Mathematics 136, section 2 – Advanced Placement Calculus Review Sheet for Exam 1 September 23, 2009

General Information

The first exam for the course will be given next Thursday, October 1 at 7:30pm in our regular classroom. You will have until 9:00pm to work on the exam if you need that much time. (Note: This is one day later than announced in the course syllabus because of the Hanify-Howland Lecture). It will cover the material discussed in class from the start of the semester through and including the material on implicit differentiation, derivatives of inverse functions and related rates problems from class on September 23. There will be 4 or 5 problems, some possibly with several parts. Some may ask for a graph or the result of a calculation; others may ask for a precise definition of a term or concept we have used, or a short description or explanation of some phenomenon.

Calculators

As announced in the course syllabus, I will be providing a basic scientific numerical calculator (TI model TI-30Xa) for you to use on this exam. These calculators can compute +, −, ×, ÷ and values of all functions we have talked about, but have no graphing or symbolic computation features.

Review Session

Since we are taking an extra day for the exam, class on Tuesday, September 29 or Wednesday, September 30 will be an optional review session (choice of which day will be up to the class). We will work some example problems, review the background for some topics, and so forth. I think you will find this to be valuable, but you are not obligated to attend if you feel well-prepared.

Suggested Practice Problems

From the text:

• Chapter 1 Review problems: 5, 7, 9-15, 24-26

• Chapter 2 Review problems: 1, limit problems like 3-18, 30, 32, 35, 37ab, 39, 41, 43

• Be able to differentiate any functions like p. 248-9/1 - 36 from the Chapter 3 Review problems (you don’t need to do all of these; just make sure you can handle all the different functions involved, can spot which derivative rule(s) apply, etc. Also you might need to compute a derivative like one of these as part of a larger problem.)

• Chapter 3 Review problems: 41, 47ab, 53-60, 65

• Section 4.1/3, 5, 11.

Sample Exam (The numbers next to each part are the possible points on that part; they add up to 100. Disclaimer: The actual exam questions may be posed differently, may combine these topics in somewhat different ways, etc.)

I. A. (5) Plot the graph of the piecewise-defined function:

\begin{cases}t + 1 & if t < 0 \\ −3 & if t ≥ 0. \end{cases}

B. (10) On one set of axes, plot the graphs y = 2f(t − 2) and y = f(−t) + 1. Label which is which.

C. (5) Is f continuous at t = 0? Explain why or why not.

D. (5) Is f continuous at t = 2? Explain why or why not.

II. A population of wildebeest grows in a habitat that can support no more than P_{1} animals. At time t = 0, the number of wildebeest present is P_{0} < P_{1}. The population grows more and more rapidly at first, but then the rate of growth decreases and the population approaches P_{1}, without exceeding it.

A. (10) Sketch a graph of the population of wildebeest as a function of time that fits this description. For full credit your graph should show proper slope and concavity.

B. (10) Give a qualitative graph of the derivative function of your population function.

III. A. (5) What is the exact mathematical definition of the derivative of a function f(x) at x = a?

B. (10) Use the definition (not the “shortcut rules”) to compute the derivative of f(x) = 4x^{2} − 2x + 2 at a general x.

C. (5) What is the equation of the tangent line to y = 4x^{2} − 2x + 2 at the point (1, 4)?

IV. Compute derivatives using the “shortcut rules”:

A. (5) f(x) = 2x^{3} + 6x^{2/3} + e^{π}. For which x is f differentiable?

B. (5) f(x) = \frac{sin(x)}{1+cos(x)}

C. (5) g(x) = tan^{−1}(e^{x} − ln(x))

D. (5) Find \frac{dy}{dx} by implicit differentiation if xy^{6} + e^{y} = 1.

V. (15) A cube of ice is taken out of a freezer and left on a kitchen counter to melt, losing volume at a rate of 2 cubic inches per minute. At what rate is the side of the cube changing when the volume is 64 cubic inches?