I recommend that, after you finish the homework for Chapter 1, 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. Feel free to use one sheet per problem if that is necessary to be able to put the problems in order when you finish. Turn in this copy of the test along with your solutions.

1. The air in a factory is being filtered so that the quantity of a pollutant P (measured in mg/liter), is decreasing according to the equation , where t represents time in hours. If 15% of the pollution is removed in the first six hours,

a. What percent of the pollution is left after 10 hours?

b. How long will it take before the pollution is reduced by 50%?

2. A population of animals varies sinusoidally between a low of 1000 on January 1 and a high of 1600 exactly halfway through the year.

a. Graph the population against time. Measure time in units of days, and assume this is a year with 366 days.

b. Find a formula for the population as a function of time, t, measured in days since the start of the year.

3. A business had annual retail sales of $110,000 in 1981 and $224,000 in 1984. Assuming that the annual increase in sales followed a linear pattern,

a. Write the equation of a line that describes this relationship. Make sure that you have defined your variables.

b. Use your equation to find the retail sales in 1983.

4. Solve for t:

5. For and

a. Find

b. Find

6. If , what do you know about the values of a, b, and c if:

a. (3,26) is on the graph of .

b. (1,2) is the vertex (Recall that the axis of symmetry of such a parabola is .)

c. The y intercept of the graph is (0,8).

d. Find a quadratic function that satisfies all three conditions.

7. In each of the following problems, sketch a graph that illustrates all of the info given.

a. A drug is injected into a patient's bloodstream over a five-minute interval. During this time, the quantity in the blood increases linearly. After five minutes the injection is discontinued, and the quantity then decays exponentially. Sketch a graph of the quantity versus time.

b. When there are no other steroid hormones (for example, estrogen) in a cell, the rate at which steroid hormones diffuse into a cell is fast. The rate slows down as the amount in the cell the builds up. Sketch a possible graph of the quantity of steroid hormone in the cell against time, assuming that initially there are no steroid hormones in the cell.

8. The cost, C, of producing q articles is given by the function

.

a. Find a formula for the inverse function.

b. Explain in practical terms what the inverse function tells you.

9. (Extra Credit: 10 points) A catalyst in a chemical reaction is a substance which speeds up the reaction but which does not itself change. If the product of a reaction is itself a catalyst, the reaction is said to be autocatalytic. Suppose the rate, r, of a particular autocatalytic reaction is proportional to the quantity of the original material remaining times the quantity of the product, p, produced. If the initial quantity of the original material is A, and the expression for the amount remaining is

a. Express r as a function of p.

b. What is the value of p when the reaction is proceeding the fastest?

Last updated December 25, 1997. mparker@austincc.edu