Math 165

Spring 2001

Exam 3

1. (32 pt) Compute the following limits:

a) b) c) d)

2. (15 pt) Sketch the graph of the function (for your convenience the first two derivatives are ).

3. Sketch the graph of the following two functions. I have listed the derivatives for your convenience:

a) (15 pt) .

b) (10 pt).

4. (15 pt) You are given 100p square feet of material. You are instructed to construct a cylindrical can with a bottom, but no top, of maximum volume. What is the volume of this “maximal can”?

5. (15 pt) You wish to cut a piece of wire of length L into two pieces. The first piece you will form into a circle and the second piece will be formed into a square. How do you make the cut to maximize and minimize the total area of the formed by the square and circle?

6. (8 pt) Show that there is a unique solution to the equation (do not attempt to find it exactly…hint: Rolle’s Theorem).