Calculus Basic Skills Pre-Test



Test #5 – Applications of Derivatives I - Practice Problems Name ____________________

Advanced Placement Calculus

Mr. Honigs

Multiple Choice Section (Weight = 50%)

Circle the letter of the most appropriate choice for each multiple choice item. The most appropriate choice will be written in exact simplified form.

1. Determine the critical values for the function [pic].

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

2. Determine on which interval(s) the function is increasing. The function [pic] is defined on [pic].

a. [pic]

b. [pic]

c. [pic]

d. [pic] and [pic]

e. [pic] and [pic]

3. Determine the interval(s) for which [pic]is concave down. The function [pic] is defined for all real numbers.

a. [pic]

b. [pic]

c. [pic] and [pic]

d. [pic]

e. [pic]

4. For which of the following functions f does the Mean Value Theorem guarantee that there will be c in [pic]such that [pic]?

I. [pic] II. [pic] III. [pic]

a. I only

b. II only

c. III only

d. II and III only

e. I, II, and III

5. The function [pic] is defined on [pic]. On what inverval(s) is [pic] decreasing?

a. [pic]

b. [pic]

c. [pic]

d. [pic] and [pic]

e. [pic] and [pic]

6. If [pic], then [pic] could be

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

7. Locate the extrema of the following function and classify each extrema: [pic].

a. local max @ [pic] and local min @ [pic]

b. local max @ [pic] and local min @ [pic]

c. local max @ [pic] and local min @ [pic]

d. local max @ [pic] and local min @ [pic]

e. no local extrema; the function never turns around

8. A function [pic], differentiable on [pic] has only one critical number [pic]. What feature must the graph of f have at [pic] if [pic] and [pic]?

a. a relative minimum

b. a relative maximum

c. a point of inflection

d. a zero

e. none of the above

9. At which point on the graph shown below are both the first and second derivatives of [pic] negative?

a. A

b. B

c. C

d. D

e. E

10. Let f be a function defined for all real numbers. Which of the following statement about

f must be true?

a. If [pic], then [pic].

b. If [pic], then -3 is in the range of f.

c. If [pic], then [pic] exists.

d. If [pic], then [pic] does not exist.

e. If [pic] does not exist, then [pic] does not exist.

Free Response Section (Weight = 50%)

Show all your work for problems in this section. Justify answers with full sentences. Use complete and correct notation on each and every item you work.

1. Consider the profit function, given by [pic]

a. Determine the critical points (values) of [pic]. Create and complete a sign chart for [pic].

b. Determine all intervals of increasing and decreasing for [pic].

c. Classify the local extrema for [pic]. Justify your answers. Discuss the real-life implications of the local extrema in the context of this profit function.

2. Consider the functions [pic], [pic], and [pic], defined on the indicated intervals.

[pic] [pic] [pic]

a. Find the value(s) of c guaranteed by Rolle’s Theorem in the indicated interval for [pic], if the theorem can be applied.

b. Find the value(s) of c guaranteed by the Mean Value Theorem in the indicated interval for [pic], if the theorem can be applied.

c. Locate the absolute extrema of the function [pic] on the indicated interval.

3. Consider the following information describing a function [pic].

[pic] when x < -2 [pic] when –2 < x < 0

[pic] when x > 0 [pic] when x = 0

[pic] when x = -2 [pic] when x < -2

[pic] when -2 < x < 0 [pic] when x > 0

[pic] when x = 0 [pic] when x = -2

a. Construct a sign chart for [pic].

b. Construct a sign chart for [pic].

c. Construct a sketch of one possible graph of [pic].

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