I recommend that, after you finish the homework for Chapter 2, 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, in order, on blank paper. Your work, as well as your answers, is graded. Please use a separate sheet for each problem (six of them). It's OK if it takes more than one sheet for a problem. Work on the problems in any order, but put them in the order on the test when you turn them in. Notice that the last problem, number 5, should take you about 1/3 of the class time. Allow enough time for that.

The point values are indicated for each problem and each part of each problem.

Usually, you will turn in your homework for the chapter on the day of the test. If you did not have it organized to turn in today, bring it in next time.

1. (25 points) Consider the graph of the function .

a. Find the equation of the tangent line to f at x=2.

b. Use that tangent line to approximate the function value at x=2.15.

c. How good is that approximation? (Write a complete sentence to answer this, and in the sentence mention the value of the approximation, the value of the function, and your conclusion about whether it is a good approximation.)

d. Find the x and y intercepts of the line in part a.

a. (12) Find the equation of the tangent line to f at .

b. (4) Use that tangent line to approximate the function value at .

c. (3) How good is that approximation? (Write a complete sentence to answer this, and in that sentence mention the value of the approximation, the value of the function, and your conclusion about whether it is a good approximation.)

d. (6) Find the x and y intercepts of the line in part a.

2. (15 points) For the function , find the derivative function algebraically, as we did in this chapter. (Don't use any shortcut formulas, even if you know them.) (Be sure to use correct and complete notation and to carefully all of show your work.. About half the credit for this problem is for the careful and complete notation.)

3. (15 points) For the function , as graphed here , (Graph to be added to the Web document when I learn to do that! In the meantime, look at some of the examples in the text.)

a. (3) For what x values is the function continuous?

b. (3) For what x values is the function differentiable?

c. (9) Sketch the graph of its derivative.
4. ( 15 points) Suppose that is the cost to heat my house, in dollars per day, when the outside temperature is T degrees Fahrenheit.

a. (5) What does mean? (Hint: What are the units of this derivative? )

b. (5) If and , approximately what is the cost to heat my house when the outside temperature is 21 degrees Fahrenheit?

c. (5) What does mean?
5. (30 points) Jamie is a swimmer who prides herself in having a smooth backstroke. Let be her position in an Olympic size pool, as a function of time ( is measured in meters, t is in seconds). (The pool is 50 meters long.)

In this table, we list some values for for a recent swim, in which she swam across the pool and back again. Notice how the position values indicate this. Also notice that the t values are not evenly spaced.

t	0	3.0	8.6	14.6	20.8	27.6	31.9	38.1	45.8	53.9	60
	0	10	20	30	40	50	40	30	20	10	0

a. (10) Sketch possible graphs for Jamie's position and velocity. Put scales on your axes and sketch the graphs rather carefully so that they have reasonably accurate numerical values at the various points.

b. (3) Find Jamie's average velocity over the entire swim.

c. (3) Find Jamie's average speed over the entire swim.

d. (5) Based on these data, is it possible to say whether or not Jamie's instantaneous speed was ever greater than 3 meters per second? Why?

e. (4) Give a brief verbal description of the graph of Jamie's position (i.e. describe where the position is increasing, decreasing, concave up or down.)

f. (5) Explain those features of the graph in terms of Jamie's swimming behavior.

6. (Extra credit - 6 points) The height of an object above the ground at time t is given by where represents the initial velocity and g is a constant called the acceleration due to gravity.

a. (1) At what height is the object initially?

b. (2) How long is the object in the air before it hits the ground?

c. (1) When will the object reach its maximum height?

d. (2) What is that maximum height?