Mathematics 36, section 1 -- AP Analysis

Review Sheet for Exam 1

September 19, 1997

General Information

The first exam for the course will be given next Friday, Sept. 26, as announced in the course syllabus. It will cover the material discussed in class from the start of the semester it through and including the material from Monday, September 22. 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 (similar to the questions from Lab Day assignments).

Review Session

If there is interest, I will be happy to schedule a review session outside of class time to help you get ready for the exam. I will be available any time between 4:00 p.m. and 7:30 p.m. on Wednesday, or any time after 4:00 p.m. Thursday (including evening).

Suggested Practice Problems

From the text:

• Chapter 1 Review Problems: p. 86-90/7, 13, 14, 26 (use what you know about the formulas, don't just graph each of them with your graphing calculator!), 28-36, 37, 40, 42
• Chapter 2 Review Problems: p. 146-7/2 (Note: log = log_{10} -- the idea of the problem is to approximate the derivative without using the formula. You can gauge the accuracy of your approximation by using the formula too!), 10,11,12,13,16
• Be able to differentiate any functions like the ones in p. 237/1 - 21, 24. (The derivative computation might come as part of a larger problem.)

Other problems:

• Imagine you are zooming in on the graphs of the following functions near the origin:
1. y = sin(x) - tan(x)
2. y = e^x - 1
3. y = x²/(x² + 1)
4. y = x/(x² + 1)
5. y = x/(x + 1)
6. y = |x| (the absolute value function).

Which of them will eventually look indistinguishable? Which would be locally linear? For those that are, find the equations of the lines without carrying out the zooming process.

• State the precise definition of the derivative of a function f at x = a. Use the definition to compute f'(x) for f(x) = 2x² - 4, and f(x) = sqrt(x) (the square root function).