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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Chapter 1 • AP ® Practice Exam, Big Idea 1: Limits 159
16. Let f be the function defined by
⎧
5x + 17 if −5 ≤ x < −3
⎪⎨
4 if x =−3
f (x) =
⎪⎩
8 − x 2 if −3 < x < 3
x − 4 if x ≥ 3
For what values of x is f NOT continuous?
(A) none (B) –3 only (C) 3 only (D) –3 and 3
k
17. If lim = 0, which of the following combinations of p and k
x→∞ x p
is not possible?
(A) p = –1, k = 1 (B) p = 0, k = 0
(C) p = 2, k = –1 (D) p = 3, k = –2
18. Which of the following functions does not satisfy
lim f (x) =∞?
x→∞
(A) f (x) = e 2x (B) f (x) = ln(x − 5)
(C) f (x) = 3e −x (D) f (x) = √ x + 4
19. If f (x) = x2 − 9
which of the following is false?
x − 3
(A) lim f (x) = 6
x→3
(B) f is continuous for all real numbers x.
(C) lim f (x) =∞
x→∞
(D) The domain of f is {x|x = 3}
20. Let f be the function given by f (x) = x2 − 3x
. For what value
x − a
of a is f continuous for all real numbers x?
(A) None (B) a = –3 (C) a = –1 (D) a = 3
Section 2: Free Response
Show all of your work. Be sure to indicate clearly the methods you use
to obtain your results.
1. The tax on gross income in a small country is computed using
the table below.
Gross Income, i
Tax, T
$0 up to but not
$0
including $30,000
$30,000 up to but not 10% of gross income
including $50,000
$50,000 and higher $5000 + 20% of gross income
in excess of $50,000
(a) Use a piecewise-defined function to represent T as a
function of i.
(b) Find lim
i→0 + T (i).
(c) Determine the numbers, if any, at which T is discontinuous.
(d) Use the definition of continuity to explain your answer to (c).
16. B
17. A
18. C
19. B
20. A
Free Response
1. (a)
⎧0 if 0≤ i < 30,000
⎪
Ti () = ⎨0.1i
if 30,000 ≤ i < 50,00
⎪
⎩⎪
0.2i− 5000 if i > 50,00
(b) 0
(c) T is discontinuous at 30,000.
(d) Answers will vary.
Chapter 1 • AP® Practice Exam, Big Idea 1: Limits
159
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