MATH 2431 Summer Session 2005





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 cm2/sec, how fast is the long leg changing?



2. (20 points) Evaluate the following limits:











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.