 3
0
(x + 2) dx −
∫ 3
0
[(x − 1) 2 + 1] dx 21 2 − 6 9 2 .
x
6
4
g
g(x) − f (x)
2
g
f
△x
f
x
1 2 3
f (x) (x − 1) 2 + 1 (x) x + 2
[0, 3]

(x) − f (x) Δx
A ((x) − f (x))Δx.
[0, 3]
A ≈
n∑
((x i ) − f (x i ))Δx,
i1
n →∞
A
∫ 3
0
((x) − f (x)) dx.
∫ 3
0
(x) dx −
∫ 3
0
f (x) dx
∫ 3
0
((x) − f (x)) dx.
y (x) y f (x) (a, (a)) (b, (b)) x
a ≤ x ≤ b (x) ≥ f (x) A ∫ b
a
((x) − f (x)) dx.
y √ x y 1 4 x
y 12 − 2x 2 y x 2 − 8
y f (x) cos(x) (x) sin(x)
x f

y x 3 − x y x 2
x y 2 − 1 y x − 1
x x y +1
y + 1 y 2 − 1 y 2 − y − 2 0
y −1 y 2 (0, −1)
(3, 2)
x
x −1 x 0
2
x = y 2 − 1
2
x = y 2 − 1
2
x = y 2 − 1
1
1
1
△y
-1
1 2 3
y = x − 1
-1
1 2 3
y = x − 1
-1
1 2 3
x = y + 1
x y 2 − 1 y x − 1
y
y y −1 y 2
x y + 1
x y 2 − 1 Δy
A [(y + 1) − (y 2 − 1)]Δy,
y [−1, 2]
A ≈
n∑
[(y i + 1) − (y 2 i
− 1)]Δy.
i1

A
∫ y2
[(y + 1) − (y 2 − 1)] dy.
y−1
y
y
Δy
A 9 2
Δx
x (y) x f (y) ((c), c) ((d), d) y
c ≤ y ≤ d (y) ≥ f (y)
A
∫ yd
yc
((y) − f (y)) dy.
x y 2 x 6 − 2y 2
x 1 − y 2 x 2 − 2y 2
x y x 2 y 2 − x
x y 2 − 2y y x

y
f
h
△y
x
x 0 x 1 x 2 x 3 △x
L slice
y f (x)
a x 0 b x 3
y f (x)
x a x b L
Δx L
h L
√
L ≈ h (Δx) 2 + (Δy) 2 .
(Δx) 2
√
L ≈ (Δx) 2 + (Δy) 2
√
( )
(Δx) 2 1 + (Δy)2
(Δx) 2
√
1 + (Δy)2
(Δx) 2 · Δx.
n →∞ Δx → 0 Δy
Δx → dy
dx f ′ (x)
√
L ≈ 1 + f ′ (x) 2 Δx.

n →∞
f [a, b] L
y f (x) x a x b
∫ b √
L 1 + f ′ (x) 2 dx.
a
y x 2 x −1 x 1
y √ 4 − x 2 −2 ≤ x ≤ 2
y xe 3x [0, 1]
y f (x) 0.1x 2 + 1
x y
y f (x)
t 4
t 0 (0, f (0))
x a x b
A ((x) − f (x))Δx,
A
∫ b
a
((x) − f (x)) dx.
Δx Δy
y
x

L y f (x) x a x b
∫ b √
L 1 + f ′ (x) 2 dx.
a
y x 1/2 y x 1/4 0 ≤ x ≤ 1
y 7 sin x y 10 cos x [0,π]
A
x + y 2 42 x + y 0
x y
f (x) 9 √ x 3 x 5 x 8
x y(y − 2) x −(y − 1)(y − 3)
[ π 4 , 3π 4 ]
x y 2 − y − 2 y 2x − 1
y mx y x 2 − 1 m
f (x) 1 − x 2 (x) ax 2 − a a
a f
f (x) 2−x 2 f
∫
[a, b] 1 b
b−a a
f (x) dx
f (x) 2 − x 2 [0, √ 2] r
y f (x) y r
[0, √ 2] y f (x)
y r y r y f (x)

y
x
x
y
V πr 2 h
r 2 Δx
V π · 2 2 · Δx.
2
y
x
3
x
Δx
Δx → 0
V
∫ 3
0
π · 2 2 dx.
∫ 3
0
4π dx 12π 12π

y f (x)
Δx
x
x 0 x 5
x
3
y
x
5
x
Δx
V 1 3 πr2 h.
(0, 2) (3, 2) x
y 3 − 3 5
x x 0 x 5
x

R
y 4 − x 2 x x
y 4 − x 2 x (−2, 0)
(2, 0) R x
x
x x −2
x 2
Δx
y 4 − x 2
V π(4 − x 2 ) 2 Δx,
r
h V πr 2 h
y = 4 − x 2
y
x
Δx
V
∫ 2
−2
π(4 − x 2 ) 2 dx.
V 512
15 π
x x

y r(x) [a, b]
x
V
∫ b
a
πr(x) 2 dx.
R y 4 − x 2 y x + 2 x
y 4 − x 2 y x + 2
y
x + 2 4 − x 2
x 2 + x − 2 0,
x −2 x 1 (−2, 0)
(1, 1)
R x
y
R(x)
x
r(x)
r(x) R(x)

x x −2
x 1 r(x) y x + 2
r(x) x + 2 R(x) y 4 − x 2
R(x) 4 − x 2
πR(x) 2 Δx − πr(x) 2 Δx π[R(x) 2 − r(x) 2 ]Δx,
V π[(4 − x 2 ) 2 − (x + 2) 2 ]Δx.
V
∫ 1
−2
π[(4 − x 2 ) 2 − (x + 2) 2 ] dx.
V 108
5 π
y R(x) y r(x) [a, b]
R(x) ≥ r(x) x [a, b]
x
V
∫ b
a
π[R(x) 2 − r(x) 2 ] dx.
S x y √ x x 4
S x
S y y √ x y 2
S x

S y √ x y x 3 S
x
S y 2x 2 + 1 y x 2 + 4 S
x
S y y √ x y 2
S y
y
y
Δy
y
R y √ x y x 4 y
x 1 (1, 1)
R y
y
R(y)
r(y)
x
r(y) R(y)
Δy

y y 0
y 1 y √ x
x y x x y 2 r(y)
y x 4 x
y x 4 √ y
V π[R(y) 2 − r(y) 2 ] π[ 4 √ y
2 − (y 2 ) 2 ]Δy.
y 0
y 1
V
∫ y1
y0
π [ 4√ y
2 − (y 2 ) 2] dy.
V 7
15 π
S y y √ x y 2
S y
S x y √ x x 4
S y
S y 2x y x 3
S x
S y 2x y x 3
S y
S x (y − 1) 2 y x − 1 S
y
y 0 x 0

S y x 2 y x y −1
(0, 0) (1, 1)
y
−1
y x 2 y x
r(x) r(x) x 2 + 1
R(x) x + 1
y
x
V π[R(x) 2 − r(x) 2 ]Δx π (x + 1) 2 − (x 2 + 1) 2 Δx.
V
∫ 1
0
π (x + 1) 2 − (x 2 + 1) 2 dx 7 15 π.
S y 2x y x 3
S y −2
S y 4
S x −1
S x 5

Δy y
x
x y
y e 2x , y 0, x −2, x 0 x
y x 6 y 1 y y
y x 6 y 1 y x
y x 6 y 1 y y −5
y 4
y x 2 , y 1
x −5
y x 2 , x y 2

f (x) 3 cos( x3
4
)
y x f (x) 0 R
f x y
f
R
R
R x
R y
y sin(x) y cos(x)
R y y sin(x) y cos(x)
x
R
R x
R y
R y 2
R x −1
R y 1 + 1 2 (x − 2)2 y 1 2 x2
x 0
R y −1
R y
R

v(t)
t
∫ b
n∑
v(t) dt ≈ v(t i )Δt,
a
v(t) t
i1
· .
∫ b
a
v(t) dt
c(x) 200 + 100e −0.1x
x
∫ 2
0 (200 + 100e−0.1x ) dx
c(x) · Δx
∫ 2
n∑
c(x) dx ≈ c(x i )Δx?
0
i1

∫ 2
0 c(x) dx ∫ 2
0 200 + 100e
−0.1x
dx
B
x
B(x) B(x) 0.5 + 1
(x+1) 2
B(x) · Δx
∫ 36
n∑
B(x) dx ≈ B(x i )Δx?
12
∫ 72
0
B(x) dx ∫ ( )
72
0 0.5 +
1
(x+1) dx
2
i1
3
3
1500
0.0023 3 .
3
d m V
d m , m d · V.
V
m d · V
·
A l · w

Δt
v(t)Δt v(t)
· f
Δx
f (x) Δx f (x)Δx
ft/sec
y
y = v(t)
y = f (x)
v(t)
sec
f (x)
x
△t
△x
v(t)Δt
f (x)Δx
b x 0
2 ρ(x)
x ρ(x)
x
Δx
2 ρ(x)
3
x

Δx
Δx
2 1Δx 3
≈ ρ(x)
3 · 1Δx 3 ρ(x) · Δx .
n∑ ∫ b
ρ(x i )Δx ≈ ρ(x) dx,
i1
0 b
x
ρ(x)
ρ(x) · Δx Δx
·
ρ(x)
M x a x b
M
∫ b
a
0
ρ(x) dx.
2
ρ(x) 2e −0.2x x
ρ(x)
3

x
ρ(x) 400+ 200 3
1+x 2
x
2
ρ(x) 1
25 (x − 15)2
z
n a 1 a 2 ... a n
a 1 + a 2 + ···+ a n
n
f [a, b]
[a, b]
∫
1 b
f (x) dx.
b − a a
,
3.3 + 3.7 + 2.7 + 2.7
4
3.1.

3.3 · 5 + 3.7 · 4 + 2.7 · 3 + 2.7 · 3
5 + 4 + 3 + 3
3.16.
x 0
x 1 0 x 2 6
x
x 1 0
x 2 2x 3 4 x 4 6 x
x
x 1 0 x 2 2 x 3 3 x 4 6
x 1 0 x 2 2 x 3 4 x 4 6
x
x 3 5 x
x 3 4
x
x

n m 1 ... m n
x 1 ... x n
x x 1m 1 + x 2 m 2 + ···+ x n m n
m 1 + m 2 + ···+ m n
.
ρ(x)
Δx x i m i
m i ≈ ρ(x i )Δx
n
x ≈ x 1 · ρ(x 1 )Δx + x 2 · ρ(x 2 )Δx + ···+ x n · ρ(x n )Δx
.
ρ(x 1 )Δx + ρ(x 2 )Δx + ···+ ρ(x n )Δx
x ≈
∑ n
i1 x i · ρ(x i )Δx
∑ n
i1 ρ(x i)Δx .
n →∞
ρ(x) x a x b
∫ b
a
xρ(x) dx
x ∫ b
a ρ(x) .
dx

x
x
ρ(x) 4 + 0.1x x 0
ρ x
M
x
p(x) 4e 0.020732x
x 0 ρ(x) p(x)
p(x) 4e 0.020732x
D V m m
D · V.
ρ(x)
x a x b
m
∫ b
a
ρ(x) dx.
m 1 ,...,m n
x 1 ,...,x n x
x
∑ n
i1 x im i
∑ n
i1 m i
.
ρ(x)
x a x b
x
∫ b
a
xρ(x) dx
∫ b
a ρ(x) .
dx

x
x
δ(x) 4 + 5x
Dx Δx
Σ
δ(x) 7 + sin(x) x x 0 x π.
δ(x) ( ( 350 2 + sin 4 √ )) x + 0.175 , x
0 ≤ x ≤ 20
Dx Δx
Σ
a ρ(x) 10e −0.1x x
ρ
a
x

ρ(x) 1
1+x 2 p(x) e −0.1x
f (x) ρ(x) + p(x)
∫ 10
0
xf(x) dx
y f (x) 2xe −1.25x + (30 − x)e −0.25(30−x)
x 0 ...30 y 0 ...3
x y
x
x y
x
x
x ρ(x)
0.6π f (x) 2 f
x 0
x 30

y = f (x)
y
y = v(t)
ρ(x)
f (x)
v(t)
t
△x
a
△x
b
a
△t
b
y f (x) y v(t)
y ρ(x)
A l · w
y f (x) x [a, b]
Δx A f (x)Δx
A
∫ b
a
f (x) dx.
y v(t) [a, b] v(t)

d r · t
r Δt v(t)
d v(t)Δt
Δt
d
∫ b
a
v(t) dt.
M D · V
x y ρ(x)
Δx M ρ(x)Δx Δx → 0
M
∫ b
a
ρ(x) dx.
A l·w d r·t
M D · V
W F · d 20 · 4 80 .
W F · d
d
B h
B(h)
B(0) 20 B(50) 12

h
B(h)
B(5)Δh Δh 2
h 5 h 7
h 5 h 7
B(22)Δh Δh 0.25
W B(h)Δh
h Δh
∫ 50
0
B(h) dh
W F · d F
h B(h)
12 + 8e −0.1h
h Δh
W B(h)Δh (12 + 8e −0.1h )Δh.
Δh
W
∫ 50
0
B(h) dh
∫ 50
0
(12 + 8e −0.1h ) dh.
W ∫ 50
0 (12 + 8e−0.1h ) dh ≈ 679.461

x F(x)
a b
W
∫ b
a
F(x) dx.
F(h) h
B(h)
h
h B(h) 25 + 15e −0.05h
h 0 h 100
x
F(x) kx k
F(1/3) 5 k

y +
(0,1.5)
Δx
(4,0.75)
x +
x

y
y f (x) (0, 1.5) (4, 0.75) x
f (x) 1.5−0.1875x
x
V π f (x) 2 Δx π(1.5 − 0.1875x) 2 Δx.
3
F 62.4 · V 62.4π(1.5 − 0.1875x) 2 Δx.
x
x
d x + 9.
W F · d 62.4π(1.5 − 0.1875x) 2 Δx · (x + 9),
x 0 x 4
Δx → 0
∫ 4
W 62.4π(1.5 − 0.1875x) 2 (x + 9) dx,
0
W
10970.5π

3
F ma m a
9.81 3 3
x + y +
3
3
3

P F , F P · A,
A
P F A
F PA
A
1 × 1 d
V 1 · 1 · d 3
62.4d
d P 62.4d 2
d
P 62.4d d
F PA
x

45
x
y +
x − 5 y = f (x)
△x
x + (25,30)
x
y f (x)
f y f (x) 45 − 3 5
x
A 2 f (x)Δx 2(45 − 3 5 x)Δx.
x
(x − 5)
P 62.4d
P 62.4(x − 5).
F PA
F P · A 62.4(x − 5) · 2(45 − 3 5 x)Δx.
x
x 5

x 30
F
∫ x30
x5
62.4(x − 5) · 2(45 − 3 x) dx.
5
F ≈ 1.248 × 10 6
x + y +
3

W F · d
F P · A
δ 1080 / 3 .
50 3
δ 62.4 3
3

1000 3
9.8 2
f (x) 3 cos( x3
4
)
y x f (x) 0 R
f x y x
y
x
y R
x
x
y R
x
h
3

t a t b t
∫ b
a
0.3e −0.3t dt.
∫ 3
2
0.3e −0.3t dt −e −0.3t
3
2
−e −0.9 + e −0.6
≈ 0.1422.
t 2 t 3
p(t) 0.25e −0.25t
t a
t b
∫ b
a
p(t) dt.

F(b) 0 b
F(b)
F(b)
lim b→∞ F(b)
b
∫ b
0
0.3e −0.3t dt
∫ b
lim 0.3e −0.3t dt,
b→∞ 0
∫ ∞
0
0.3e −0.3t dt.

y
y
b
t
···
t
p(t) 0.3e −0.3t [0, b]
b →∞···
∞
∫ ∞
1
∫
1
0
x 2 dx,
−∞
∫
1
∞
1 + x 2 dx,
−∞
e −x2 dx
∞
∫ ∞
0
f (x) dx
∫ ∞
∫ b
f (x) dx lim f (x) dx.
0
b→∞ 0
∫ b
0
f (x) dx
∫ ∞
1
1
x dx ∫ ∞
1
1
x 3/2 dx
∫ ∞
1
1
x dx
∫ 10 1
1 x dx ∫ 1000 1
1 x
dx
∫ 100000 1
1 x
dx

∫ b
1
b
∫ ∞
1
1
x 3/2 dx
∫ b
lim
b→∞ 1
1
x dx.
∫ 10
1
∫ 100000
1
∫ b
1
b
1
x
dx
1
dx ∫ 1000 1
x 3/2 1
dx
x 3/2
1
dx
x 3/2
∫ b
lim
b→∞ 1
1
dx.
x3/2 1
x 3/2 dx
y 1 x y 1 x
x 3/2
0 ...10 x
∫ ∞ 1
1 x dx ∫ ∞ 1
1
dx
x 3/2
x ≥ 1
f
[1, ∞] f (x) 1 1
x
f (x)
∫
x 3/2
b
lim b→∞ 1
f (x) dx
f (x) x ≥ a ∫ ∞
a
∫ b
lim f (x) dx
b→∞ a
∫ ∞
a
f (x) dx
f (x) dx
f lim x→∞ f (x) 0

f 0 x →∞ ∫ ∞
a
f (x) dx
∫ ∞
1
∫ ∞
1
x 2 dx
0
e −x/4 dx
∫ ∞
2
9
(x+5) 2/3 dx
∫ ∞
4
3
(x+2) 5/4 dx
∫ ∞
0
xe −x/4 dx
∫ ∞ 1
1 x
dx p
p
∫ 1
0
1
√ x
dx,
f (x) √ 1
x
x 0 f [0, 1]
y
y
f (x)= 1 √ x
1
f (x)= 1 √ x
1
a
x
x
f (x) √ 1
x
[a, 1]
a → 0 +

∫ 1
√
1
0 x
dx 0 a a
∫ 1
∫
1
1
1
√ dx lim √ dx,
0 x a→0 +
a x
∫ 1
a
1
√ x
dx
∫ 1
∫ 1
0
0
∫
1
1
√ dx lim
x a→0 +
a
lim 2√ x 1
a→0 + a
1
√ x
dx
lim
a→0 + 2√ 1 − 2 √ a
2,
1
√ x
dx
∫ 3
1
1
(x − 2) 2 dx.
1
(x−2) 2
− 1
x−2
1 3
1
f (x)
(x−2) 2
y
y = 1
(x−2) 2
1 2 3
x
1
f (x)
x 2
(x−2) 2

−2
f (x) 1 x 2
(x−2) 2
∫ 3
1
∫
1
a
dx lim
(x − 2) 2 a→2 − 1
∫
1
3
dx + lim
(x − 2) 2 b→2 + b
1
(x − 2) 2 dx
a 2
∫ 2
1
∫
1
a
dx lim
(x − 2) 2 a→2 − 1
lim
a→2 − − 1
1
(x − 2) 2 dx
a
(x − 2)
1
lim − 1
a→2 − (a − 2) + 1
1 − 2
∞,
1
a−2 →−∞ a ∫ 2
∫ 3
2
∫ 3
1
1
1
1
(x−2) 2 dx
dx
(x−2) 2 1
dx
(x−2) 2
∫ 1
0
∫ 2
1
x 1/3 dx
0 e−x dx
∫ 4
1
1
√
4−x
dx
∫ 2
−2
1
x 2 dx
∫ π/2
0
tan(x) dx
∫ 1
0
1
√
1−x 2 dx
∫ b
a
f (x) dx a b ±∞
f x c c
a ≤ c ≤ b

∫ ∞
∫ b
f (x) dx lim f (x) dx,
a
b→∞ a
∫ ∞
1
1
x p dx
p
p > 1 0 < p ≤ 1
∫ 1 1
0 x
dx 0 < p < 1 p ≥ 1
p
∫ 3
−8
0 x √ x dx
∫ ∞
2
1x 2 e −x3 dx
∫ 1
−∞
e 5x
1+e 5x dx
∫ 1
−1
1
v dv
y 1
cos 2 (t)
t 0 t π/2
∫ ∞
e
∫ ∞
e
ln(x)
x
dx
1
x ln(x) dx

∫ ∞
e
∫ ∞
e
∫ 1
1
x(ln(x)) 2 dx
1
x(ln(x)) p dx p
ln(x)
x
0
dx
∫ 1
0
ln(x) dx
∫ ∞ 1
1
dx
1+x 3
1
1+x 3
x > 0 1 + x 3 > x 3
∫ b
1
1 + x 3 < 1 x 3 .
∫
1
b
1 1 + x 3 dx < 1
1 x 3 dx
b > 1 b →∞ ∫ ∞
1
1
1+x 3 dx
∫ ∞ 1
1
dx
x 3
∫ ∞ 1
1
dx
1+x 3
x 2 + x + 1 > x 2 x ≥ 1 ∫ ∞
∫ ∞ 1
1
dx
x 2
x > 1 ln(x) < x
∫ b
2
∫
1
b
x dx <
b > 2 ∫ b
√
x 4 +1
x 4
2
2
1
ln(x) dx
1
ln(x)
dx
1
1
dx
x 2 +x+1
> 1 x > 1
∫ √ ∞
1
1 x · x 4 + 1
x 4 dx

s(t)
s t v(t)
4t + 1
[0, 4]

s(4) t
4 s(4)
s(4)
s(0) 7 s(2)
s(0) 3 s(2)
v(t) 4t + 1
s(t) 2t 2 + t − 4
s(t)
v(t) 4t + 1
s(0) s(t)
dy
dx x sin x
y(x)
t
ds
dt
4t + 1.
A(t)
t
dA/dt 0.03A

dA
dt 0.03A.
ds
dt
4t + 1
t
s(t) 4t + 1
A
A(t)
◦
◦
m
ma m,
dv
dt ,
v 9.8

6
v
6
v
5
5
4
4
3
3
2
2
1
1
1 2 3
t
1 2 3
t
dv/dt v 0.5, 1.0, 1.5, 2.0 2.5
dv/dt

1.5
dv
dt
1.0
0.5
1 2 3 4 5
v
-0.5
-1.0
-1.5
dv/dt v 3.5, 4.0, 4.5 5.0

ds
dt
4t + 1,
s(t) s(t) 2t 2 + t
s(t) 2t 2 + t + 4
2x 2 − 2x 2x + 6 x 3
x □
2□ 2 − 2□ 2□ + 6.
x 3
3 □
2□ 2 − 2□ 2 · 3 2 − 2 · 3 12,
2□ + 6 2 · 3 + 6 12.
x 3
dv
dt
d□
dt
1.5 − 0.5v,
1.5 − 0.5□.
v(t) 3 − 2e −0.5t v
dv
dt d□
dt d dt [3 − 2e−0.5t ] −2e −0.5t · (−0.5) e −0.5t
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 .
dv
dt 1.5 − 0.5v t v 3 − 2e−0.5t

dv
dt
1.5 − 0.5v.
v(t) 1.5t − 0.25t 2
v(t) 3 + 2e −0.5t
v(t) 3
v(t) 3+ Ce −0.5t C
C v(t)
3 + Ce −0.5t
C
6
5
4
3
2
1
v
3
2
1
0
−1
−2
−3
1 2 3
t
dv
dt
1.5 − 0.5v
C v(0) v(0)
3 + C
v(t)
dv
1.5 − 0.5v, v(0) 0.5
dt

v
v v
dv
1.5 − 0.5v
dt
d 2 y
dt 2 −10y
dv
1.5 − 0.5v
dt
dv/dt
dy
1.5t − 0.5y,
dt

d 2 y
dx 2
9y
dy
dx 3y
d 2 y
dx 2
−9y
dy
dx −3y
y e −3x y e 3x
y e 3x
y 3 sin(x)
y sin(3x) y 3 sin(x)
y e −3x
y sin(3x)
k y cos(kt)
k
d 2 y
+ 4y 0.
dt2 A k
dy
dt
k(A − y)
T(t)
T t
dT
dt − 1 15 T + 5.
T(0) 105
dT
dt | T105
T t 0
t 1
dT/dt T

5
4
3
dT
dt
2
1
-1
30 60 90 120
T
-2
-3
T T T T
T(t) 75 + 30e −t/15
T(0) 105
P(t)
P
f (P)
dP
dt f (P)
dP
dt
P
1 2 3 4

P
P
P(0) 3
0 < P(0) < 1
1 < P(0) < 3
3 < P(0)
dy
dt y − t.
y(t) t + 1 + 2e t
y(t) t + 1
y(t) t + 2
h
d 2 h
dt 2 −kh
k
h(t) 4 sin(3t)

f a
f a
f (a) f ′ (a)
f (x) ≈ f (a) + f ′ (a)(x − a).
dy
dt
t − 2, y(0) 1.
y(t) t 0
t 0 −0.25 ≤ t ≤ 0.25
y(t)
t 1, 2, 3

y(t)
0 ≤ t ≤ 3
y(t)
y(t)
3
2
1
-1
-2
y
1 2 3
t
dy
dt
t − 2. t 0
dy/dt 0 − 2 −2
y
y
t 0 −2
3
2
1
y
-1
1 2 3
t
-2
t 0

t 1
dy/dt 1 −
2 −1
y
y t 1 −1
3
2
1
-1
-2
y
1 2 3
t
t 1
t 2 dy/dt 0
t 3 dy/dt 1
3
2
y
1
-1
-2
1 2 3
t
t 2 t 3

3
2
1
-1
-2
y
1 2 3
t
y(0) 1
3
2
1
-1
-2
y
1 2 3
t

3
y
3
y
2
2
1
1
-1
1 2 3
t
-1
1 2 3
t
-2
-2
3
y
3
y
2
2
1
1
-1
1 2 3
t
-1
1 2 3
t
-2
-2
y(0) 1
dy
dt
t−2
y(0)
dy
dt
t − 2

dy
dt −1 (y − 4).
2
dy
dt
y
y y y y
3
2
dy
dt
1
y
-1 1 2 3 4 5 6 7
-1
-2
y(0) 0 y(0) 2
y(0) 4 y(0) 6.
y(t) 4+2e −t/2
y(0) 6.
y(0) 4
7
6
5
4
3
2
1
y
-1 1 2 3 4 5 6 7
-1
t
dy
dt
− 1 2
(y − 4)
dy/dt f (y) dy/dt 0 f (y)

dy
dt
f (y) − 1 2
(y − 4)
f (y) − 1 2
(y − 4) 0
y 4
y f (y) 0 dy
dt
f (y)
dy
dt −1 y(y − 4).
2
dy
dt
y
y y y y
2
dy
dt
1
y
-2 -1 1 2 3 4 5 6
-1
-2
y(0)
−1, 0, 1,...,5
6
5
y
4
3
2
1
t
-1
-1
1 2 3 4 5 6 7
-2
dy
dt
− 1 2
y(y − 4)

y y
y lim t→∞ y(t) y
y
y
y(t)
y(0)
dy/dt
f (y) y f (y) 0
dy/dt f (y) y
y
y
dy
dt
= f (y)
dy
dt
= f (y)
y
y
y
y
dy
dt
y
dy
dt
y

y

y x
y x 0
y ′ 0
y ′
y ′ 2xy + 2xe −x2
y ′ 2 sin(3x) + 1 + y
y ′ 2y − 2
y ′ (2x + y)
−
(2y)
x ′ (t) x 4 − 5x 3 − 2x 2 + 24x + 0
dy
dt t − y.

y(0) −4, −3,...,4
y(0)
−1
t y
4
3
y
2
1
t
-4 -3 -2 -1
-1
1 2 3 4
-2
-3
-4
P(t) P
f (P)
dP
dt f (P)
dP
dt
P
1 2 3 4

P(0) > 0
P(0) > 1
P(0) < 1
P(t) P
t
dP
f (P) P(6 − P).
dt
f (P) P(6 − P)
P(t)
dP/dt
P
h
y(t) t
C
M(y) C
y
2 + y

M(y) C 1
M(y) y y
C 1
C 60

dy
dt 1 (y + 1), y(0) 0.
2
y(t) t 0
t 0
0 ≤ t ≤ 2
y(2)
t 2
7
6
5
4
3
2
1
y
t
1 2 3 4 5 6 7
y(2) y(2)
y(t) t 2

t 2
2 ≤ t ≤ 4
y(4) t 4
y(6)
Δt
m Δy
Δt
y
Δy
Δy mΔt.
Δt
t
dy
t − y, y(0) 1.
dt
y(t) t − 1 + 2e −t
(t i , y i ) y i ≈ y(t i ) Δt 0.2
y(0) 1 (t 0 , y 0 ) (0, 1)
Δt 0.2 (t 1 , y 1 ) t 1 t 0 + Δt 0.2
e

m dy
dt
0 − 1 −1.
(0,1)
Δt 0.2
Δy mΔt −1 · 0.2 −0.2.
y(0.2) ≈
y 1 y 0 + Δy 1 − 0.2 0.8
1.2
0.8
0.4
y
(t 0 ,y 0 )
(t 1 ,y 1 )
t
0.4 0.8 1.2
(t 1 , y 1 ) (0.2, 0.8)
m dy
dt
0.2 − 0.8 −0.6.
(0.2,0.8)
Δt t 2
t 1 + Δ 0.4
Δy −0.6 · 0.2 −0.12.
y 2 y 1 +Δy 0.8−
0.12 0.68 y(0.2) ≈ 0.68
1.2
0.8
0.4
y
(t 0 ,y 0 )
(t 1 ,y 1 )
(t 2 ,y 2 )
t
0.4 0.8 1.2

(t i , y i )
y(t) y(t)
1.2
y
1.2
y
0.8
0.8
0.4
0.4
0.4 0.8 1.2
t
0.4 0.8 1.2
t
(t i , y i )
t i y i dy/dt Δy
0.0000 1.0000
m dy/dt dy/dt
Δy y Δy
mΔt
t i y i dy/dt Δy
0.0000 1.0000 −1.0000 −0.2000
t i Δt y i Δy
t i y i dy/dt Δy
0.0000 1.0000 −1.0000 −0.2000
0.2000 0.8000

t i y i dy/dt Δy
0.0000 1.0000 −1.0000 −0.2000
0.2000 0.8000 −0.6000 −0.1200
0.4000 0.6800 −0.2800 −0.0560
0.6000 0.6240 −0.0240 −0.0048
0.8000 0.6192 0.1808 0.0362
1.0000 0.6554 0.3446 0.0689
1.2000 0.7243 0.4757 0.0951
Δt 0.2
dy
dt
2t − 1, y(0) 0
Δt 0.2 t i 0.2, 0.4, 0.6, 0.8
1.0 (t i , y i )
t i y i dy/dt Δy
0.0000 0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
0.2
-0.2
-0.6
y
0.4 0.8 1.2
t
-1.0
t i
y 5
∫ 1
0 (2t−
1) dt

y(0) 1
dy
dt
6y − y 2
8 y
t
6
4
2
2 4 6 8
y(0)
1
y(0) 1 Δt 0.2
t i 0.2, 0.4, 0.6, 0.8 1.0
(t i , y i )

t i y i dy/dt Δy
0.0 1.0000
0.2
0.4
0.6
0.8
1.0
8
6
4
2
y
0.4 0.8 1.2
t
Δt 0.2
y(0)
6
dy
dt
y, y(0) 1
t 0
y(t) e t y(1) e
Δt E Δt
Δt 0.2
y(1) ≈ E 0.2 2.4883
y(1) − E 0.2 e − 2.4883 ≈ 0.2300.
Δt

Δt E Δt
0.200 2.4883 0.2300
0.100 2.5937 0.1245
0.050 2.6533 0.0650
0.025 2.6851 0.0332
0.2
Error
0.1
0.1 0.2
Δt
Δt
Δt.
Δt
t
Δt
K
y(t) − E Δt ≈ KΔt
t
y(t) t Δt
Δt
y ′ −x − y
Δx 0.1 y x 1.4
y(1) 1 y(1.4) ≈
Δx 0.1 y x 2.4
y(1) 0 y(2.4) ≈

y ′ −3y y(0) 1.5
Δx 0.2 x 0.2, 0.4,...,1
x
y ≈
y 1.5e −3x y ′ −3y y(0) 1.5
B 1100 t 0
Δt 1 B(1) ≈
Δt 0.5 B(1) ≈
dB
dt 0.04B
Δt 0.25 B(1) ≈
B
T(t) T r
dT
dt −k(T − T r),
k
◦
◦
k 0.5 k 0.1
T r 70 + 10 sin t.
Δt 0.1
0 ≤ t ≤ 50

Δt E Δt
t
y(t) − E Δt ≈ KΔt
K
Δt
y(t) − E Δt KΔt.
dy/dt y y(0) 1
Δt 0.2 Δt 0.1
y(1) − 2.4883 0.2K
y(1) − 2.5937 0.1K.
y(1) K
y(1) y(1)
e 2.71828 ....
E 0.05 2.6533 E 0.025 2.6851
dy
dt
t − y, y(0) 0.
y(0.3) E 0.1 E 0.05
y(0.3) 0.0408
Δt
dy/dt y y(0) 1
Δt 0.2 0 ≤ t ≤ 0.2 t 0

y(0.2) ≈ y 1 1 + 1(0.2) 1.2
0 ≤ t ≤ 0.2
t 0 t 0.2
(1 + 1.2)/2 1.1
y(1) y 1 1 + 1.1(0.2) 1.22
t i y i (t i+1 , y i+1 )
0.0 1.0000 1.2000 1.1000
0.2 1.2200 1.4640 1.3420
0.4 1.4884 1.7861 1.6372
.
.
.
.
.
.
.
.
y(1) e
Δt 0.1 y(1)
Δt
Δt

dy
dt ty.
t y t
y dy/dt
y
1 dy
y dt t.
dy/dt
dy
dt t − y,
y + dy
dt t.
y dy
dt

(y) dy
dt
h(t)
dy
dt −3y
dy
ty − y
dt
dy
dt t + 1
dy
dt t2 − y 2
+C
∫
xdx x2
2 + C?
f
∫
∫
f ′ (x) dx xdx.
f
y
dy
dt t
y 2 .
y 2 dy
dt t.

t
∫
y 2 dy ∫
dt dt tdt.
t y dy dy
dt
dt
∫ ∫
y 2 dy tdt.
C C
y 3
3 t2
2 + C.
y 3 /3 t 2 /2
y t
y(t) 3 √
3
2 t2 + 3C.
3C
C
y(t) 3 √
3
2 t2 + C
C
C
dy
dt t , y(0) 2,
y2 y(t) 3 √
3
2 t2 + C C
C y(0) 2
2 y(0) 3 √
3
2 02 + C 3 √
C,
dy/dt

C 2 3 8
y(t) 3 √
3
2 t2 + 8.
dy
dt
(y) · h(t)
1 dy
(y) dt h(t),
t
y y t
dy
dt 3y.
1 dy
y dt 3.
t
∫
∫
1 dy
y dt dt
3 dt,
∫ 1
y dy ∫
3 dt.
ln |y| 3t + C.
y
|y| e 3t+C e 3t e C .
C e C
e x x y
+ −C
y(t) Ce 3t .

y 0
y y 0
C 0
P(t) t
P(t)
k
105 ◦
75 ◦
dT
dt
−k(T − 75),
k
k
t →∞
80 ◦
dy
dt
− (2 − t)y 2 − t

1 dy
t dt
e t2 −2y
y ′ 2y + 2 y(0) 2
y ′ 2y 2 y(−1) 2
dy
dt
−2ty
t 2 +1 y(0) 4
t
y
dy
dx x2 y (x, y) (1, 3)
dy
dt
0.5(y − 250)
y 70 t 0
y
y 5 t 1
y
dy
dt y2 (5 + t),
u
du
dt e2u+8t .

u(0) 13.
u
dy
dx 45yx4
y 5
y(x)
M(t) k
M 0 M(t)
k
dy
dt − t y , y(0) 8
t
y
y
h(t)
k

k −1 k −10
k
t
P(t)
( )
dP P
dt −P ln −P(ln P − ln 3).
3
P(t) 0 ≤ P ≤ 6
P(t) P(0) 1
P(t)
P(t) P(0) 6
P(t)

5 · 10 12
5 · 10 10
3 · 10 −8
P(t) P
t
P(t)
P(t)
5 · 10 10
3 · 10 −8
(
5 · 10 10 m3
)(
3 · 10 −8 m3
m 3
)
1.5 · 10 3 .
P(t)
P 5 · 10 12
P
.
5 · 1012 5 · 10 10
(
P m 3 )(
)
5 · 10 12 m 3 5 · 10 10 m3
P .
100
P
dP
dt 1.5 · 103 −
P
100
1
100 (1.5 · 105 − P).

P(t)
dP/dt P
dP/dt
P
1.5 · 10 5 P
t
dP
dt
P dP
dt
1
100 (1.5 · 105 − P)
dP
1
100 (1.5 · 105 − P)
dt
P 1.5 · 10 5
1.5 · 10 5
dP
dt 1
100 (1.5 · 105 − P), P(0) 0.
1 dP
1.5 · 10 5 − P dt 1
100 .
t
∫
∫
1 dP
1.5 · 10 5 − P dt dt
1
100 dt,
∫
∫
dP
1.5 · 10 5 − P
1
100 dt

− ln |1.5 · 10 5 − P| 1
100 t + C
−1
1.5 · 10 5 − P Ce −t/100 .
C P 0 t 0
1.5 · 10 5 − 0 Ce 0 C.
C 1.5 · 10 5
C
P
P
P(t) 1.5 · 10 5 (1 − e −t/100 ).
P
1.5 · 10 5
t
P(t)
dP
dt
1
100 (1.5 · 105 − P)
dP
dt
P
1.5 · 10 5

t 0
dP
dt − 1
100 P, P(0) 1.5 · 105 .
P(t)
1.5 · 10 5 e −t/100
T P(T) 0.75 · 10 5
0.75 · 10 5 1.5 · 10 5 e −T/100 ,
1
2 e−T/100 .
( ) 1
T −100 ln ≈ 69.3
2
A(t)
t
A t
A
M(t)
k
M 0

M(t)
k
M(t) M(t)
2860 0.04
6
y(0)
S
S ′ f (t, S)
S(t)
S(78)
50 1000 0.025
10

220
5 1100
t t
5
1510
320
58
100 88
k
0.05
S(t) k
S(t)
k 1820000 47
k

≈ 9.8
k
k
v(t)
k
S(t) t
t
S
S(t)
t
S(t)

dP
dt 1 2 P.
4
P
3
2
1
t
2 4
dP
dt
1 2
P

P(0) dP
dt
1 2 P
dP
dt 1 P(3 − P).
2
4
P
3
2
1
t
2 4
P(0)
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
dP
dt kP
k
P ′ (0)
t 0
P(0)
k
k
dP
dt
kP, P(0) 6.084.
k
dP
dt
kP k
k dP/dt
P .
k

P
k dP
dt
P t 0
P
0.015 per capita growth rate
0.014
0.013
0.012
0.011
0.010
6.0 6.2 6.4 6.6 6.8 7.0
P
0.015 per capita growth rate
2 4 6 8 10 12 14 16 18 20
0.03
per capita growth rate
0.014
0.013
0.012
0.011
0.010
6.0 6.2 6.4 6.6 6.8 7.0
0.02
0.01
P
-0.01
P

dP/dt
P
0.025 − 0.002P.
P
dP
dt
P(0.025 − 0.002P).
dP/dt P
dP
dt
P
0.10
dP
dt
0.05
2 4 6 8 10 12 14 16 18 20
P
-0.05
-0.10
dP
dt
P dP
dt
P(0.025 − 0.002P)
dP
dt
P(0.025 − 0.002P)
P 0
P 12.5
P(0) > 0
dP
dt
kP(N − P).

P 0 1 − P N
0 P N
P N
1 dP
P(N − P) dt k,
∫
∫
1
P(N − P) dP kdt.
1
P(N − P) 1 [ ]
1
N P + 1
.
N − P
∫ [ ]
1 1
N P + 1
N − P
∫
dP
kdt.
N 1 N
1
(ln |P| − ln |N − P|) kt + C.
N
N
ln P
N − P
kNt + C.
e C C C
P
N − P CekNt .
C P
P(0) P 0 C P 0
N−P 0
P
N − P P 0
N − P 0
e kNt .
P (N − P)(N − P 0 )
P(N − P 0 ) P 0 (N − P)e kNt
P 0 Ne kNt − P 0 Pe kNt .

P
P 0 Ne kNt P(N − P 0 ) + P 0 Pe kNt
P
P(N − P 0 + P 0 e kNt ).
P 0 Ne kNt
N − P 0 + P 0 e kNt .
1 P 0
e −kNt
P(t)
N
( N−P0
)
P 0
e −kNt + 1 .
dP
dt kP(N − P), P(0) P 0,
N
P(t) ( N−P0
)
P 0
e −kNt + 1 .
k 0.002, N 12.5, P 0 6.084.
P(t)
12.5
1.0546e −0.025t + 1 ,
15
12
P
9
6
3
t
40 80 120 160 200

N k P 0
dP
dt
kP(N − P)
dP
dt
P
dP
dt
N
P
N/2
dP
dt
P
P
P

N
P(t) ( N−P0
)
P 0
e −kNt + 1 .
P(0) P 0 lim t→∞ P(t) N
P dP/dt 3P − 3P 2
P
P(0)
P(0)
P(0)
P
P
P
P P

P P
P →
P
dP/dt P
dP/dt
P
dP/dt
P
dP/dt
P
dP/dt
P
dP/dt
P
P
P ′ (1980) ≈
P ′ (1985) ≈
P ′ (1990) ≈
P

P
L
L
75
P
1 + 316.75e −0.699t .
L
t k 2
P
P
dP
dt
− .
dP
dt aP2 − bP a, b > 0.
dP/dt P dP/dt
dP/dt < 0 P
dP/dt > 0 P
dP/dt
P b/a

P(0) > b/a P
P →
P(0) b/a P
P →
P(0) < b/a P
P →
p(t) t
dp
0.2p(1 − p)
dt
p
p(0) 0.1
p(t)
p(t)
p
b(t)
b t
db/dt
b
b
db
dt
dP
0.1P(10 − P),
dt
P t

1, 1, 2, 3, 5, 8,...
1, 3, 6, 10, 15, 21, 28, 36, 45, 55,...

$5000
8%
0.08
12 · P P
( ) 0.08
5000
12
$33.33 $5033.33
P n
n
(
P n 5000 1 + 0.08 ) n
12
0 $0.00 $5000.00
1 $33.33 $5033.33
2
3
4
5
6
7
8
9
10
11
12

s 1 , s 2 ,...,s n ...
s 1 , s 2 , s 3 ,...
f
s 1 s 2 s 3 ...
f f (n) s n n
s 1 , s 2 , s 3 ,...
{s n } s n s(n) n
n
s n n 1, 2, 3,... s n
(n, s n ) n
n
n
x →∞

{s n } n →∞
s n n 1, 1 2 , 1 3 ,... s n
x →∞ {s n } n →∞
s n n 2, 3 2 , 4 3 , 5 4 ,... s n
x →∞
{s n } n →∞
f L x →∞
{s n } n →∞
1+n
2+n
n →∞
{s n }
n {s n } n
L
lim
n→∞ s n L
{s n } {s n }
{s n } L s k
k ≥ n L n
n →∞
x →∞

1+2n
3n−2
5+3 n
10+2 n
10 n
n! ! n! n(n − 1)(n − 2) ···(2)(1)
n 0!
{s n } L
s k k ≥ n L n
n →∞
s n n(n + 1) − 1
s n 1/(n + 1)
s n 3 − 1/n
s n n sin(n)/(n + 1)
s n (n + 1)/n
±∞
∞
s n n ≥ 1
s n
n →∞
4n+8
n
4 n

4n+8
n 2
sin n
4n
f (t) sin t t
f (t) Δt
s n f (nΔt) f
sin(1/10) sin(2/10) sin(3/10)
f (t) (t − 0.5) 2 .
Δt 0.25
ln(n)
n
f
f (x) ln(x)
x
f (x) [0, 10]
{ }
ln(n)
n
lim x→∞ f (x) L
f (x) ∞ ∞
x
lim x→∞ f (x)
ln(n)
lim x→∞ f (x) lim n→∞ n
P r%
r
12
r
8% 0.08
12
P
( ) r
P
12
P
)
)
P 1 P + P P
( r
12
(
1 + r
12

P 1
P 2
(
P 2 P 1 1 + r ) (
P 1 + r ) 2
12 12
P 3
P
P n
P
P n
A
t A 2
t + 5
A 0 100
A 1 A 1
A 2 A 2
A 3 A 3
A 4 A 4
A n 5n A n
{A n }

f f (x) sin(4x) [0, 10]
f
1.0
0.5
2 4 6 8 10
x
-0.5
-1.0
f (x) sin(4x) [0, 10]
f [0, 10]
f
[0, 10]
f
f [0, 10]

n
Q(n)
(n + 1)
Q(1) 5 × 0.08
Q(2) (5 + Q(1)) × 0.08
Q(2) (5 × 0.08) (1 + 0.08)
Q(3) (5 + Q(2)) × 0.08
Q(3) (5 × 0.08) 1 + 0.08 + 0.08 2
Q(4) (5 + Q(3)) × 0.08
Q(4) (5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3
Q(n) n

Q(n) n
Q(n) Q(n)
n
Q(1) 0.40
Q(2)
Q(3)
Q(4)
Q(5)
Q(6)
Q(7)
Q(8)
Q(9)
Q(10)
Q(n) n
(5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3 + ···+ 0.08 n−1
a + ar + ar 2 + ···+ ar n−1
a 5 × 0.08 r 0.08 r
a r r 1
S n a + ar + ar 2 + ···+ ar n−1
S n n
S n r
rS n S n
S n − rS n a − ar n

S n S n
n
S n
S n a + ar + ar 2 + ···+ ar n−1
a r r 1 S n
S n a + ar + ar 2 + ···+ ar n−1 a(1 − rn )
1 − r
Q(n) (5 × 0.08) 1 + 0.08 + 0.08 2 + 0.08 3 + ···+ 0.08 n−1
Q(n) a 5 × 0.08 0.4 r 0.08
( 1 − 0.08
n )
Q(n) 0.4
1
1 − 0.08 2.3 (1 − 0.08n )
n 0.08 n
1
lim Q(n) lim
n→∞ n→∞ 2.3 (1 − 0.08n ) 1
2.3 ≈ 0.435
1
2.3
n
n
n
∞∑
a + ar + ar 2 + ··· ar n
n0
r
n + 1 ar n n ar n−1 r
N 0.121212

N 0.12 + 0.0012 + 0.000012 + ···
( ) ( )( ) ( )( ) 12 12 1 12 1 2
+
+
+ ···
100 100 100 100 100
a 12
100 r 1
100
r 1 a
S a + ar + ar 2 + ···ar n−1 + ···
n
S n a + ar + ar 2 + ···+ ar n−1
S n a 1 − rn
1 − r
n S n S
lim n→∞ r n |r| > 1 |r| < 1
|r| < 1 S n
S S
N
12
100 + 12
1
100 100 + 12
1
2
100 100 + ··· a
12
100 r 1
100
N 12 100
1
1 − 1
100
12
100
( ) 100
4
99 33
S n
∞∑
ar k
k0
∑n−1
S n a + ar + ar 2 + ···+ ar n−1 ar k
S n n
k0

∞∑
a + ar + ar 2 + ··· ar n
n0
a r r 0
n S n
S n a + ar + ar 2 + ···+ ar n−1
|r| < 1 S n
a 1−rn
1−r
S
S lim S n lim a 1 − rn
n→∞ n→∞ 1 − r a
1 − r
n 0
n 0
∞∑ ( ) 1 k ( ) ( ) 1 1 2 ( ) 1 3
(2) (2) + (2) + (2) + ···
3 3 3 3
k1
(2) 1
3
a r −1 < r < 1
∞∑
ar k ar 3 + ar 4 + ar 5 + ···
k3
ar 3
a r −1 <
r < 1 n
∞∑
ar k ar n + ar n+1 + ar n+2 + ···
kn

ar n
∞∑
ar k
a r r 0
∑ ∞
k0 ark n
k0
∑n−1
S n ar k
k0
n
S n a 1 − rn
1 − r
|r| < 1 ∑ ∞
k0 ark
a
1−r
7 th
−3, −16.5, −90.75, ...
1 + 1 3 + 1 9 + ... + 1 + ...
3n−1
∞∑ ( 3 n + 5 n )
9 n
n1
−16 + 4 − 4 4 + 4
16 − 4
64 + 4
256 −···
∑14
n
n3
1
2

1 $0.01 $0.01
2 $0.02 $0.03
3
4
5
6
7
8
9
10
n
n
h
3 4 h n
n h 0 h
h 1 h
h 2 h
h 3 h
h n h

H
t
H − 1 2 t2
H
1 6
( )( )( ) ( ) 5 5 1 5 2 ( ) 1
6 6 6 6 6
( )( )( )( )( ) ( ) 5 5 5 5 1 5 4 ( ) 1
6 6 6 6 6 6 6
( )
1 5 2 ( ) ( ) 1 5 4 ( ) 1
6 + + + ···
6 6 6 6
k
n
P

P
P
0.75P(1 + 0.75 + 0.75 2 + ···)
P
r
M
r
P r
M
P
r r
12
P

0.0r
12 P M
M − r
12
P
P 1
( ( ) ) ( r
P 1 P − M − P 1 + r )
P − M
12
12
P 1
P 1 r
P 2
P 2 ()P 1 − ()M
P 2
1 + r 2
12 P − 1 + 1 +
r
12 M
P 3
P 3
(
1 + r ) 3
P −
12
[
1 +
(
1 + r ) (
+ 1 + r ) 2
]
M
12 12
n
P n
(
1 + r ) n
P −
12
⎡
∑n−1
⎢
⎣ k0
(
1 + r ) k ⎤ ⎥⎥⎥⎥⎦
M
12
∑n−1
k0
(
1 + r ) k
12
P n
P n
P(t) t
(
P(t) P − 12M )(
1 + r ) 12t
+ 12M
r 12 r

M 0
M P(t) 0
P(t) 0 M
M
12t
rP 1 + r
( 12
12 1 +
r 12t
)
12 − 1

n
N 0.1212121212 ··· 12
100 + 12
100 · 1
100 + 12
100 · 1
100 2 + ···
N
N 4
33
e
e π ln(2)
f (x) e x
f x 0 e
L(x) L(1) L(1) ≈ f (1) e
e x e
e
e x x
f (x) f ′ (x)
x 0 P 2 (x) f (x) e x
x 0 P 2 (0) f (0) P ′ 2 (0) f ′ (0) P ′′
2 (0) f ′′ (0)
P 2 (x) 1+x+ x2
2 P 2(0) f (0) P ′ 2 (0) f ′ (0) P ′′
2 (0) f ′′ (0)
P 2 (x) e P 2 (1) ≈ f (1)

e
f
f x
P 3 (x) 1+x + x2
2 + x3
6 P 3(0) f (0) P ′ 3 (0) f ′ (0) P ′′
3 (0) f ′′ (0)
P ′′′
3 (0) f ′′′ (0) P 3 (x) e
P 2 (x)
e
e 2.5
e
1 + 1 2
1 + 1 + 1 2
2.5
1 + 1 + 1 2 + 1 6
2.6
1 + 1 + 1 2 + 1 6 + 1
24
2.7083
1 + 1 + 1 2 + 1 6 + 1
24 + 1
120
2.716
e
1 + 1 + 1 2 + 1 6 + 1
24 + 1
120 + ···+ 1 n!
n
e ≈ 1 + 1 + 1 2 + 1 6 + 1 24 + 1
120 + ···+ 1 n∑
n! 1
k!
n n
e
∞∑ 1
e
k!
k0
1
n!
0! 0! 1
k0

a 1 + a 2 + ···+ a n + ···
∞∑
a k
k1
a 1 a 2 ...
{a n } n≥1
∑
k≥1
a k
∞∑
a k
k1
a 1 +a 2 +···+a n +···
∞∑
k1
1
k 2
n n
2∑
k1
1
k 2

1∑
k1
2∑
k1
3∑
k1
4∑
k1
5∑
k1
1
k 2 1
1
k 2
1
k 2
1
k 2
1
k 2
6∑
k1
7∑
k1
8∑
k1
9∑
k1
10∑
k1
1
k 2
1
k 2
1
k 2
1
k 2
1
k 2
∑ ∞
k1
1
k 2
n S n ∑ n
k1
{S n }
{S n }
∑ ∞
k1
1
k 2
n
n ∑ ∞
k1 a k S n ∑ n
k1 a k
n S n n
S n a 1 + a 2 + ···+ a n
S 1 , S 2 ,...,S n ,...
∞∑ 1
k 2
k1
1
k 2

∞∑
k1
a k
{S n }
S n
n∑
a k
k1
lim n→∞ S n S S ∑ ∞
k1 a k
∞∑
k1
a k lim
n→∞
S n S
∞∑
k1
∞∑
a k
a k
k1
a k k
m
∞∑
a k (a 1 + a 2 + ···+ a m ) +
k1
∞∑
km+1
a 1 + a 2 + ···+ a m ∑ ∞
k1 a k
∑ ∞
km+1 a k
∑
a k
a k

∑ ∞
k0 ark a r r 1
n S n
S n 1 − rn
1 − r
|r| < 1
r
∑ a k
{a n }
∑ ∞
k1 a k L
n S n ∑ ∞
k1 a k
(n − 1) S n−1 ∑ ∞
k1 a k
n (n − 1)
∑ ∞
k1 a k
∑ ∞
k1 a k L lim n→∞ S n
lim n→∞ S n−1
lim n→∞ a n
lim n→∞ (S n − S n−1 )
∑ ∞
k1 a k {a k } k
{a k } n
∑ ∞
k1 a k
lim k→∞ a k 0 ∑ a k

∑ k
k+1
∑ (−1) k
∑ 1
k
∑ a k
lim k→∞ a k 0 ∑ a k
lim k→∞ a k 0
∞∑
k1
1
k
1∑
k1
2∑
k1
3∑
k1
4∑
k1
k1
1
k 1
1
k 1.5
1
k 1.833333333
1
k 2.083333333
6∑
k1
7∑
k1
8∑
k1
9∑
k1
k1
1
k 2.450000000
1
k 2.592857143
1
k 2.717857143
1
k 2.828968254
5∑ 1
∑10
k 2.283333333 1
k 2.928968254
∑ ∞
k1
∑ ∞ 1
k1 k
∑ 1000
n1 1 k ≈ 7.485470861
1
k

∑ ∞ 1
k1 k
n S n
∑ ∞ 1
k1 k n∑ 1
S n
k
k1
1 + 1 2 + 1 3 + ···+ 1 n
( ) ( ) ( )
1 1 1
1(1) + (1) + (1) + ···+ (1)
2 3 n
S n
1 m
1.00
0.75
0.50
a k
0.25
k
0.00
1 2 3 4 5 6 7 8 9 10
f f (x) 1 x
S 9
n
n ∑ ∞ 1
k1 k
∫ ∞ 1
1 x
dx
∫ ∞
1
1
x
dx
∑ ∞
k1
1
k
f

a k f (k) k ∑ ∞
k1 a k
n∑
S n
a k
k1
f (x)
a k
f
S n >
∫ n
1
f (x) dx
n
∞∑
a k >
k1
∫ ∞
1
f (x) dx
∫ ∞
1
f (x) dx ∑ ∞
k1 a k
1.00
a k
0.75
0.50
0.25
k
0.00
1 2 3 4 5 6 7 8 9 10
1.00
a k
0.75
0.50
0.25
k
0.00
1 2 3 4 5 6 7 8 9 10
f [1, n]
∫ ∞
∞∑
f (x) dx > a k
1
k2

∫ ∞
1
f (x) dx ∑ ∞
k2 a k ∑ ∞
f f x
c a k f (k) k
∫ ∞
c
∫ ∞
c
k1 a k
f (x) dx ∑ ∞
k1 a k
f (x) dx ∑ ∞
k1 a k
∑ 1
k
p
p
p p 1
p
∫ ∞ 1
1
dx ∑ ∞ 1
x 2
k1
k 2
∫ ∞ 1
1 x
dx p > 1 p
p
∑ ∞ 1
k1 k
p
∫ ∞ 1
x
dx p < 1
p
p ∑ ∞
k1
1
1
k
p
p ∑ ∞
k1
1
k p
p
∑ k 2 +1
k 4 +2k+2
p
∑ k+1
k 3 +2
k

k
k+1 k 100 k 1000
k 3 +2
k+1
k
k
k 3 +2 k 3
a k k+1
k 3 +2 b k k
k 3
a k
lim
k→∞ b k
a k b k k
∑ k
k 3
∑ k+1
k 3 +2
∑ a k ∑ b k
∑ a k
k a k b k
k ∑ a k ∑ b k
c
b k
lim c
k→∞ a k
b k ≈ ca k k
∑ ∑ ∑
b k ≈ ca k c a k
∑ a k ∑ b k
∑ a k ∑ b k
b k
lim c
k→∞ a k
c ∑ a k ∑ b k
∑ p(k)
q(k)
p(k) m q(k) l
∑ p(k)
q(k)
∑ k m
k l

∑ 3k 2 + 1
5k 4 + 2k + 2
∑ 1
k 2
∞∑
k1
2 k
3 k − k
∑ a k a k+1
a k
2k
3 k −k
∑ 2 k
3 k −k
a k
k
5
10
20
21
22
23
24
25
a k+1
a k
∑ 2 k
3 k −k
a k+1
a k
k

∑ 2 k
3 k −k
k ∑ 2 k
3 k −k
∑
ak
a k+1
≈ r
a k
k a k+1 ≈ ra k k ∑ a k
∑ ar k k r
−1 < r < 1 ∑ a k
a k+1
lim r
k→∞ a k
r |r| < 1
∑ a k
|a k+1 |
lim
k→∞ |a k |
0 ≤ r < 1 ∑ a k
1 < r ∑ a k
r 1
r
p
p
n

∑ k
2 k
∑ k 3 +2
k 2 +1
∑ 10 k
k!
∑ k 3 −2k 2 +1
k 6 +4
∞∑
a 1 + a 2 + ···+ a n + ···
a k k
n S n ∑ ∞
k1 a k n
n∑
S n a 1 + a 2 + ···+ a n a k
{S n } ∑ ∞
k1 a k
◦ ∑ ∞
k1 a k {S n }
∞∑
k1
k1
k1
a k lim
n→∞
S n
◦ ∑ ∞
k1 a k {S n }
a k
A n 80
8 n
∞∑
(A n )
n1
{A n }
∞∑
n1
10
n + 2 s n
s n
n∑
i1
10
i + 2 .

s 4 s 8
s 4
s 8
2n
a n
10n + 7 .
∞ −∞
∞∑ 2n
10n + 7
n1
{
2n
10n + 7
}
∫
∞ −∞
∞
3 dx
1 x 2 + 1
∞∑ 3
n 2
+ 1
n1
b n
n!
b
∑ 10 k
k!
∑ b k
k!
b
b n
n!
b n
n!
∑ a k a n → 0 n →∞
n√
an → r
n a n ≈ r n n
∑ a k n
∑ a k
r ∑ a k
n√ a n → r n →∞

∑ n
k0 a k ∑ n
k0 b k
n∑ n∑
a k + b k
k0 k0
n∑
(a k + b k )
k0
a n 1 + 1
2
b n n −1 n
∑ ∞
k0 a k ∑ ∞
k0 b k
∑ ∞
k0 (a k + b k )
∞∑ ∞∑ ∞∑
a k + b k (a k + b k )
k0
k0
∑ ∞
k0 a k ∑ ∞
k0 b k
∑ ∞
k0 (a k + b k )
∑ a k ∑ b k ∑ (a k + b k )
k0
∑ ∑ ∑
(a k + b k ) a k + b k
A n B n n ∑ ∞
k1 a k ∑ ∞
k1 b k
n∑
A n + B n (a k + b k )
k1
∞∑
(a k + b k )
k1
∞∑ ∞∑
a k + b k
k1 k1
∑ ∞ 2 k +3 k
k0
5 k

∑ 1
k 2 ∑ 1
k 2 + k
∑ 1
k 2 p p 2 > 1 ∑ 1
k 2
∑ 1
k 2
∑ 1
k 2 +k a k 1 b
k 2 k 1
k 2 +k
S n n ∑ 1
S 1 T 1
T
k 2 n n ∑ 1
k 2 +k
S 2 S 1 + a 2 T 2 T 1 + b 2
a 2 b 2 S 2 T 2
S 3 S 2 + a 3 T 3 T 2 + b 3
a 3 b 3 S 3 T 3
a n b n S n T n
∑ 1
∑ 1
k 2 k 2 +k ∑ 1
k 2 +k
k
0 ≤
0 ≤ b k ≤ a k
∑ ∑
b k ≤ a k
∑ a k ∑ b k
∑ b k ∑ a k
∑ 1
k−1
∑
k
k 3 +1

n
2 − 4 3 + 8 9 −···+ 2 (
− 2 3
) n
+ ···
a 2 r − 2 3
S
a
1 − r 2
1 −
− 2 6 5
3
e
ln(2)
f (x) ln(1 + x)
f x 0 ln(2)
L(x) f (x)
L(1) ≈ f (1) ln(2)
ln(1 + x) ln(2)
ln(2)
ln(1 + x) x

x 0 P 2 (x) f (x)
ln(1 + x) x 0 P 2 (0) f (0) P ′ 2 (0) f ′ (0)
P ′′
2 (0) f ′′ (0)
P 2 (x) x − x2
2 P 2(0) f (0) P ′ 2 (0) f ′ (0) P ′′
2 (0) f ′′ (0)
P 2 (x) ln(2) P 2 (1) ≈ f (1)
ln(2)
f
f x P 3 (x)
x − x2
2 + x3
3 P 3(0) f (0) P ′ 3 (0) f ′ (0) P ′′
3 (0) f ′′ (0)
P ′′′
3 (0) f ′′′ (0) P 3 (x)
ln(2)
ln(1 + x)
ln(2)
ln(2)
1 − 1 2
1 2
ln(2)
ln(2)
∞∑
(−1) k+1 1 k
k1
(−1) k+1
1 1
1 − 1 2
0.5
1 − 1 2 + 1 3
0.83
1 − 1 2 + 1 3 − 1 4
0.583
1 − 1 2 + 1 3 − 1 4 + 1 5
0.783
ln(2) ≈ 0.6931471806
ln(2)
ln(2)

∞∑
(−1) k a k
a k > 0 k
k0
∞∑
(−1) k+1 a k
k1
k 1
∞∑
(−1) k+1 1 k
k1
1∑
(−1) k+1 1 6∑
k
(−1) k+1 1 k
k1
2∑
(−1) k+1 1 k
k1
3∑
(−1) k+1 1 k
k1
4∑
(−1) k+1 1 k
k1
k1
7∑
(−1) k+1 1 k
k1
8∑
(−1) k+1 1 k
k1
9∑
(−1) k+1 1 k
k1
5∑
(−1) k+1 1 ∑10
k (−1) k+1 1 k
k1
k1
∑ ∞
k1 (−1)k+1 1 k

lim k→∞ a k 0 ∑ a k
∞∑
(−1) k+1 a k
k1
a k
n S n
n∑
S n (−1) k+1 a k
k1
S 2 a 1 − a 2 a 1 > a 2 0 < S 2 < S 1
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
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
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
n
a n
S 2 S 4 S 6
... S n−1 S n S 5 S 3 S 1
(n − 1) S n−1
n S n
|S n − S n−1 | a n
{a n }
∑
(−1) k a k
{a k } k →∞

{a k }
∑ ∞ (−1) k
k1 k 2 +2
∑ ∞
k1
(−1) k+1 2k
k+5
∑ ∞ (−1) k
k2 ln(k)
n S n
S
S
∞∑
(−1) k a k
k1
S
|S − S n | < |S n+1 − S n | a n+1
a n+1
S n S
S
∞∑
(−1) k+1 a k
k1
S n
n∑
(−1) k+1 a k
k1
n
∞∑
(−1) k+1 a k − S n ≤ a n+1
k1

S 100
∞∑ (−1) k+1
k1
k
S ∑ ∞
k1
|S 100 − S| < a 101
(−1) k+1
k
a 101 1
101 ≈ 0.0099
∞∑ (−1) k+1
S
ln(2)
k
S ≈ 0.693147
S 100
∑100
k1
k1
(−1) k+1
k
≈ 0.6881721793
S S 100 0.0049750013
0.0099
∞∑ (−1) k+1
k1
k 4
1 − 1 4 − 1 9 + 1 16 + 1 25 + 1 36 − 1 49 − 1 64 − 1 81 − 1 + ···
100

1 − 1 4 − 1 9 + 1
16 + 1 25 + 1 36 − 1
49 − 1 64 − 1 81 − 1
100 + ···
∞∑
k1
1
k 2
1 − 1 4 − 1 9 + 1
16 + 1 25 + 1 36 − 1
49 − 1 64 − 1 81 − 1
100 + ···
∞∑
− 1 k 2
k1
1 − 1 4 − 1 9 + 1
16 + 1 25 + 1 36 − 1
49 − 1 64 − 1 81 − 1
100 + ···
∑ a k
∑ |a k |
∑ a k ∑ |a k | ∑ a k
p ∑ 1
k 2
∑ (−1) k+1 1 k
∑ 1
k
∑ a k
∑ a k ∑ |a k |
∑ a k ∑ |a k |
∑ a k

∑ (−1) k ln(k)
k
∑ (−1) k ln(k)
k 2
ln(k) < √ k
k p
{a n }
∑ a k ∑ (−1) k a k
∑ ar k r −1 < r < 1
|r| ≥ 1
∞∑
k0
ar k
a n ∑ a k
lim n→∞ a n
0
f [c, ∞) a k f (k)
k ≥ c
∫ ∞
c
f (t) dt ∑ a k
∫ ∞
c
f (t) dt ∑ a k
a
1−r

f (x)
0 ≤ a k ≤ b k k
∑ b k ∑ a k
∑ a k ∑ b k
a n b n
a k
lim L
k→∞ b k
L ∑ a k ∑ b k
a k 0 k
|a k+1 |
lim
k→∞ |a k |
r
r < 1 ∑ a k
r > 1 ∑ a k
r 1
a k ≥ 0 k
lim
k→∞
r < 1 ∑ a k
r > 1 ∑ a k
r 1
k√
ak r
a k k
a n lim
n→∞
a n 0
∑ (−1) k+1 a k
a n
{a n }

S n
n∑
(−1) k+1 a k n
k1
∞∑
(−1) k+1 a k a n > 0 n a n
k1
lim S n S |S − S n | < a n+1
n→∞
S n
∑ ∞
k3
∑ ∞
k1
∑ ∞
k0
∑ ∞
k0
∑ ∞
k1
∑ ∞
k1
2
√
k−2
k
1+2k
2k 2 +1
k 3 +k+1
100 k
k!
2 k
5 k
k 3 −1
k 5 +1
∑ ∞
k2
∑ ∞
k2
∑ ∞
k1
∑ ∞
k2
3 k−1
7 k
1
k k
(−1) k+1
√
k+1
1
k ln(k)
n
n S n
∑ ∞ (−1) n
n2 ln(n)
∑
(−1) k a k
a k k
∑ ∞
k1 (−1)k a k {S n }
n∑
S n (−1) k a k
k1
n

∑ ∞
k1 (−1)k a k
{a n } n
n − 1 S n−1 n S n
∑ ∞
k1 (−1)k a k |S n − S n−1 | a n
S
|S − S n | < a n
n ∑ ∞
k1 (−1)k a k
a n
∞ ∑
∞ ∑
n1
n1
(−1) n
3 n
(−1) n
3n
∞∑
a n 0.45 − (0.45)3 + (0.45)5 − (0.45)7 + ...
3! 5! 7!
n1
∞∑
a n 1 − (0.4)2 + (0.4)4 − (0.4)6 + (0.4)8 − ...
2! 4! 6! 8!
n1
∞∑
a n 1 − 1 10 + 1
100 − 1
1000 + ...
n1
π
4 1 − 1 3 + 1 5 − 1 ∞∑
7 + ··· (−1) k 1
2k + 1
π
k0

S n n
π 4 S 100
π 4
∑ (−1) k+1 a k S
S n
S n +S n+1
2
S S n
S n+S n+1
2
S n + 1 2 (−1)n+2 a n+1
n 20 ln(2)
∞∑
ln(2) (−1) k+1 1 k
k1
S 20 S 20 155685007
232792560
1 − 1 + 1 2 − 1 4 + 1 3 − 1 9 + 1 4 − 1 16 + ···
{ }
1
n −
1
n 2
n n
( 1
(1 − 1) +
2 − 1 ) ( 1
+
4 3 − 1 ) ( 1
+
9 4 − 1 )
+ ···
16
{a n }
∑ ∞ (−1) k+1
k1 k
∞∑ (−1) k+1
ln(2)
k
k1

∑ ∞
k1
1
2k
∑ ∞ 1
k1 2k+1
∑ ∞ 1
kC 2k ∑ ∞ 1
kC 2k+1
C
a + b b + a
a b
S
∑ ∞ (−1) k+1
k1 k
S
S
S
S
P 1
P 1
∑n 1
k1
1
2k + 1 1 + 1 3 + 1 5 + ···+ 1
2n 1 + 1
S
S 1 P 1
S 1
N 1
∑m 1
1
N 1 −
2k −1 2 − 1 4 − 1 6 −···− 1
2m 1
k1
S 2 S 1 + N 1 ≤ S
S 2
P 2
P 2
∑n 2
kn 1 +1
1
2k + 1 1
2(n 1 + 1) + 1 + 1
2(n 1 + 2) + 1 + ···+ 1
2n 2 + 1
S 3 S 2 + P 2 ≥ S

S 3
N 2 −
∑m 2
km 1 +1
1
2k − 1
2(m 1 + 1) − 1
2(m 1 + 2) −···− 1
2m 2
S 4 S 3 + N 2 ≤ S
{S n }
lim n→∞ S n S

1 + 1 2 + 1 4 + ···+ 1 ∞∑
2 + ··· 1
k 2
k
x
r 1 2
x
k0
1 + x + x 2 + ···+ x k + ···
∞∑
x k
k0
r |r| < 1
a
1−r
|x| < 1
1 + x + x 2 + ···+ x k + ··· 1
1 − x
1
1−x
k
1 + x + x 2 + x 3 ≈ 1
1 − x
x
x

e
ln(2)
e x f (x) e x
P 1 (x) f (x) x 0
P 1 P 1 (x)
f (x) x 0 f P 1 x 0
f (x) e x
f x 0
P 1 (x)
P 2 (x) P 1 (x) + c 2 x 2
c 2 c 2
P 2 (x) f (x) x 0
P 1 (x) f (x) 0
P 1 (0) f (0) P ′ 1 (0) f ′ (0)
c 2
P 2 (0) f (0) P ′ 2 (0) f ′ (0) P ′′
2 (0) f ′′ (0)
P 2 (x) P 1 (x) + c 2 x 2
P 2 (0) P 2 (0) f (0)
P ′ 2 (0) P′ 2 (0) f ′ (0)
P ′′
2 (x) c 2 P ′′
2 (0) f ′′ (0)
P ′′
2 (0) f ′′ (0)
P 2 P 2 f x 0
x

f x a
P 1 (x) f
x a f a P 1 (x)
f (a, f (a))
P 1 (x) f (a) + f ′ (a)(x − a)
P 2 (x) P 1 (x) + c 2 (x − a) 2
P 2 (x) f (x) x a
P 2 (x)
P 2 (x) P 1 (x) + c 2 (x − a) 2
P 2 (a) P 1 (a) f (a)
P ′ 2 (x) P′ 1 (x) + 2c 2(x − a)
P ′ 2 (a) P′ 1 (a) f ′ (a)
P ′′
2 (x) 2c 2 P ′′
2 (a) 2c 2
P 2 (x) f (x) P 1 (x) P 2 (x) f (x)
x a
P ′′
2 (a) f ′′ (a)
2c 2 f ′′ (a)
c 2 f ′′ (a)
2
P 2 (x) f x 0
P 2 (x) f (a) + f ′ (a)(x − a) + f ′′ (a)
(x − a) 2
2!
P 3 (x) P 2 (x) + c 3 (x − a) 3
P 4 (x) P 3 (x) + c 4 (x − a) 4
P 5 (x) P 4 (x) + c 5 (x − a) 5
P n (x) P n−1 (x) + c n (x − a) n

n P n (x)
n f x a
k n
P (k)
n (a) f (k) (a)
P n (x) c 0 + c 1 (x − a) + c 2 (x − a) 2 + ···+ c n (x − a) n
P (0)
n (a) c 0
P (1)
n (a) c 1
P (2)
n (a) 2c 2
P (3)
n (a) (2)(3)c 3
P (4)
n (a) (2)(3)(4)c 4
P (5)
n (a) (2)(3)(4)(5)c 5
P (k)
n (a) (2)(3)(4) ···(k − 1)(k)c k k!c k
P (k)
n (a) f (k) (a) k!c k f (k) (a)
c k f (k) (a)
k!
k c k n
f
n f x a
P n (x) f (a) + f ′ (a)(x − a) + f ′′ (a)
(x − a) 2 + ···+ f (n) (a)
(x − a) n
2!
n!
n∑ f (k) (a)
(x − a) k
k!
k0
n f (x) x a
(a) f (k) (a) k 0 ...n
P (k)
n
f (x) e x
n f x 0

f ′ (x) e x f ′′ (x) e x f ′′′ (x) e x
f (0) f ′ (0) f ′′ (0) f ′′′ (0) 1
f (x) e x x 0
P 3 (x) f (0) + f ′ (0)(x − 0) + f ′′ (0)
2!
1 + x + x2
2 + x3
6
(x − 0) 2 + f ′′′ (0)
(x − 0) 3
3!
f f (k) (x) e x k
k n f (x) x 0
f (k) (0)
(x − 0) k 1 k!
k! xk
n f (x) e x x 0
P n (x) 1 + x + x2
2! + ···+ 1 n∑ x k
n! xn
k!
k0
n a 0
e x
n∑ x k
k!
k0
f (x) 1
1−x
f (x) x 0
1
P 4 (x)
1−x
f (k) (0)
f (x) cos(x)
f (x) x 0
P 4 (x) cos(x)
f (k) (0)
k k
f (x) sin(x)

f (x) x 0
P 4 (x) sin(x)
f (k) (0)
k k
n n
P 2 (x)
sin(x) P 2 (x) x
P 4 (x) sin(x)
P 4 (x) x − x3
3!
f (k) (0)
P 5 (x) x − x3
3! + x(5)
5!
P 7 (x) x − x3
3! + x(5)
− x(7)
5! 7!
P 9 (x) x − x3
3! + x(5)
− x(7)
+ x(9)
5! 7! 9!

3
2
y
3
2
y
1
x
-4 -2 2 4
-1
-2
1
x
-4 -2 2 4
-1
-2
-3
-3
3
2
y
3
2
y
1
x
-4 -2 2 4
-1
-2
-3
1
x
-4 -2 2 4
-1
-2
-3
x 0 f (x) sin(x)
P 1 (x) x
{P n (x)}
f
f
f (x)
∞∑
k0
f (k) (a)
(x − a) k
k!

f x a
f x a T f (x)
T f (x)
∞∑
k0
f (k) (a)
(x − a) k
k!
a 0
f n
e x e x
∞∑
k0
x k
k!
f (x) 1
1−x
f (x) cos(x)
f (x) sin(x)
n f (x) 1
1−x
x 0
x
e x e x
x
cos(x)
cos(x) x cos(x)
1
1−x
x
1
1−x e x

cos(x)
1
1−x
e x sin(x) cos(x) 1
1−x
e x cos(x) sin(x)
1
1
x
1−x
1−x
1
1
x
1−x
1−x
(−1, 1) x
f (x) x f
x
f
f (x) x
e x
x x
∞∑ x k
k!
k0
x
∞∑
x k
∞∑
k!
|x| k
k!
k0
a k+1
lim
k→∞ a k
lim
k0
|x| k+1
(k+1)!
k→∞ |x| k
k
|x| k+1 k!
lim
k→∞ |x| k (k + 1)!
|x|
lim
k→∞ k + 1
0
x x
x

e x x
e x x
x
∞∑
c k (x − a) k
k0
x a a k |c k (x − a) k |
a
lim k+1
k→∞ a k
< 1
a k+1
|x − a| |c k+1|
a k |c k |
a k+1
lim lim |x − a| c k+1
k→∞ a k k→∞ c k
c k f (k) (a)
k!
c k+1
lim
k→∞ c k
L
a k+1
lim
k→∞ a k
|x − a| · L
L L
L x
L 0 (−∞, ∞)
L x a
L x
|x − a| · L < 1
x
|x − a| < 1 L
(
a − 1 L , a + 1 )
L

|x − a| · L 1 a ± 1 L
x
x a
f (x) 1
1−x
x 0
f (x) cos(x) x 0
f (x) sin(x) x 0
x
f
f (x)
f
sin(x)
n P n (x)
f (x)
f
P n (x) f P n (x)
f (x)
P n (x) f (x)
a E n (x) f (x)
P n (x)
E n (x) f (x) − P n (x)
|E n (x)| P n f
P (k)
n (0) f (k) (0)

0 ≤ k ≤ n
E (k)
n (0) 0
0 ≤ k ≤ n P n (x) n
E (n+1)
n
x
P (n+1)
n (x) 0
(x) f (n+1) (x) − P (n+1)
n (x)
E (n+1)
n (x) f (n+1) (x)
f (x) c P n (c)
| f (n+1) (t)| M [0, c]
f (n+1) (t)
≤ M
0 ≤ t ≤ c
E(n+1) n
t c
(t)
f (n+1) (t)
≤ M
− M ≤ E (n+1)
n (t) ≤ M
[0, c] t 0 t x
∫ x ∫ x
∫ x
−M dt≤ E (n+1)
n (t) dt ≤ Mdt
0
x [0, c] E (n)
n (0) 0
x [0, c]
0
−Mx ≤ E (n)
n (x) ≤ Mx
x [0, c]
∫ x
n
0
−Mt dt ≤
−M x2
∫ x
0
E (n)
n (t) dt ≤
0
∫ x
2 ≤ E(n−1) n (x) ≤ M x2
2
−M
xn+1
(n + 1)! ≤ E n(x) ≤ M xn+1
(n + 1)!
0
Mt dt

x [0, c]
|E n (x)| ≤ M |x|n+1
(n + 1)!
x [0, c] E n
a 0
a
P n (x) f n + 1
M f (n+1) (x) ≤ M
[a, c] P n (x) n f (x) x a
Pn (c) − f (c) |c − a|n+1
≤ M
(n + 1)!
P 10 (x) sin(x)
sin(2)
f (x) sin(x) c 2 a 0 n 10
M f (x) sin(x) ± sin(x) ± cos(x)
f (n+1) (x)
≤ 1
n x M
P10 (2) − f (2) |2 − 0|11
≤ (1)
(11)!
211
(11)! ≈ 0.00005130671797
P 10 (2) sin(2)
P 10 (2) ≈ 0.9093474427 sin(2) ≈ 0.9092974268
P n (x) n sin(x) x 0
n P n (2) sin(2)
sin(x) sin(x) x
f (x) sin(x)
f (n+1) (x)
≤ 1

n x M 1
|P n (x) − sin(x)| ≤ |x|n+1
(n + 1)!
x
∑ ∞ x k
k0 k!
x
lim
n→∞
x n+1
(n + 1)! 0
x n →∞
x
lim |P n(x) − sin(x)| 0
n→∞
sin(x)
∞∑
n0
(−1) n x 2n+1
(2n + 1)!
cos(x) cos(x)
x
e x
e x e x
x
e x e x
x
e x e x
x
e x x
e x e x x
P n (x) n e x
n P n (5) e 5

n
x a f
P n (x) f (a) + f ′ (a)(x − a) + f ′′ (a)
(x − a) 2 + ···+ f (n) (a)
(x − a) n
2!
n!
n∑ f (k) (a)
(x − a) k
k!
k0
x a f
∞∑
k0
f (k) (a)
(x − a) k
k!
n a f n
a n f
f x
f
f (x)
∞∑
k0
f (k) (a)
(x − a) k
k!
P n (x) n
f x a M f (n+1) (x)
[a, c]
Pn (c) − f (c) |c − a|n+1
≤ M
(n + 1)!
n cos(3x) x
n 2 P 2 (x)
n 4 P 4 (x)
n 6 P 6 (x)
(7) −1, ′ (7) −1
′′ (7) 1 ′′′ (7) −4
7

P 2 (x)
7
P 3 (x)
(7.1)
P 2 (7.1) ≈
P 3 (7.1) ≈
5 a −2
x
5
x
cos(x) a
−π/4
cos(x)
f (x) ln(x)
a 3
ln(x)
a 0 f (x) x 3 − 2x 2 + 3x − 1
a 0 f (x) x 3 − 2x 2 + 3x − 1
f (x) m n
a 0 n
a
f (x) sin(x) f
x π 2 f (x) x π 2
f (x) ln(x) f
x 1 f (x) x 1

ln(2)
sin(x 2 )
sin(x 2 ) a 0
f (x) sin(x)
∞∑
(−1) k x 2k+1
(2k + 1)!
k0
x 2 x
sin(x 2 )
sin(x 2 )
f f (x)
⎧⎪
f (x) ⎨
e −1/x2 x 0,
⎪
⎩0 x 0.
f ′ (0) 0
f (n) (0) 0 n ≥ 2
f
f
x
f (x)

x
e x 1 + x + x2
2! + x3 xn
+ ···+
3! n! + ···
e x
1 + 2x + 3x 2 + 4x 3 + ···
x
f
f (x) a 0 + a 1 x + a 2 x 2 + ···+ a k x k + ···
a 0 a 1 ...
y ′ ky
k
k 1

y f (x) f (x)
y f (x)
∞∑
a k x k
a k
k0
f (x) x
∞∑
ka k x k−1
k1
∞∑
a k x k
k0
f (x) ∑ ∞
k0 a kx k
a 1 a 0
a 2 a 1 a 2 a 0
a 3 a 2 a 3 a 0
a 4 a 3 a 4 a 0
a k
a 0
y a 0
x

x a
∞∑
c k (x − a) k
k0
{c k } x
x
f x
∞∑
f (x) c k (x − a) k
k0
∞∑ x
f (x) k
2 k
f (1) f 3
2 f (x)
k0
f x 1
∞∑ 1
2 k
k0
1 2
f (1) 1
1 − 1 2
2
f (3/2)
∞∑ ( ) 3 k
1
4 1 − 3 4
k0
4
f (x) x 2
f (x)
∞∑ ( ) x k
1
2 1 − x 2
k0
2
2 − x

−1 < x 2
< 1
f −2 < x < 2
x
x
f (x) ∑ ∞
k1
x k
k 2
n∑ x
S n (x) k
k 2
n ≥ 1 S 10 (x) S 25 (x) S 50 (x)
k1
10
y
5
x
-2 -1 1 2
-5
-10
∑ ∞ x k
k1 k 2
S 50
(−1, 1)
−1 < x < 1
x
|a k+1 |
lim
k→∞ |a k |
a k xk
k 2
|x| k+1
(k+1) 2
k→∞ |x| k
k 2
lim

(
k
lim |x|
k→∞ k + 1
|x| lim
|x|
k→∞
(
k
k + 1
f (x)
|x| < 1 |x| > 1 |x| 1
x 1 x −1
x 1
f (1)
p p > 1 x −1
∞∑
k1
1
k 2
) 2
) 2
f (−1)
∞∑ (−1) k
k1
k 2
{ 1
n 2 }
x −1
−1 ≤ x ≤ 1
∑ ∞
k1
(x−1) k
3k
∑ ∞
k1 kxk
∑ ∞ k 2 (x+1) k
k1
4 k ∑ ∞
k1
x k
(2k)!
∑ ∞
k1 k!xk
sin(x)
e x

f (x) 1
1+x 2
x 0
f (x) 1
1+x 2
f (x)
f (x)
(x) 1
1−x
(−x 2 ) f (x) −x 2 x
(x) (−x 2 ) f (x)
f (x)
x f (x)
sin(x)
cos(x)
∞∑
k0
(−1) k x 2k+1
(2k + 1)!
∞∑ (−1) k x 2k
k0
(2k)!
−∞< x < ∞
−∞< x < ∞
∞∑
e x x k
k!
k0
−∞< x < ∞
1
∞∑
1 − x x k − 1 < x < 1
k0
sin(x 2 ) e 5x3 cos(x 5 )
f
f (x)
∞∑
(−1) k x2k
(2k)!
k0

f (x) 1 − x2
2! + x4
4! − x6
x2k
+ ···+ (−1)k
6! (2k)! + ···
f ′ (x)
f (x) f ′ (x)
f ′′ (x) f ′′ (x)
f
f ′
f ∫ f (x) dx
f (x)
f (x)
∞∑
c k x k
k0
f (x) −r < x < r
∑ ∞
k1 kc kx k−1 f (x)
f ′ (x) −r < x < r
f ′ (x)
∞∑
kc k x k−1 , |x| < r
k1
∑ ∞
k0 c k xk+1
k+1
f (x)
−r < x < r
∫
∞∑ x
f (x) k+1
dx c k + C, |x| < r
k + 1
k0

x 0 arctan(x)
arctan(x)
x 0 arctan(x)
1
∞∑
1 + x 2 (−1) k x 2k
−1 < x < 1
∫
k0
d
dx [arctan(x)] 1
1 + x 2
1
dx arctan(x) + C
1 + x2 1
1+x 2
arctan(x)
∫ ∞∑
arctan(x) (−1) k x 2k dx
k0
∞∑ (∫
)
(−1) k x 2k dx
k0
∞∑
(−1) k x2k+1 + C
2k + 1
k0
−1 < x < 1
C arctan(0) 0
0 arctan(0) ∞∑
(−1) k 02k+1 + C C
2k + 1
k0
C 0
−1 < x < 1
arctan(x)
∞∑
(−1) k x2k+1
2k + 1
∞∑
(−1) k x2k+1
2k + 1
k0
k0

x −1 x 1
1
2k+1
arctan(x) −1 ≤ x ≤ 1
ln(1 + x) x 0
∞∑
a k x k
k0
4
∞∑
(1 − 10x) f (x) c n x n
c 0
c 1
c 2
c 3
c 4
R
n0
f (x) 4x arctan(6x)
∞∑
f (x) c n x n .
n0
c 0

c 1
c 2
c 3
c 4
R
R
∫ 1
0 sin(x2 ) ds
sin(x) sin(x 2 )
sin(x 2 )
sin(x 2 )
∫ sin(x 2 ) dx
∫ 1
0 sin(x2 ) dx
∫ 1
0 sin(x2 ) dx
f
f (x)
∞∑
a k x k
k0
f a k
f (n) (0) n!a n n
{ f n }
f 0 0, f 1 1, f 2 1, f 3 2, f 4 3, f 5 5, f 6 8, f 7 13, ···
f 0 0 f 1 1 n ≥ 2
f n f n−1 + f n−2
F(x)
∞∑
f k x k
k0

xF(x) ∑ ∞
k1 f k−1x k
x 2 F(x) ∑ ∞
k2 f k−2x k
F(x) − xF(x) − x 2 F(x) x
F(x) − xF(x) − x 2 F(x) x
F(x)
x
1−x−x 2
y ′′ − xy 0
x y
x
y f (x)
y
∞∑
a k x k
k0
y y ′
y ′′
∞∑
∞∑
(k − 1)ka k x k−2 − a k x k+1 0
k2
k0
x k−2 x k+1
∞∑
∞∑
(k − 1)ka k x k−2 (k + 1)(k + 2)a k+2 x k
k2
k0
∞∑
a k x k+1
k0
∞∑
a k−1 x k
k1
y ′′ ± k 2 xy 0

∞∑
∞∑
(n + 1)(n + 2)a n+2 x n − a n−1 x n 0
n0
n1
x
∞∑
2a 2 + [(k + 1)(k + 2)a k+2 − a k−1 ] x k 0
k1
a 3k+2 0 k
1
a 3k
(2)(3)(5)(6)···(3k−1)(3k) a 0 k ≥ 1
1
a 3k+1
(3)(4)(6)(7)···(3k)(3k+1) a 1 k ≥ 1
y a 0
1 +
∞∑
k1
+ a 1
x +
k1
x 3k
(2)(3)(5)(6) ···(3k − 1)(3k)
∞∑ x 3k+1
(3)(4)(6)(7) ···(3k)(3k + 1)
a 0 a 1

∫ du
1 a 2 +u 2 a arctan u a + C
∫
√
du
ln |u + √ u 2 ± a 2 | + C
u 2 ±a 2
∫ √ √
u 2 ± a 2 du u 2 u 2 ± a 2 ± a2
2 ln |u + √ u 2 ± a 2 | + C
∫
√
u 2 du
√ u
u 2 ±a 2 2 u 2 ± a 2 ∓ a2
2 ln |u + √ u 2 ± a 2 | + C
∫ du
u √ − 1 u 2 +a 2 a ln a+ √ u
2 +a 2
u
+ C
∫ du
u √ 1 u 2 −a 2 a sec−1 u a + C
∫
√
du
arcsin u
a 2 −u 2
a + C
∫ √ √
a 2 − u 2 du u 2 a 2 − u 2 + a2
2 arcsin u a + C
∫ √
u 2
du − √ u
a 2 −u 2 2 a 2 − u 2 + a2
2 arcsin u a + C
∫ du
u √ − 1 a 2 −u 2 a ln a+ √ a
2 −u 2
u
+ C
∫
√
du
u 2√ − a 2 −u 2
+ C
a 2 −u 2 a 2 u

u
x a