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

190 Chapter 2 • The Derivative and Its Properties

85. Tangent Lines and Derivatives Let f and g be two functions, 87. A function f is defined for all real numbers and has the following
each with derivatives at c. State the relationship between their three properties:
tangent lines at c if:
f (1) = 5 f (3) = 21 f (a + b) − f (a) = kab + 2b 2
1
′
(a) f (c) = g (c) (b) f (c) = − g (c) 6= 0
′
′
′
g (c) for all real numbers a and b where k is a fixed real number
′
independent of a and b.
Challenge Problems
(a) Use a = 1 and b = 2 to find k.
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86. Let f be a function defined for all real numbers x. Suppose f has
′
(b) Find f (3).
the following properties:
(c) Find f (x) for all real x.
′
′
f (u + v) = f (u) f (v) f (0) = 1 f (0) exists
88. A function f is periodic if there is a positive number p so
that f (x + p) = f (x) for all x. Suppose f is differentiable. Show
(a) Show that f (x) exists for all real numbers x.
′
that if f is periodic with period p, then f is also periodic with
′
(b) Show that f (x) = f (0) f (x).
′
′
period p.
R
Preparing for the AP Exam
R
AP Practice Problems
Multiple-Choice Questions
2
x − ax if x ≤ 1
PAGE PAGE
183 1. The function f (x) = , where a and b 183 5. If f (x) = |x|, which of the following statements
ax + b if x > 1
about f are true?
are constants. If f is differentiable at x = 1, then a + b =
I. f is continuous at 0.
(A) −3 (B) −2 (C) 0 (D) 2 II. f is differentiable at 0.
III. f (0) = 0.
PAGE
180 2. The graph of the function f , given below, consists of three line
(A) I only (B) III only
f (3 + h) − f (3)
segments. Find lim .
h→0 h (C) I and III only (D) I, II, and III
y PAGE
186 6. The graph of the function f shown in the figure has horizontal
6 tangent lines at the points (0, 1) and (2, −1) and a vertical
(0, 4) tangent line at the point (1, 0). For what numbers x in the open
4
interval (−2, 3) is f not differentiable?
(22, 2)
2
(6, 0) y
24 22 2 4 6 x 4
2
2 3
(A) −1 (B) − (C) − (D) The limit does not exist.
3 2
 2 22 2 x
 x − 25
 if x 6= 5 22
PAGE x − 5
186 3. If f (x) =

5 if x = 5

which of the following statements about f are true? (A) −1 only (B) −1 and 1 only
I. lim f exists. (C) −1, 0, and 2 only (D) −1, 0, 1, and 2
x→5
II. f is continuous at x = 5. PAGE f (1 + h) − f (1)
186 7. Let f be a function for which lim = −3.
III. f is differentiable at x = 5. h→0 h
Which of the following must be true?
(A) I only (B) I and II only I. f is continuous at 1.
(C) I and III only (D) I, II, and III II. f is differentiable at 1.
III. f is continuous at 1.
′
PAGE
186 4. Suppose f is a function that is differentiable on the open
interval (−2, 8). If f (0) = 3, f (2) = −3, and f (7) = 3, (A) I only (B) II only
which of the following must be true? (C) I and II only (D) I, II, and III
I. f has at least 2 zeros.
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II. f is continuous on the closed interval [−1, 7]. 180 8. At what point on the graph of f (x) = x − 4 is the tangent line
III. For some c, 0 < c < 7, f (c) = −2. parallel to the line 6x − 3y = 2?
(A) I only (B) I and II only (A) (1, −3) (B) (1, 2) (C) (2, 0) (D) (2, 4)
(C) II and III only (D) I, II, and III
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