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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
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116 Chapter 1 • Limits and Continuity
y
3
2
1
22 21
21
1 2
The graph indicates that f has zeros
at x =− 1 and x = 1 in the interval
( −2,2).
99. (a) Since f ( −2) and f (2) are both
positive, there is no guarantee that
the function has a zero in the interval
( −2,2).
(b)
y
2
1
x
112. f (x) = 3x 3 + 5x − 40; interval: [2, 3]
113. f (x) = x 4 − 2x 3 + 21x − 23; interval [1, 2]
114. f (x) = x 4 − x 3 + x − 2; interval: [1, 2]
115. Intermediate Value Theorem Use the Intermediate Value
Theorem to show that the function f (x) = x 2 + 4x − 2 has a
zero in the interval [0, 1]. Then approximate the zero correct to
one decimal place.
116. Intermediate Value Theorem Use the Intermediate Value
Theorem to show that the function f (x) = x 3 − x + 2 has a zero
in the interval [−2, 0]. Then approximate the zero correct to two
decimal places.
117. Continuity of a Sum If f and g are each continuous at c,
prove that f + g is continuous at c.(Hint: Use the Limit of a
Sum Property.)
118. Intermediate Value Theorem Suppose that the functions f
and g are continuous on the interval [a, b]. If f (a) g(b), prove that the graphs of y = f (x) and
y = g(x) intersect somewhere between x = a and x = b.
[Hint: Define h(x) = f (x) − g(x) and show h(x) = 0 for some
x between a and b.]
Challenge Problems
119. Intermediate Value Theorem Let f (x) = 1
x − 1 + 1
x − 2 .
Use the Intermediate Value Theorem to prove that there is a real
number c between 1 and 2 for which f (c) = 0.
120. Intermediate Value Theorem Prove that there is a real
number c between 2.64 and 2.65 for which c 2 = 7.
f (a + h) − f (a)
121. Show that the existence of lim
implies f is
h→0 h
continuous at x = a.
122. Find constants A, B, C, and D so that the function below is
continuous for all x. Sketch the graph of the resulting function.
⎧
x 2 + x − 2
if −∞ < x < 1
x − 1
⎪⎨ A if x = 1
f (x) =
B (x − C) 2 if 1 < x < 4
D
⎪⎩
if x = 4
2x − 8 if 4 < x < ∞
123. Let f be a function for which 0 ≤ f (x) ≤ 1 for all x in [0, 1].
If f is continuous on [0, 1], show that there exists at least one
number c in [0, 1] such that f (c) = c.
[Hint: Let g(x) = x − f (x).]
2 1 1 2
x
The graph indicates that f does not
have a zero in the interval ( −2,2).
100. If the functions intersect, then any
point of intersection must be a solution
3 2
of fx ( ) = x + x − 1=
0. This f is
continuous on [0,1] , with f (0) < 0 and
f (1) > 0, so IVT guarantees a solution,
and hence an intersection of the two
original functions, on (0,1).
(b) (0.755,0.430)
(c)
y
1.0
0.8
0.6
0.4
0.2
101. See TSM.
(0.755, 0.430)
0.2 0.4 0.6 0.8 1.0
102. h is continuous anywhere on
[ ab , ] except where fx ( ) = 0 . If
fx ( ) ≠ 0 anywhere on [ ab , ] , then
h is continuous everywhere on that
interval.
103. (a) x =− 2, x = 2, x = 3
(b) p = 5
(c) h is an even function.
104. This is because lim fx ( ) does not
x → 0
exist (because lim fx ( ) ≠ lim fx ( )).
−
x→0 x→+
0
105. Answers will vary. Sample answer:
fx ( ) = x
2 −1,
gx ( ) = x − 3, c = 3.
106. Answers will vary. Sample answer:
2
x −9
fx ( ) = has a removable
x −3
discontinuity at x = 3, and
⎧
⎪ x+ 3, x≤3
gx ( ) = ⎨
has a
2
⎩⎪ 9 − x , x>
3
nonremovable discontinuity at x = 3.
x
AP® Practice Problems
PAGE
103 1. The graph of a function f is shown below. Where on the open
interval (−3, 5) is f discontinuous?
23
21
(A) 3 only (B) −1 and 3 only
(C) 1 only (D) −1, 2, and 3
y
3
1
y 5 f (x)
2 3 5 x
PAGE
103 2. The graph of a function f is shown below.
22 21
y
3
1
y 5 f (x)
1 2 3 4
If lim x→c
f (x) exists and if f is not continuous at c, then c =
(A) −1 (B) 1 (C) 2 (D) 3
107. (1.125,1.25) 108. (1.625,1.75)
109. (0.125,0.25) 110. (0.375,0.5)
111. (3.125,3.25) 112. (2.125,2.25)
113. (1.125,1.25) 114. (1.25,1.375)
115. Since f is continuous on [0, 1],
f (0) < 0, and f (1) > 0, IVT guarantees that f
has a zero on (0, 1). Correct to one decimal
place, the zero is x = 0.8.
116. Since f is continuous on [−2, 0], f ( − 2) < 0,
and f (0) > 0, IVT guarantees that f has a zero
on ( −2,0). Correct to two decimal places, the
zero is x =− 1.52.
117. See TSM.
118. See TSM.
119. See TSM.
120. See TSM.
x
PAGE
104 3. If the function f (x) = x2 − 25
then f (−5) =
x + 5
is continuous at −5,
(A) −10 (B) −5 (C) 0 (D) 10
⎧ √ √
⎨ 2x + 5 − x + 15
if x = 10
PAGE
105 4. If f (x) = x − 10
⎩
k if x = 10
and if f is continuous at x = 10, then k =
1
(A) 0 (B) (C) 1 (D) 10
10
PAGE
105 5. If lim f (x) = L, where L is a real number, which of the
x→c
following must be true?
(A) f is defined at x = c. (B) f is continuous at x = c.
(C) f (c) = L. (D) None of the above.
PAGE
111 6. If f (x) = x 3 − 2x + 5 and if f (c) = 0 for only one real
number c, then c is between
(A) −4 and −2 (B) −2 and −1 (C) −1 and 1 (D) 1 and 3
PAGE
111 7. The function f is continuous at all real numbers, and f (−8) = 3
and f (−1) =−4. If f has only one real zero (root), then which
number x could satisfy f (x) = 0?
(A) −10 (B) −5 (C) 0 (D) 2
121. See TSM.
122. A = 3, B = 1 , C = 4, D = 0.
3
123. See TSM.
y
6
4
2
22 2 4 6
Answers to AP® Practice
Problems
1. D 2. C 3. A 4. B 5. D
6. A 7. B
x
116
Chapter 1 • Limits and Continuity
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