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MA 125, CALCULUS I
December 03, 2007
Name (Print last name first):...................................................
Student Signature:............................................................
TEST IV
No calculators are permitted!
PART I - Basic Skills
Part I consists of 6 questions. Clearly write your answer in the space provided after each question.
Each question is worth 6 points.
Question 1
Find the absolute minimum value of the function f (x) = x^3 − 3 x + 1 on the closed interval [0, 2]. (Be sure to give the y-coordinate!)
Answer:.....................
Question 2
The function f (x) = x −
x^2 satisfies the hypotheses of the Mean Value Theorem on the
interval [0, 2]. Find the number c that satisfies the conclusion of the Mean Value Theorem.
Answer:..................... Question 3
Find the critical number(s) of the function f (x) =
x^3 +
x^5.
Answer:..................
PART II - Problem Solving Skills
Each problem is worth 16 points.
Part II consists of 5 problems. You must show your work to get full credit. Displaying only the final answer (even if correct) without the relevant steps will not get full credit.
Problem 1
Suppose that the derivative of a function f is given by
f ′(x) = (x − 2)^7 (x^2 − 4)
Answer all the following questions.
(a) Find all the critical numbers of the function f.
(b) On what interval(s) is the function f increasing? (Justify your answer!)
(c) On what interval(s) is the function f decreasing? (Justify your answer!)
Problem 2
This problem has two separate questions. (Answer all the questions.)
(1) Find the dimensions of a rectangle with perimeter 80 ft and whose area is as large as possible.
(2) Find two positive numbers whose product is 49 and whose sum is minimal.
Problem 4
Consider the function f given by
f (x) = xe−x^
which can also be written as f (x) = x ex
Answer all the following questions.
(a) Find the x and y-intercept(s) of the curve.
(b) Find, if any, the vertical and horizontal asymptote(s) of the curve. [Hint: L’Hospital’s Rule might prove useful here!]
(c) Find the (open) interval(s) of increase, and the (open) interval(s) of decrease. [Hint: Factoring out might prove useful in your calculations!]
(d) Find, if any, all local maximum and minimum value(s). [Be sure to give the y- coordinate(s)!]
(e) Find the open interval(s) where the function is concave down, and the open interval(s) where it is concave up. [Hint: Factoring out might prove useful in your calculations!]
(f) Find the inflection point. [Be sure to give the x and the y coordinate!]
(g) Use the information from parts (a)–(f) above to sketch the graph.
