634 Solutions to exercises
- (i)ln 16 (ii)ln 1223 (iii)ln 132
(iv)ln 12
- (i)ln 1 x
2
(ii)ln(2x 1 − 1 3) (iii) 0
(iv)x (v)x
2
- (i)0.2310 s (ii)0.6931 1 × 110
5
s
Chapter 4
Section 4.2
1.∆y 1 = 13 x
2
∆x 1 + 13 x(∆x)
2
1 + 1 (∆x)
3
- (i)
(ii)
- (i)
(ii)
Section 4.3
4.x 1 = 1 − 1 , essential 5.x 1 = 10 , removable
6.x 1 = 13 , essential;x 1 = 10 , removable
Section 4.4
- 0 8.±∞ 9. 123 10. 122
- 1 12. 0 13.∞ 14.− 4
- 2 16.−ln 12 17.−ln 13
Section 4.5
∆
∆
=− +
∆
−=−
−− −
y
x
e
x
e
dy
dx
e
xx x2
- ,
∆
∆
=
+∆+∆
+∆ +
=
y
x
xxx x
xx x
dy
dx
x
33 3
2
22
32 32
1
()
()
,
22
∆
∆
=−
+∆
+∆
=−
y
x
xx
xx x
dy
dx
x
42 4
22 3
()
,
dy
dx
= 4 x
3
∆
∆
=+∆+∆+∆
y
x
46 4xxxxx x
32 2 3
()(),
∆
∆
=++∆ =+
y
x
xx
dy
dx
432 , 43 x
lim
()
∆p
pK
Kp
→
∆
∆
=
0
1
2
θ
∆=
∆
++∆
θ
Kp
11()Kp K p p
lim
∆x
y
x
x
→
∆
∆
=
0
3
2
∆
∆
=+∆+∆
y
x
33 xxx x
22
()
p
p
= RT
−
ooγ
exp [(μμ) 2 ]
h
RT
Mg
p
p
=−
ln
0
Section 4.6
- 3 x
2
24.(5 2 4)x
124
25.(1 2 3)x
− 223
26.− 32 x
4
27.− 21 + 16 x 1 − 112 x
2
1 + 151 cos 1 x 1 + 161 sin 1 x 1 + 17 e
x1 − 182 x
28.nRT(2nB 1 − 1 V) 2 V
3
29.−(1 1 − 14 x
2
) 1 sin 1 x 1 − 18 x 1 cos 1 x
30.(5 1 + 13 x)e
x31.(cos 1 x 1 − 1 sin 1 x)e
x
11 + 1 ln 1 x
35.−cosec
2
x 36.(1 1 − 1 ln 1 x) 2 x
2
37.5(1 1 + 1 x)
4
- 41 cos 14 x 42.− 2 e
− 2 x45.−(4x 1 − 1 3)sin(2x
2
1 − 13 x 1 + 1 1)
46.cos 1 xe
sin 1 x47.−tan 1 x
52.cos 1 x 1 cos 12 x 1 − 121 sin 1 sin 12 x
- 41 cos
2
12 x 1 sec
2
14 x 1 − 121 tan 14 x 1 sin 14 x
Section 4.7
xy
xxy
−
(ln)1+
−
32 +
3
2
x
y
−
x
y
2
1
4
x
−x
2
14
2
- x
1
xx() 1 +
2
1
4
x
+x
2
14
2
− x
−
−
−
1
2
32
2 na
V
nRT
()Vnb
−
()Vnb−
p
−
V
nRT V nB
3
() 2
1
43 y−
34
2
2
232
xx
x
()
()
22
22 3
2xxe
x(1+
+
)
62
3
22
12
2
12
xx
x
x
()
()
++
sin cos
sin sin
22 2
2
2
xx
xx
1
2
1
3
xx−
6sin(3
cos(3xx e
22 x2
2−+
)
)
43
231
2
x
xx
−
−+
(4 3)xe
xx−
231 −+
2−−
−−
34 3
22 3 1
232
()
()
x
xx
2
3
22
x
()−x
x
2 x
2
−
814 +−
2
2
xx x x
x
sin ( )cos
sin
618 3 4 3
3
234
32
+−−−
xx x x
()x