**MATH 2431 – Summer
Session 2005**

**NAME_______________________________________________________**

**You may NOT use a calculator on this portion of the test.
Show work and write clearly.**

1. (20 points) At a given instant
the legs of a right triangle are 5 cm and 12 cm long. If the short leg is
increasing at the rate of 1 cm/sec and the area is decreasing at the rate of 2 cm^{2}/sec, how fast is the long leg changing?

2. (20 points) Evaluate the following limits:

a. _{}

b. _{}

c. _{}

3. (12 points) The cost of producing *x* units of a commodity is given by _{}.

a. Find the marginal cost function.

b. Find the minimum value of the marginal cost.

4. (12 points) Find intervals of concavity:
_{}.

5. (12 points) Find the global (absolute) maximum and the
global (absolute) minimum of_{} on [–1, 3].
Explain (in detail) how you know there is a global
maximum and a global minimum.

6. (12 points) Given _{}

a. Find the
critical numbers of *f*(*x*).

b. Find the
intervals of increase/decrease of *f*(*x*).

7. (12 points) The graphs of _{} and _{} are provided below.
Find:

a. approximate critical numbers and inflection points of _{}. Explain.

b. approximate intervals of increase/decrease for _{}. Explain.

c. approximate intervals of concavity for _{}. Explain.

d. approximate maximum and minimum points of _{}. Explain.