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MA 125-8B, CALCULUS I
April 3, 2008
Name (Print last name first):..........................................
Your signature:...........................
TEST III
PART I
Part I consists of 6 questions. Clearly write your answer (only) in the space provided after each question. You do not need not to show your work for this part of the test. No partial credit is awarded for this part of the test!
Each question is worth 5 points.
Question 1
Differentiate the function y = e−x^ cos x.
Answer:.....................
Question 2
Find the derivative of the function f (x) =
ln x 2 x^2
Answer:.....................
Question 3
Differentiate the function y = ln(cos x).
Answer:..................
Question 4
Suppose h(x) = e−xg(x) where g(0) = 2 and g′(0) = −4. Find the numerical value of h′(0).
Answer:..................
Question 5
Find the derivative of the function y = tan−^1 (2x).
Answer:..................
Question 6
Evaluate (^) xlim→∞
ln(x + 2) 3 x − 5
Answer:..................
Problem 2
Consider the function f (x) = x^3 − 12 x on the interval [− 3 , 3].
(1) Find all critical numbers of f in the given interval.
(2) Find the maximum and minimum values of f on [− 3 , 3].
Problem 3
(1) Differentiate the function, and simplify completely by expressing your answer as a single fraction.
g(x) = ln
3
x + 1 x − 1
(2) Differentiate the function y = e−x^ cos^ x.
Problem 5
(1) Find the absolute maximum value of f (x) = x^3 − 3 x on the interval [0, 3].
(2) Show that the equation 2x + 8 − cos x = 0 has exactly one real root.
