HW 1.6
AP Calculus
IVT
Name:
Date:
Collaborator:
1.) Draw a function 𝑓 that is defined on a closed interval, and whose values at the endpoints
have opposite signs, but for which the equation 𝑓(𝑥) = 0 has no solution in the interval.
2.) Draw a function 𝑓 that is defined on a closed interval, has two vertical asymptotes, and
whose values at the endpoints have opposite signs, but for which the equation 𝑓(𝑥) = 0 has no
solution in the interval.
3.) Draw a function 𝑓 that is defined on a closed interval whose values at the endpoints have
opposite signs, but for which the equation 𝑓(𝑥) = 0 has at least one solution in the interval.
4.) Explain what the three examples above have to do with the intermediate value theorem.
Specifically mention the difference between continuous vs. discontinuous functions and the
intermediate value theorem.
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HW 1.6
AP Calculus
IVT
Name:
Date:
Collaborator:
5. Use the table below for the continuous function f (x ) to answer the questions that follow:
x
f (x )
-5
0
-4
-2
-3
-8
-2
1
-1
2
0
5
1
-1
2
-3
3
4
a. What is the minimum number of times that f (x ) must have a value of f ( x )  0 ? Justify
your answer with an explanation.
b. What is the maximum number of times that f (x ) could equal zero on the domain given in the
table? Justify your answer with an explanation.
6.) The table below contains input/output values for the continuous function 𝑔(𝑥)
x
𝑔(𝑥)
-10
0
-6
-2
-5
-8
-2
1
0
2
2
5
5
-1
8
-3
14
4
a.) Determine the intervals over which 𝑔(𝑥) must equal 3 at least once. Justify your answer
with an explanation.
b.) Determine the intervals over which 𝑔(𝑥) must equal 2.5 at least five times. Justify your
answer with an explanation.
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HW 1.6
AP Calculus
IVT
Name:
Date:
Collaborator:
7 a.) Verify that the Intermediate Value Theorem applies to f ( x )  x 2  x  12 on the interval
[−1,5].
b.) Find the value of c such that f ( c )  0 on that interval.
8.) Suppose that a function 𝑓 is continuous everywhere and that 𝑓(−2) = 3, 𝑓(−1) = −1,
𝑓(0) = −4, 𝑓(1) = 1, 𝑎𝑛𝑑 𝑓(2) = 5. On which of the following intervals does the Intermediate
Value-Theorem guarantee that 𝑓 has a root? You must explain your answer to receive credit.
a.) [−2, −1]
b.) [−1,0]
c.) [−1, 1]
d.) [0, 2]
9.) Show that the equation 𝑥 3 + 𝑥 2 − 2𝑥 = 1 has at least one solution in the interval [−1, 1].
3
HW 1.6
AP Calculus
IVT
Name:
Date:
Collaborator:
10.) Suppose that 𝑓 is a continuous function defined for all real numbers 𝑥 and
𝑓(−5) = 3 𝑎𝑛𝑑 𝑓(−1) = −2. If 𝑓(𝑥) = 0 for one and only one value of 𝑥, then which of the
following could be the value of 𝑥?
a.) −7
b.) −2
c.) 0
d.) 1
e.) 2
11.) If 𝑓(𝑥) = 𝑥 3 − 𝑥 + 3 and if 𝑐 is the only real number such that 𝑓(𝑐) = 0, then 𝑐 is between
a.)
b.)
c.)
d.)
e.)
−2 𝑎𝑛𝑑 − 1
−1 𝑎𝑛𝑑 0
0 𝑎𝑛𝑑 1
1 𝑎𝑛𝑑 2
2 𝑎𝑛𝑑 3
𝑥+1
12.) Using the intermediate value theorem, determine whether 𝑔(𝑥) = 1−𝑥 has a root / xintercept / zero somewhere between [−3, 0].
𝑥+1
13.) Verify if the Intermediate Value Theorem applies to ℎ(𝑥) = 1−𝑥 on the interval [0, 5]. If
so, determine if there is a value of 𝑐 such that ℎ(𝑐) = −2 on the interval. Then determine the
value of 𝑐 that would make ℎ(𝑐) = −2.
14.) Use the intermediate value theorem to show that 𝑖(𝑥) = 𝑥 3 + 𝑥 + 1 crosses the 𝑥 − 𝑎𝑥𝑖𝑠
somewhere between 𝑥 = −2 and 𝑥 = 0.
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