45.
y = 2 sec x + tan x
dy
⇒
= 2 sec x tan x + sec2 x
dx
= sec x(2 tan x + sec x)
2 sin x + 1
= sec x(
)
cos x
⇒ x satisfies sec x = 0 (impossible) or 2 sin x + 1 = 0
⇒ sin x = − 21
⇒ x = 7π
or 11π
6
6
52.
(1)
dy du dx
dy
=
dt
du dx dt
1
= {3 · [ (1 + u)]2 } · (− sin x) · 2
2
1
= −3 · [ (1 + cos 2t)]2 · sin 2t
2
= −3 · (cos2 t)2 · (2 sin t cos t)
= −6 sin t cos5 t
1
1
1
(2) y = [ (1 + u)]3 = [ (1 + cos x)]3 = [ (1 + cos 2t)]3 = [cos2 t]3 = cos6 t
2
2
2
dy
5
5
⇒
= 6 · cos t · (− sin t) = −6 sin t cos t
dt
56.
(a)
d
sin x · dx
(cos x) − cos x ·
d
d cos x
(cot x) =
(
)=
dx
dx sin x
sin2 x
− sin2 x − cos2 x
=
sin2 x
−1
=
sin2 x
= − csc2 x
1
d
(sin x)
dx
(b)
(c)
d
d
1
1
d
(sec x) =
(
)=
· (− (cos x))
2
dx
dx cos x
cos x
dx
1
=
· sin x
cos2 x
= sec x tan x
d
d
1
1
d
(csc x) =
(
)=
· (− (sin x))
2
dx
dx sin x
dx
sin x
1
=
· (− sin x)
sin2 x
= − csc x cot x
67.
1
(a) f (x) = x sin( )
x
d
1
d
1
⇒ f 0 (x) =
(x) · sin( ) + x · (sin( ))
dx
x
dx
x
1
1
−1
= sin( ) + x · cos( ) · ( 2 )
x
x
x
1
1
1
= sin( ) − cos( )
x
x
x
1
g(x) = xf (x) = x2 sin( )
x
d 2
1
d
1
⇒ g 0 (x) =
(x ) · sin( ) + x2 · (sin( ))
dx
x
dx
x
1
1
−1
2
= 2x · sin( ) + x · cos( ) · ( 2 )
x
x
x
1
1
= 2x sin( ) − cos( )
x
x
1
1
1
1
(b) lim g 0 (h) = lim [2h sin( ) − cos( )] = 0 − lim cos( ) = − lim cos( ),
h→∞
h→∞
h→∞
h→∞
h
h
h
h
the limit does not exist.
⇒g 0 (x) is not continuous at x = 0.
69.
2
2π
.
(a) g(x) is differentiable at x =
3
2π
⇔ g(x) is continuous at x =
and lim − g 0 (x) = lim + g 0 (x)
3
x→ 2π
x→ 2π
3
3
Continuity:
√
2π
3
, lim + g(x) = lim + (ax + b) =
a+b
lim − g(x) = lim − sin x =
2π
2π
2π
2
3
x→
x→
x→
x→ 2π
3
3
3
3
√
2π
3
⇒
a+b=
3
2
Differentiability:
lim − g 0 (x) = lim − cos x =
x→ 2π
3
⇒a=
x→ 2π
3
−1
, lim + g 0 (x) = lim + a
2 x→ 2π3
x→ 2π
3
−1
2
√
2π
−1
3 π
Therefore, g(x) is differentiable at x =
if a =
,b =
+ .
3
2
2
3
(b)
1.0
0.5
1
2
3
4
5
-0.5
-1.0
71.
y = A sin ωt + B cos ωt
dy
⇒
= Aω cos ωt − Bω sin ωt
dt
d2 y
⇒
= −Aω 2 sin ωt − Bω 2 cos ωt
dt2
3
6
Thus,
d2 y
2
2
2
+
ω
y
=
−Aω
sin
ωt
−
Bω
cos
ωt
+ ω 2 (A sin ωt + B cos ωt) = 0.
dt2
4