Mathematics 36, section 5 – AP Calculus Information for Exam 3 November 23, 1999

General Information

As per the syllabus given out at the beginning of the semester, it’s almost time for the third midterm exam. This exam will be given the Friday of the week after Thanksgiving, December 3. The format will be similar to that of the other exams this semester.

Topics to be Covered

This exam will cover the material we have discussed since Exam 2, starting from applications of integrals (also covered on Exam 2, but worth a second look!), through the material on differential equations and growth models. The topics are:

1) Applications of integration (setting up problems via Riemann sums; in limit a definite integral is obtained) – volumes by slices, arclengths, physical examples like mass from non-constant density functions, etc. (A copy of the table of integrals from the text will be provided with the exam; any integral you need to compute will be do-able by some combination of substitution, integration by parts, and/or consultation with the table).

2) Differential equations – slope fields and solutions

3) Euler’s Method for numerical solutions of differential equations

4) Solving differential equations via separation of variables and integration,

5) Exponential and logistic population growth models.

Review Session

If there is sufficient interest, I would be happy to hold a review session in the afternoon or evening of Thursday, December 2. I have a commitment Wednesday evening (an orchestra rehearsal in Boston), so that time is not good for me. We can discuss this in class on Monday, November 29 and set it up.

Some good review problems

From the text:

• From the review problems from Chapter 8 (p. 405): 1-8,10,16,17,18

• From the review problems from Chapter 10 (p. 584-8): 1 - 22, 30,31,32,36

Sample Exam

THE EXAM PROBLEMS WILL DEAL WITH DIFFERENT GEOMETRIC AND PHYSICAL SITUATIONS, AND DIFFERENT DIFFERENTIAL EQUATIONS MOD-ELING BEHAVIOR. BE SURE YOU UNDERSTAND THE METHOD USED TO DE-RIVE THE SOLUTION – DO NOT JUST MEMORIZE THE SOLUTIONS OF THESE PARTICULAR PROBLEMS. It may help, as a part of your preparation, to set aside 50 minutes or so and take this as a practice test. Solutions will be distributed via the course homepage so that you can check your work.

I. A wire in the shape of the graph y = x^{2}, x ∈ [−1, 2] has density 4 − x at the point (x, x^{2}). A) Set up a Riemann sum approximating the total mass of the wire, and explain how

you got it. B) What definite integral computes the total mass? C) Evaluate your integral (using the table from the text as needed).

II. The amount of radioactive carbon-14 in a sample of living tissue from a plant produces about 13.5 atom disintegrations per minute per gram of carbon. In 1977, a charcoal fragment found near Stonehenge in England recorded 8.2 disintegrations per minute per gram of carbon. Assuming that the rate of change of the amount of carbon-14 is proportional to the amount at all times, and that it takes 5370 years for 50% of any sample to disintegrate, how old is the charcoal?

III. The population of fish in a lake is attacked by a microscopic water-borne parasite at t = 0, and as a result the population declines at a rate proportional to the square root of the population from that time on. A) Express this statement about the rate of growth of the population P as a differential

equation. B) There should be a constant of proportionality, say −k, in your equation. Setting

−k = −1, sketch the slope field for the corresponding differential equation for t ∈ [0, 4], P ∈ [0, 4], indicating the slope field segment at each point with integer coordinates.

C) Use n = 4 steps of Euler’s Method to approximate the value of the solution of your

differential equation with P (0) = 200 at t = 1. D) Use separation of variables to find the analytic solution of your differential equation. E) At t = 0 there were 900 fish in the lake; 441 were left after 6 weeks. When did the

fish population disappear entirely?