5128_Ch03_pp098-184
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182 Chapter 3 Derivatives<br />
29. y csc 1 sec x, 0 x 2p<br />
30. r ( 1 sin<br />
u<br />
2<br />
1<br />
sin<br />
sin<br />
1<br />
2<br />
1 cos<br />
u <br />
1<br />
cos<br />
cos <br />
(1<br />
cos <br />
) 2<br />
)<br />
In Exercises 31–34, find all values of x for which the function is<br />
differentiable.<br />
31. y ln x 2 For all x 0 32. y sin x x cos x For all<br />
real x<br />
1 x<br />
33. y <br />
1 2 x For all x 1 34. y 2x 7 1 x 5<br />
For all x 7 In Exercises 35–38, find dydx.<br />
2 <br />
35. xy 2x 3y 1 y <br />
<br />
2 36. 5x 45 10y 65 15 1<br />
37. xy 1 y x 3<br />
3 (xy)1/5<br />
x or 1<br />
x2<br />
38. y 2 x<br />
<br />
x 1 2y(x<br />
1 1) 2<br />
In Exercises 39–42, find d 2 ydx 2 by implicit differentiation.<br />
39. x 3 y 3 1 2 x<br />
y5 40. y 2 1 2 x 1 2xy x 4 <br />
2<br />
y3<br />
41. y 3 y 2 cos x 42. x 13 y 13 4<br />
In Exercises 43 and 44, find all derivatives of the function.<br />
43. y x 4<br />
3 2 2 x<br />
5<br />
x2 x 44. y <br />
1 20<br />
In Exercises 45–48, find an equation for the (a) tangent and<br />
(b) normal to the curve at the indicated point.<br />
45. y x 2 2<br />
2x, x 3 (a) y x 3 (b) y 3<br />
x 5 3<br />
<br />
3<br />
2 2<br />
46. y 4 cot x 2 csc x, x p2 (a) y x 2 2 (b) y x 2 2<br />
47. x 2 2y 2 9, 1, 2 48. x xy 6, 4, 1<br />
In Exercises 49–52, find an equation for the line tangent to the curve<br />
at the point defined by the given value of t.<br />
49. x 2 sin t, y 2 cos t, t 3p4 y x 22<br />
50. x 3 cos t, y 4 sin t, t 3p4<br />
51. x 3 sec t, y 5 tan t, t p6 y 1 0<br />
x 53<br />
3<br />
52. x cos t, y t sin t, t p4 y (1 2)x 2 1 4 <br />
53. Writing to Learn<br />
or y 2.414x 3.200<br />
(a) Graph the function<br />
x, 0 x 1<br />
f x { 2 x, 1 x 2.<br />
41. 2 (3y2 1) 2 cos x 12y sin 2 x<br />
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<br />
3<br />
(a) [1, 0) (0, 4]<br />
2x 3, 1 x 0<br />
57. f x { (b) At x 0<br />
x 3, 0 x 4<br />
(c) Nowhere in its domain<br />
x 1<br />
, 2 x 0 (a) [2, 0) (0, 2]<br />
x<br />
58. gx <br />
{ x (b) Nowhere<br />
1<br />
, 0 x 2 (c) Nowhere in its domain<br />
x<br />
In Exercises 59 and 60, use the graph of f to sketch the graph of f .<br />
59. Sketching f from f<br />
y<br />
60. Sketching f from f<br />
61. Recognizing Graphs The following graphs show the<br />
distance traveled, velocity, and acceleration for each second<br />
of a 2-minute automobile trip. Which graph shows<br />
(a) distance iii? (b) velocity? i (c) acceleration? ii<br />
(i)<br />
(iii)<br />
0 1 2<br />
–3<br />
t<br />
–2<br />
(ii)<br />
1<br />
–1<br />
–1<br />
y<br />
0 1<br />
1<br />
y = f(x)<br />
2 3<br />
y = f(x)<br />
2<br />
t<br />
x<br />
x<br />
(b) Is f continuous at x 1? Explain.<br />
(c) Is f differentiable at x 1? Explain.<br />
54. Writing to Learn For what values of the constant m is<br />
sin 2x, x 0<br />
f x { mx, x 0<br />
(a) continuous at x 0? Explain.<br />
(b) differentiable at x 0? Explain.<br />
In Exercises 55–58, determine where the function is<br />
(a) differentiable, (b) continuous but not differentiable, and<br />
(c) neither continuous nor differentiable.<br />
55. f x x 45 (a) For all x 0 56. gx sin x 2 1<br />
(b) At x 0 (c) Nowhere<br />
(b) Nowhere (c) Nowhere<br />
(a) For all x<br />
43. y2x 3 – 3x – 1, y6x 2 – 3, y 12x, y (4) 12, and the rest are all zero.<br />
x4<br />
44. y , 2 4<br />
y x 3<br />
6 , y x 2<br />
, y (4) x, y (5) 1, and the rest are all zero.<br />
2<br />
0 1 2<br />
62. Sketching f from f Sketch the graph of a continuous<br />
function f with f 0 5 and<br />
2, x 2<br />
f x { 0.5, x 2.<br />
63. Sketching f from f Sketch the graph of a continuous<br />
function f with f 1 2 and<br />
2, x 1<br />
f x <br />
{ 1, 1 x 4<br />
1, 4 x 6.<br />
47. (a) y 1 4 x 9 (b) y 4x 2<br />
4<br />
48. (a) y 5 4 x 6 (b) y 4 5 x 1 1<br />
<br />
5<br />
t