5128_Ch03_pp098-184

182 Chapter 3 Derivatives
29. y csc 1 sec x, 0 x 2p
30. r ( 1 sin
u
2
1
sin
sin
1
2
1 cos
u
1
cos
cos
(1
cos
) 2
)
In Exercises 31–34, find all values of x for which the function is
differentiable.
31. y ln x 2 For all x 0 32. y sin x x cos x For all
real x
1 x
33. y
1 2 x For all x 1 34. y 2x 7 1 x 5
For all x 7 In Exercises 35–38, find dydx.
2
35. xy 2x 3y 1 y
2 36. 5x 45 10y 65 15 1
37. xy 1 y x 3
3 (xy)1/5
x or 1
x2
38. y 2 x
x 1 2y(x
1 1) 2
In Exercises 39–42, find d 2 ydx 2 by implicit differentiation.
39. x 3 y 3 1 2 x
y5 40. y 2 1 2 x 1 2xy x 4
2
y3
41. y 3 y 2 cos x 42. x 13 y 13 4
In Exercises 43 and 44, find all derivatives of the function.
43. y x 4
3 2 2 x
5
x2 x 44. y
1 20
In Exercises 45–48, find an equation for the (a) tangent and
(b) normal to the curve at the indicated point.
45. y x 2 2
2x, x 3 (a) y x 3 (b) y 3
x 5 3
3
2 2
46. y 4 cot x 2 csc x, x p2 (a) y x 2 2 (b) y x 2 2
47. x 2 2y 2 9, 1, 2 48. x xy 6, 4, 1
In Exercises 49–52, find an equation for the line tangent to the curve
at the point defined by the given value of t.
49. x 2 sin t, y 2 cos t, t 3p4 y x 22
50. x 3 cos t, y 4 sin t, t 3p4
51. x 3 sec t, y 5 tan t, t p6 y 1 0
x 53
3
52. x cos t, y t sin t, t p4 y (1 2)x 2 1 4
53. Writing to Learn
or y 2.414x 3.200
(a) Graph the function
x, 0 x 1
f x { 2 x, 1 x 2.
41. 2 (3y2 1) 2 cos x 12y sin 2 x
42. 2 (3y 2 1) 3 3 x4/3 y 1/3 2 3 x5/3 y 2/3 8 3 x5/3 y 1/3 50. y 4 x 42
3
(a) [1, 0) (0, 4]
2x 3, 1 x 0
57. f x { (b) At x 0
x 3, 0 x 4
(c) Nowhere in its domain
x 1
, 2 x 0 (a) [2, 0) (0, 2]
x
58. gx
{ x (b) Nowhere
1
, 0 x 2 (c) Nowhere in its domain
x
In Exercises 59 and 60, use the graph of f to sketch the graph of f .
59. Sketching f from f
y
60. Sketching f from f
61. Recognizing Graphs The following graphs show the
distance traveled, velocity, and acceleration for each second
of a 2-minute automobile trip. Which graph shows
(a) distance iii? (b) velocity? i (c) acceleration? ii
(i)
(iii)
0 1 2
–3
t
–2
(ii)
1
–1
–1
y
0 1
1
y = f(x)
2 3
y = f(x)
2
t
x
x
(b) Is f continuous at x 1? Explain.
(c) Is f differentiable at x 1? Explain.
54. Writing to Learn For what values of the constant m is
sin 2x, x 0
f x { mx, x 0
(a) continuous at x 0? Explain.
(b) differentiable at x 0? Explain.
In Exercises 55–58, determine where the function is
(a) differentiable, (b) continuous but not differentiable, and
(c) neither continuous nor differentiable.
55. f x x 45 (a) For all x 0 56. gx sin x 2 1
(b) At x 0 (c) Nowhere
(b) Nowhere (c) Nowhere
(a) For all x
43. y2x 3 – 3x – 1, y6x 2 – 3, y 12x, y (4) 12, and the rest are all zero.
x4
44. y , 2 4
y x 3
6 , y x 2
, y (4) x, y (5) 1, and the rest are all zero.
2
0 1 2
62. Sketching f from f Sketch the graph of a continuous
function f with f 0 5 and
2, x 2
f x { 0.5, x 2.
63. Sketching f from f Sketch the graph of a continuous
function f with f 1 2 and
2, x 1
f x
{ 1, 1 x 4
1, 4 x 6.
47. (a) y 1 4 x 9 (b) y 4x 2
4
48. (a) y 5 4 x 6 (b) y 4 5 x 1 1
5
t