QUIZ #8: CALCULUS 1A (Stankova) Wednesday, March 17, 2004 Section 10:00–11:00 (Voight)

Name:

Please complete the following problem(s) in the space provided. You may not use a calculator. You will have 15 minutes to complete the quiz.

Please include all relevant intermediate calculations and explain your work when appropriate.

Problem 1. Verify that the function

f(x) = x^{3} + 2x − 2

satisfies the hypotheses of the Mean Value Theorem on the interval [0, 1].

Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.

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Problem 2. Let f(x) = x^{2}/(x − 2). Show that there is no value of c such that

f(3) − f(1) = f′(c)(3 − 1).

Does this contradict the Mean Value Theorem? Why or why not?

QUIZ #8: CALCULUS 1A (Stankova) Wednesday, March 17, 2004 Section 11:00–12:00 (Voight)

Name:

Please complete the following problem(s) in the space provided. You may not use a calculator. You will have 15 minutes to complete the quiz.

Please include all relevant intermediate calculations and explain your work when appropriate.

Problem 1. Let f(x) = xe^{x}.

(a) On what intervals is f increasing or decreasing? (Open or closed intervals are acceptable.) Explain your work. [Hint: e^{x} > 0 for all x.]

(b) Find the local maximum and minimum values of f. Explain.

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(c) Find the intervals of concavity and the inflection points of f. Explain.

(d) Draw the graph of f.