Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 41
Answer to Exercise 4.4
1. (a) 364
3
(b) −2 (c) 16 3
(d) 88
3
(e) 7 8
(f) 12
5
(g) 0 (h) 2 (i) Does not exist
(j) 3 (k) ln3 (l) e+e −1 −2 (m) 28
ln2
(q) 5 2
(r) 2− √ 2
2
(s) − 11 6
(t) 10.7
(n) 3e 3 −12 (o) π2
9 +2√ 3 (p) π 2
2. (a) f ′ (x) = x 2 −3x+2 (b) g ′ (x) = √ 1+2x (c) g ′ (y) = y 2 siny
(d) F ′ (x) = −cos(x 2 ) (e) f ′ (x) = (e −x4 +1)2x (f) f ′ (x) = −ln(x 2 +1)
(g) h ′ (x) = −arctan(1/x)/x 2 (h) y ′ = cos√ x
2x
(j) y ′ = 3x 7/2 sin(x 3 )−(sin √ x)/(2 4√ x)
Evaluate the definite integral, if it exists.
1.
3.
5.
7.
9.
11.
13.
15.
∫ 2
0
∫ 1
0
∫ 1
0
∫ 4
1
∫ 3
(x−1) 25 dx 2.
x 2 (1+2x 3 ) 5 dx 4.
Exercise 4.5
∫ 2
0
∫ 1
−1
∫ π
(i) y ′ = 3(1−3x)3
1+(1−3x) 2
x √ x 2 +1dx
x
(x 2 +1) 2 dx
4cosx
(sinx+1) 2 dx
cosπxdx 6.
π/2
√
1
1+ 1 ∫ π/2
x 2 x dx 8. cotxdx
dx
2x+3
0
∫ π/3
0
∫ 2
1
∫ a
0
10.
π/4
∫ 4
∫
sinx
13
cos 2 x dx 12.
x √ x−1dx 14.
x √ x 2 +a 2 dx (a > 0)
1
0
∫ e 4
e
x−1
√ x
dx
dx
√
3 (1+2x)
2
dx
x √ lnx
Answer to Exercise 4.5
1. 0 2. 5 3√
5−
1
3
3. 81
4
4. 0 5. 0 6. −2 7.
4 √ 2
3
− 5√ 5
12
8. 1 2 ln2 9. 1 2 ln3
10.
8
3
11. 1 12. 3 13. 16
15
14. 2 15. 1 3 (2√ 2−1)a 3