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Instructor: Name:
Exam IV
Calculus I; Fall 2011
Part I
Part I consists of 10 questions, each worth 6 points. Clearly show your work for each of the problems listed.
(1) Let f (x) = 3x^4 + 4x^3 โ 12 x^2. Find all local/absolute max/min of f (x). State both x and y coordinates.
(2) Find the absolute max/min of f (x) = x^5 โ 1 on the interval [โ 1 , 1]. Give both x and y-coordinates and justify your answer.
(3) Find two positive numbers whose sum is 10 and whose product is maximal. (You must justify your answer!)
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(4) Find the number c whose existence is guaranteed by the Mean Value Theorem for the function y = f (x) = x^2 on the interval [โ 1 , 2].
(5) If f โฒ(x) = (x โ 3)^4 (x +5)^5. Note that you are already given the derivative f โฒ(x). Find all critical points, where f (x) is increasing and decreasing, and also find the x-coordinate(s) of all local max/min.
(6) Find the most general anti-derivative of f (x) = x
(^2) sin(x)+x 3 x^2.
Part II
Part II consists of 3 problems; the number of points for each
part are indicated by [x pts]. You must show the relevant
steps (as we did in class) and justify your answer to earn credit. Simplify your answer when possible. (1) [10 pts] Find the absolute max/min of the function f (x) = (x^2 โ 2 x)^3 on the interval [โ 2 , 2].
(2) Given the function f (x) = (x
(^2) โ4) (x+1)^2 (a) [2 pts] Find the x and y intercepts of the function.
(b) [3 pts] Find all asymptotes.
(c) [4 pts] Find the open intervals where f (x) is increasing and the open intervals where f (x) is decreasing,
(d) [2 pts] Find the local maximum and local minimum value(s) of f (x). (Be sure to give the x and y coordinate of each of them).
(e) [2 pts] Find all open intervals where the graph of f (x) is concave up and all open intervals where the graph is concave down.
(f) [2 pts] Find all points of inflection (be sure to give the x and y coordinate of each point).
(g) [5 pts] Use the above information to graph the function on the next page. Indicate all relevant information in the graph.
(3) [10 pts] An oil rig is located 2 km off shore at point A. The closest point B on the shore is 15 km from an oil refinery (which is also located on the shore). If it costs $100/km to lay a pipe line in the ocean and $5/km to lay a pipe line on land, deter- mine the cheapest way to lay a pipe line from the oil rig to the refinery.
