Thomas Calculus 13th [Solutions]

182 Chapter 3 Derivatives
41.
1 2 1 2 1/2
y xsin x 1 x xsin x (1 x )
dy sin 1 2 1/2 1 1
x x 1 1
2 2
(1 x ) ( 2 x ) sin x x x
dx
sin x
2 2
1 x 1 x 1 x
42.
2
1
2 2 2
2 2 2 2
4 x
1
4 4
2
2 1 dy
1 1 1
y ln( x 4) x tan x x tan x x
x tan x x tan x
2 dx x 2 x 2 x
2
43. The angle is the large angle between the wall and the right end of the blackboard minus the small angle
1 1
between the left end of the blackboard and the wall cot x cot x .
15 3
44.
1
65 (90 ) (90 ) 180 65 65 tan 21 65 22.78 42.22
50
45. Take each square as a unit square. From the diagram we have the following: the smallest angle has a tangent
1
1
of 1 tan 1; the middle angle has a tangent of 2 tan 2; and the largest angle has a tangent
of 3
1
tan 3. The sum of these three angles is
1 1 1
tan 1 tan 2 tan 3 .
1
46. (a) From the symmetry of the diagram, we see that sec x is the vertical distance from the graph of
1
1
y sec x to the line y = and this distance is the same as the height of y sec x above the x-axis at
(b)
x; i.e.,
1 1
sec x sec ( x).
1 1
cos ( x) cos x , where 1 x 1
1 1
sec ( x) sec x
47. (a) Defined; there is an angle whose tangent is 2.
(b) Not defined; there is no angle whose cosine is 2.
48. (a) Not defined; there is no angle whose cosecant is 1 2 .
(b) Defined; there is an angle whose cosecant is 2.
49. (a) Not defined; there is no angle whose secant is 0.
(b) Not defined; there is no angle whose sine is 2.
50. (a) Defined; there is an angle whose cotangent is 1
2 .
(b) Not defined; there is no angle whose cosine is 5.
1 1 1
cos cos 1 , where x 1 or x 1
x
x
51.
du
du
dx
dx
2 2
csc 1 u 1 1 1
2 sec u d (csc ) d
dx
u dx 2
sec u 0 , u 1
u u 1 u u 1
52.
y 1
tan x tan y x d (tan ) d ( )
dx
y dx
x
2 dy
(sec y) 1
dx
dy 1 1 1 , as indicated by the
dx 2 2 2
sec y 2 1 x
1 x
triangle.
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