Solutions to exercises 635
Section 4.8
× 1 sin
122
1 x 1 cos
3
(x
2
1 + 1 1) 1 tan
123
12 x
73.6.1 1 × 110
− 4
s
Section 4.9
74.y′ 1 = 115 x
4
1 + 116 x
3
1 − 19 x
2
1 + 12 x 1 − 12
y′′ 1 = 160 x
3
1 + 148 x
2
1 − 118 x 1 + 12
y′′′ 1 = 1180 x
2
1 + 196 x 1 − 118
y
iv1 = 1360 x 1 + 196 ,y
v1 = 1360
- 3
ne
3 xSection 4.10
78.(x,y) 1 = 1 (3 22 ,− 12 4),min
79.(3,0),min;(5 23 , 322 27),
max;(7 23 , 162 27), inflection
80.(0,3),min;(− 1 ,5),
max;(− 122 ,4), inflection
81.(1,e
− 1
),max;(2, 2 e
− 2
), inflection
83.(2,−13),min;(0,3),
max;(3 22 ,− 572 8), inflection
- (i)A 1 = 1 D
eR
e12
,B 1 = 12 D
eR
e6
(ii)
- (i)
t
kk
k
k
=
−
1
21
2
1
ln
2 kT
m
UR D
R
R
R
R
()=
−
eee12 6
2
y
xn
nnnnn()
()
() cos ( )
() si
=
−
−
+
12 2
12
2
12
even
nn()2xnodd
112 6
23 4
x
xx x,− , ,−
1
2
61
4
34
2
cot tan( )
sin
xx x
x
−+
×
+−
++−
()()
()( )
11
213 21
212
213
xx
xxx
62
33 2 1
2
x
xx
−
+−
()
2
1
1
21
2
21
2
x
x
xx
+−
−
()
−
−+
−
7
33 4
3
4
13
()( )xx
x
x
23
22 2
2
22
xy
yyxy
−
−−
Section 4.11
- (i)v 1 =1K1= 14 t 1 − 13 ; a 1 =1L1= 14
(iii)v 1 = 10 whens 1 = 1 − 928
- (i)θ 1 = 1 (t
3
1 − 12 t
2
1 − 14 t) 22
(ii)ω 1 = 1 (3t
2
1 − 14 t 1 − 1 4) 22 , 71 = 13 t 1 − 12
(iv)ω 1 = 10 whent 1 = 12 ,θ 1 = 1 − 4
89.dy 1 = 12 dx 90.dy 1 = 1 (6x 1 + 1 2)dx
91.dy 1 = 1 cos 1 x 1 dx 92.dV 1 = 14 πr
2
dr
Chapter 5
Section 5.2
- 2 x 1 + 1 C 2. 3.
- 9.ln 1 A(x 1 − 1 1)
10.ln 1 A 2 (3 1 − 1 x) 11.
- 13.x
5
1 + 1 x
2
1 + 13 x 1 − 138
Section 5.3
- 2823 17. 2 18.
19.ln(5 2 4) 20.
- 1 22. 0 23.− 122
- 1423 27.− 223 π 28. 1
- (ii) 0 , 1 , 0 32. 4023 33. 1
- 0 35.− 1 36. 123
- 2 38.ln 12 39. 11 − 1 ln 12
40.odd 41.even 42.odd
43.odd 44.even
45.neither; even: 3 x
2
1 + 11 ; odd: 2 x
46.neither; even: ;
odd:
1
2
()ee
−xx−
1
2
()ee
−xx1
3
315
()ee
−−
−
3
8
−+ 422 − +
1
4
1
4
2
xxe
xsin
32
1
xx 1
x
+−+ln
1
4
sin 4 x
x
3
3
− 9
−+
−
1
2
2
eC
x1
3
3
eC
x+
−+1
4
cos 4 xC
3
2
23
xC+
−+1
2
2
x
C
3
5
53
xC+
x
C
4
4