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Unit 5 Progress Check: MCQ Part A
1. Let be the function given by . The derivative of is given by
. What value of satisfies the conclusion of the Mean Value Theorem
applied to on the interval ?
(A) 2.132 because
(B) 2.749 because
(C) 3.042 because
(D) 3.252 because
2. The derivative of the function is given by . On which of the following intervals
in is decreasing?
(A) , , and
(B) and
(C) and
(D) and
3. The temperature inside a vehicle is modeled by the function , where is measured in degrees Fahrenheit
and is measured in minutes. The first derivative of is given by . At what times , for
, does the temperature attain a local minimum?
(A) 0.354 only
(B) 1.962
(C) 3.299 only
(D) 0.354 and 3.299
4. Let be the function given by on the closed interval . Of the following intervals, on
which can the Mean Value Theorem be applied to ?
I. because is continuous on and differentiable on .
II. because is continuous on and differentiable on .
III. because is continuous on and differentiable on .
(A) None
(B) I only
(C) I and II only
(D) I, II, and III
5. Let be a differentiable function with and . Which of the following must be true for
some in the interval ?
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Unit 5 Progress Check: MCQ Part A
(A) , since the Extreme Value Theorem applies.
(B) , since the Mean Value Theorem applies.
(C) , since the Mean Value Theorem applies.
(D) , since the Intermediate Value Theorem applies.
6. Let be the function given by on the closed interval . On which of the following
closed intervals is the function guaranteed by the Extreme Value Theorem to have an absolute maximum and an
absolute minimum?
(A)
(B)
(C)
(D)
7. Let be the function defined by with domain . The function has no absolute minimum
and no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem?
(A) The domain of is not an open interval.
(B) The domain of is not a closed and bounded interval.
(C) The function is not continuous on its domain.
(D) The function is not differentiable on its domain.
8.
Selected values of a continuous function are given in the table above. Which of the following statements could be
false?
By the Intermediate Value Theorem applied to on the interval , there is a value such that
(A)
.
(B) By the Mean Value Theorem applied to on the interval , there is a value such that .
By the Extreme Value Theorem applied to on the interval , there is a value such that
(C)
for all in .
By the Extreme Value Theorem applied to on the interval , there is a value such that
(D)
for all in .
9. Let be the function defined by for . Which of the following statements
is true?
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Unit 5 Progress Check: MCQ Part A
(A) is decreasing on the interval because on the interval .
(B) is increasing on the interval because on the interval .
(C) is decreasing on the interval because on the interval .
(D) is decreasing on the interval because on the interval .
10. Let be the function defined by for . On what open interval is decreasing?
(A) only
(B)
(C)
(D) There is no such interval.
11. Let be a function with first derivative given by . At what values of does have a
relative maximum?
(A) only
(B) 0 only
(C) and 5 only
(D) , 0, and 5 only
12.
The graph of , the derivative of the function , is shown above for . Which of the following
statements is true for ?
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Unit 5 Progress Check: MCQ Part A
1. Let be the function given by . The derivative of is given by
. What value of satisfies the conclusion of the Mean Value Theorem
applied to on the interval ?
(A) 2.132 because
(B) 2.749 because
(C) 3.042 because
(D) 3.252 because
2. The derivative of the function is given by . On which of the following intervals
in is decreasing?
(A) , , and
(B) and
(C) and
(D) and
3. The temperature inside a vehicle is modeled by the function , where is measured in degrees Fahrenheit
and is measured in minutes. The first derivative of is given by . At what times , for
, does the temperature attain a local minimum?
(A) 0.354 only
(B) 1.962
(C) 3.299 only
(D) 0.354 and 3.299
4. Let be the function given by on the closed interval . Of the following intervals, on
which can the Mean Value Theorem be applied to ?
I. because is continuous on and differentiable on .
II. because is continuous on and differentiable on .
III. because is continuous on and differentiable on .
(A) None
(B) I only
(C) I and II only
(D) I, II, and III
5. Let be a differentiable function with and . Which of the following must be true for
some in the interval ?
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Unit 5 Progress Check: MCQ Part A
(A) , since the Extreme Value Theorem applies.
(B) , since the Mean Value Theorem applies.
(C) , since the Mean Value Theorem applies.
(D) , since the Intermediate Value Theorem applies.
6. Let be the function given by on the closed interval . On which of the following
closed intervals is the function guaranteed by the Extreme Value Theorem to have an absolute maximum and an
absolute minimum?
(A)
(B)
(C)
(D)
7. Let be the function defined by with domain . The function has no absolute minimum
and no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem?
(A) The domain of is not an open interval.
(B) The domain of is not a closed and bounded interval.
(C) The function is not continuous on its domain.
(D) The function is not differentiable on its domain.
8.
Selected values of a continuous function are given in the table above. Which of the following statements could be
false?
By the Intermediate Value Theorem applied to on the interval , there is a value such that
(A)
.
(B) By the Mean Value Theorem applied to on the interval , there is a value such that .
By the Extreme Value Theorem applied to on the interval , there is a value such that
(C)
for all in .
By the Extreme Value Theorem applied to on the interval , there is a value such that
(D)
for all in .
9. Let be the function defined by for . Which of the following statements
is true?
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Unit 5 Progress Check: MCQ Part A
(A) is decreasing on the interval because on the interval .
(B) is increasing on the interval because on the interval .
(C) is decreasing on the interval because on the interval .
(D) is decreasing on the interval because on the interval .
10. Let be the function defined by for . On what open interval is decreasing?
(A) only
(B)
(C)
(D) There is no such interval.
11. Let be a function with first derivative given by . At what values of does have a
relative maximum?
(A) only
(B) 0 only
(C) and 5 only
(D) , 0, and 5 only
12.
The graph of , the derivative of the function , is shown above for . Which of the following
statements is true for ?
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