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

202 Chapter 2 • The Derivative and Its Properties

85. The line x = c, where c > 0, intersects the (a) If the tangent line to the cubic at the point P is parallel to the
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cubic y = 2x + 3x − 9 at the point P and intersects the line tangent to the parabola at the point Q, find the number c.
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parabola y = 4x + 4x + 5 at the point Q, as shown in the
figure below. (b) Write an equation for each of the two tangent lines described
y x  c in (a).
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86. f (x) = Ax + B, A > 0.
(a) Find c, c > 0, in terms of A so that the tangent lines to the
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y  2x ! 3x 9 graph of f at (c, f (c)) and (−c, f (−c)) are perpendicular.
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y  4x ! 4x ! 5 (b) Find the slopes of the tangent lines in (a).
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(c) Find the coordinates, in terms of A and B, of the point of
intersection of the tangent lines in (a).
2 2 4 x
P
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Preparing for the AP Exam
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AP Practice Problems
Multiple-Choice Questions
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194 1. If g(x) = x, then g (7) = 199 9. Find f (1) if f (x) = 3e − 5x + 2 ln x − 5.
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(A) 0 (B) 1 (C) 7 (D) (A) −15 (B) 3e − 5 (C) 3e − 18 (D) 3e − 13
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3(2 + h) − 3 · 16
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197 2. The line x + y = k, where k is a constant, is a line tangent 195 10. lim =
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to the graph of the function f (x) = x − 5x + 2. h→0 h
What is the value of k? (A) 0 (B) 32 (C) 48 (D) 96
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(A) −1 (B) 2 (C) −2 (D) −4 197 11. Which is an equation of the line tangent to the graph
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of f (x) = x + 3x + 2 at the point where f (x) = 2?
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197 3. An object moves along the x-axis so that its position
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at time t is x(t) = 3t − 9t + 7. For what time t is (A) y = 2x + 2 (B) y = 2x + 2.929
the velocity of the object zero?
(C) y = 2x + 1.678 (D) y = 2x − 2.929
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(A) −3 (B) 3 (C) (D) 7
2 Free-Response Questions
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198 4. If f (x) = e , then ln( f (3)) = 197 12. For the function f (x) = x + 4
(a) Find f (1).
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(A) 3 (B) 0 (C) e 3 (D) ln 3
(b) Find an equation of the line tangent to the graph
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197 5. An equation of the line tangent to the graph of f at x = 1.
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of g(x) = x + 2x − 2x + 1 at the point where x = −2 is (c) Find f (−4).
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(A) x + 2y = 12 (B) y + 2x = 9 (d) Find an equation of the line tangent to the graph
of f at x = −4.
(C) 2x + y = −9 (D) y − 2x = 9
(e) Find the point of intersection of the two tangent lines
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195 6. The line 9x − 16y = 0 is tangent to the graph found in (b) and (d).
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of f (x) = 3x + k, where k is a constant, at a point in
( 2
the first quadrant. Find k. −x + x + 1 if x < 0
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197 13. f (x) = 1 if x = 0
3 3 3 9 x
(A) (B) (C) (D) e if x > 0
32 16 64 64
(a) Determine whether f is continuous at x = 0. Justify your
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199 7. If f (x) = 3x + ln x, find f (1). answer.
(A) 2 (B) 0 (C) 3 (D) 4 (b) Find an equation of the line tangent to the graph of f
at x = − 1.
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197 8. The cost C (in dollars) of manufacturing x units of a product (c) Find an equation of the line tangent to the graph of f
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is C(x) = 0.3x + 4.02x + 3500.
at x = 1.
What is the rate of change of C when x = 1000 units?
(A) 307.52 (B) 0.60402 (C) 604.02 (D) 1020
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