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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

200 Chapter 2 • The Derivative and Its Properties

In Problems 37–42: In Problems 49 and 50, for each function f:
(a) Find the points, if any, at which the graph of each function f (a) Find f (x) by expanding f (x) and differentiating the polynomial.
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has a horizontal tangent line. CAS (b) Find f (x) using a CAS.
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(b) Find an equation for each horizontal tangent line. (c) Show that the results found in parts (a) and (b) are equivalent.
(c) Solve the inequality f (x) > 0.
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2
49. f (x) = (2x − 1) 3 50. f (x) = (x + x) 4
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(d) Solve the inequality f (x) < 0.
(e) Graph f and any horizontal lines found in (b) on the same set
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of axes. Applications and Extensions
(f) Describe the graph of f for the results obtained in parts (c)
and (d). In Problems 51–56, find each limit.
8 8

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197 37. f (x) = 3x − 12x + 4 38. f (x) = x + 4x − 3 4 + h − 4
5
2 2 5(2 + h) − 5 · 2 5
x
39. f (x) = x + e x 40. f (x) = 2e − 1 51. lim 52. lim
h→0 h h→0 h
4
3
5
41. f (x) = x − 3x + 2 42. f (x) = x − 4x 3 √ 3(8 + h) − √ 3 · 8 5 π(1 + h) − π
10
53. lim 54. lim
h→0 h h→0 h
43. Motion on a Line At t seconds, an object moving on a line is s a(x + h) − ax 3 b(x + h) − bx n
3
n
3
meters from the origin, where s(t) = t − t + 1. 55. lim 56. lim
h→0 h h→0 h
Find the velocity of the object at t = 0 and at t = 5.
In Problems 57–62, find an equation of the tangent line(s) to the graph
44. Motion on a Line At t seconds, an object moving on a line is s
3
4
meters from the origin, where s(t) = t − t + 1. of the function f that is (are) parallel to the line L.
2
Find the velocity of the object at t = 0 and at t = 1. 57. f (x) = 3x − x; L: y = 5x
3
58. f (x) = 2x + 1; L: y = 6x − 1
Motion on a Line In Problems 45 and 46, each position function
x
59. f (x) = e ; L: y − x − 5 = 0
gives the signed distance s from the origin at time t of an object
x
moving on a line: 60. f (x) = −2e ; L: y + 2x − 8 = 0
61. f (x) = 3 ln x; L: y = 3x − 2
(a) Find the velocity v of the object at any time t.
62. f (x) = ln x − 2x; L: 3x − y = 4
(b) When is the velocity of the object 0?
3
9 63. Tangent Lines Let f (x) = 4x − 3x − 1.
2
3
45. s(t) = 2 − 5t + t 2 46. s(t) = t − t + 6t + 4
2 (a) Find an equation of the tangent line to the graph of f
at x = 2.
In Problems 47 and 48, use the graphs to find each derivative.
(b) Find the coordinates of any points on the graph of f where
47. Let u(x) = f (x) + g(x) and v(x) = f (x) − g(x). the tangent line is parallel to y = x + 12.
(c) Find an equation of the tangent line to the graph of f at any
y points found in (b).
(4, 5) (a) u (0) (b) u (4)
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(d) Graph f, the tangent line found in (a), the line y = x + 12,
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4 y f (x) (c) v (−2) (d) v (6) and any tangent lines found in (c) on the same screen.
(6, 4)
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(1, 2) (e) 3u (5) (f) −2v (3)
3
2
64. Tangent Lines Let f (x) = x + 2x + x − 1.
(1, 2) (1, 1)
(a) Find an equation of the tangent line to the graph of f
4 2 2 4 6 x at x = 0.
(5, 2)
2 (b) Find the coordinates of any points on the graph of f where
y g(x) the tangent line is parallel to y = 3x − 2.
(4, 3)
(c) Find an equation of the tangent line to the graph of f at any
48. Let F(t) = f (t) + g(t) and G(t) = g(t) − f (t). points found in (b).
(d) Graph f, the tangent line found in (a), the line y = 3x − 2,
y y g(t) (7, 6) and any tangent lines found in (c) on the same screen.
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6 (4, 6) (a) F (0) (b) F (3)
(c) F (−4) (d) G (−2) 65. Tangent Line Show that the y
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(3, 4) y e x
4 (e) G (−1) (f) G (6) line perpendicular to the x-axis
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(5, 3)
and containing the point (x, y)
y f(t) x (x, y)
2 on the graph of y = e and the
(5, 2) (2, 2)
tangent line to the graph
x
of y = e at the point (x, y)
4 2 4 6 t
intersect the x-axis 1 unit apart.
2 (5, 2) See the figure.
x  1 x x
66. Tangent Line Show that the tangent line to the graph
n
of y = x , n ≥ 2 an integer, at (1, 1) has y-intercept 1 − n.
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