I recommend that, after you finish the homework for through Chapter 5, section 3 and the indicated homework on the integral handout, you take this sample test just like a test. Make a note of which problems you feel confident 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. 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.

1. (5 pts) The graph of the rate of change of the function F is given below, where F is only defined on the interval [0,6]. If you are told that ,

a. what is the area of one square here?

b. for what value of x is F(x) the largest?

c. estimate that largest value of F(x).

d. for what value of x is F(x) the smallest?

e. estimate that smallest value of F(x).

2. (12 pts) Consider the curve .

a. Find .

b. find the equation of the tangent line to the curve at the point (0,0).

c. If , estimate y using the tangent line.

3. (10 pts) The quantity, q, of a certain camera sold depends on the selling price, p, so we write . You are given that and .

a. What does mean about the sale of cameras?

b. What does mean about the sale of cameras?

c. The total revenue R, earned by the sale of cameras is given by . Find the derivative of R with respect to the variable p and evaluate that derivative at .

d. What is the sign of your answer to part c? If the cameras are currently selling for $140, how should the price be changed to increase revenue?

4. (5 pts) Assume that a polynomial f has exactly one local maximum and one local minimum.

a. Sketch a possible graph of f.

b. What is the largest possible number of zeros f could have?

c. What is the least number of zeros f could have?

d. What is the least number of inflection points that f could have?

e. What is the smallest degree f could have?

5. (18 pts) (You must show your algebra and calculus work to get full credit for this problem, not just a graph.) Consider the function on the interval .

a. Find where f is increasing and where f is decreasing.

b. Find the largest and smallest values of f in decimal form.

c. Find all points of inflection.

d. Find where f is increasing most rapidly.

e. Sketch the graph of f.

f. Explain your answers to the following three questions with pictures:

i. How many roots are there for in this given interval?

ii. How many roots are there for in this given interval?

iii. How many roots are there for in this given interval?

Extra credit: (10 points)

Find constants a and b for the function so that the value of the function at and the function has a local maximum at . (You must show your work to earn credit.)

Part II.

Materials allowed: Scratch paper, calculator with one-line display. (NOT a graphing calculator)

Show all your work on blank paper. Your work, as well as your answers, is graded. Do 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 5 points.

In 1 - 6, find the derivative:

1.

2.

3.

4. , where k is a constant.

5.

6.

7. Consider that Find for a function defined by this equation:

8. Evaluate this integral:

9. Evaluate this integral:

10. Evaluate this integral:

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