Mathematics 36, section 1 -- AP Analysis

Review Sheet for Exam 3

November 24, 1997

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 5. 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, through the material on differential equations and growth models, and concluding with the material on parametric curves from class on Monday, December 1. The topics are:

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).
Differential equations -- slope fields and solutions
Solving differential equations via separation of variables and integration,
Exponential and logistic population growth models.
Systems of differential equations, phase plane trajectories, and parametric curves.

Review Session

If there is sufficient interest, I would be happy to hold a late afternoon or evening review session for this exam either on the afternoon of Wednesday, December 3 or during the afternoon or evening of Thursday, December 4. I have a commitment Wednesday evening (the Holy Cross Chamber Orchestra concert), so that time is not good for me. We can discuss this in class on Monday, December 1 and set it up.

Some good review problems

From the text:

From the review problems from Chapter 8 (p. 471-6): 1-8,10,12,26 (think about the shape of the apple!)
From the review problems from Chapter 9 (p. 584-8): 1,3,5,7,9,11, for these two, sketch the slope field: 20,21, also: 23,31,32,35,36
Consider the parametric curve
(x(t),y(t)) = ((e^t + e^-t)/2, (e^t - e^-t)/2)
Show that the coordinates (x,y) of every point on the curve satisfy the equation x² - y² = 1. Does the parametric curve (x(t),y(t)) pass through every point satisfying x² - y² = 1? Why or why not?
Verify that the curve
(x(t),y(t)) = (2e^4t + e^-t, -e^4t + 2e^-t)
is a phase-plane trajectory (solution) of the system dx/dt= 3x - 2y; dy/dt= -2x.

Sample Exam

THE EXAM PROBLEMS WILL DEAL WITH DIFFERENT GEOMETRIC AND PHYSICAL SITUATIONS, AND DIFFERENT DIFFERENTIAL EQUATIONS MODELING BEHAVIOR. BE SURE YOU UNDERSTAND THE METHOD USED TO DERIVE 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 solid has base equal to the triangle with corners at (0,0), (2,0), and (0,1) in the xy-plane. Its cross-sections by planes perpendicular to the x-axis are semicircles (with diameter extending the full width of the base at that point).

A) Find a formula for the cross-section area as a function of x.
B) Determine the volume of the solid.

II. A wire in the shape of the graph y = x², x in [-1,2] has density 4-x at the point (x,x²).

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).

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 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 in [0,4], P in [0,4], indicating the slope field segment at each point with integer coordinates.
C) At t = 0 there were 900 fish in the lake; 441 were left after 6 weeks. When did the fish population disappear entirely?

IV. A wheel of radius 1 rests on the x-axis in the plane and rolls without slipping to the right at a constant rate. A marked point attached to a radial spoke of length 3/2 follows the path
(x(t),y(t)) = (t + 3 sin(t)/2, 1 + 3 cos(t)/2)

A) Show that this curve is a solution of the system of differential equations dx/dt = y; dy/dt = -x + t.
B) Find the first time after t = 0 at which the dx/dt = 0 (direction of motion is vertical).
C) At how many different times is the x-coordinate of the marked point's position equal to Pi? Explain. (You do not need to find the times.)