(given in one 70-minute period for problems 1 - 6 and a second period, where unlimited time was allowed, for problems 7 - 11.)

I recommend that, after you finish the homework for Chapters 5, 6 and the handout on integration by substitution, you take this sample test just like a test. Make a note of which problems you feel confident that you worked correctly and which you didn't.

Then, since no answers or solutions are provided here, you may want to show your work to a tutor or the instructor for comments on whether you are doing the problems correctly.

Before they will discuss this with you, you must first, for each problem,

1. identify which section of the text has problems similar to this
2. find an odd-numbered problem that is similar to this one
3. work it (or find it in the homework that you have already worked)
4. check your answer
5. make a judgment about whether your solution to the sample test problem is correct

Bring all of that work with you when you come to the tutor or instructor.

Materials allowed: Calculator or Winplot software, scratch paper

Show all your work on blank paper. Your work, as well as your answers, is graded and you must use the analytic calculus techniques we learned in the course to do the problems in order to get full credit. Make sure your final answer to each part of the problem is clearly indicated.

Make sure that the problems are in order on the test or that there is a clear note in the appropriate place saying where to find the work and the solution. Each problem is worth 10 points unless otherwise indicated. (Note for the Web: There was some choice in the test. Problem 11 could replace any two of problems 1, 2, 3, 5, or 6, because the time allowed to work those was too short.)

1. A company sells running shoes to dealers at a rate of $20 per pair if less than 50 pairs are ordered. If 50 or more pairs are ordered (up to 600), the price per pair is reduced at a rate of 2 cents times the number ordered. What size order will produce the maximum amount of money for the company?

2. Consider the function . Find the upper and lower bounds on the interval . Show (completely and carefully) the calculus that is needed to do this.

3. Find the following definite or indefinite integrals.

4. (5 pts) Solve this equation and give the result correct to six decimal places. Write at least one sentence explaining the process you followed to do this. for

5. (Problem to be added to this Web document later. It was a graph of the derivative of a function. From that, the student was asked to sketch the graph of the second derivative of the function and also the graph of an antiderivative of the function. Also, the student was asked to answer several questions about the function (where are local extrema and points of inflection) from looking at the graph of the derivative.)

6. A wire 36 cm long is to be cut into two pieces. One of the pieces will be bent into the shape of an equilateral triangle and the other into the shape of a rectangle whose length is twice its width. Where should the wire be cut if the combined area is to be

a. a maximum?

b. a minimum?

7. Consider the function . Use calculus to find critical values, any local extrema, possible points of inflection, and any points of inflection. Make sure that your work is very clear. Use all of that to sketch the graph.

8. The temperature change, T, in a patient generated by a dose, D, of a drug is given by . Assume C is a positive constant.

a. What dosage maximizes the temperature change?

b. The sensitivity of the body, at dosage D, to the drug, is defined as . What dosage maximizes sensitivity?

9. On a certain planet, the acceleration due to gravity is 12 ft/sec/sec. An astronaut jumps up with an initial velocity of 16 ft/sec.

a. How high does he go?

b. When does he hit the ground?

10. (5 pts each) Find the following indefinite integrals.

a. b. c.

11. The material for the top and bottom of a cylindrical container costs $2 per square foot. The material for the curved part which is the sides costs $1 per square foot. The allotment for the total cost of the material is $48. Express the volume of the container as a function of the radius and then find the dimensions of the container that maximizes the volume.

Last updated December 23, 1997. mparker@austincc.edu