General Information

The first exam for the course will be given next Friday, October 8, as announced in the course syllabus. It will cover the material discussed in class from the start of the semester through and including the material from Monday, October 4. 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 after 4:00 pm on Wednesday, October 6. Important Note: I will be leaving campus on Thursday, October 7 at about 12:00 noon to attend a conference. The exam will be proctored by Prof. Brevik of our department. I will be available for "special office hours" 11:00 am to 12:00 noon on Thursday for last minute questions, but my regular office hours will be cancelled that day. Please start to prepare for the exam early so that you will have the opportunity to have your questions answered at the review session or on Thursday morning.

Suggested Practice Problems

From the text:

• Chapter 1 Review Problems: p. 86-90/7, 8. 9, (use what you know about the formulas, don't just graph each of them with your graphing calculator!), 11, 35-43, 46
• Chapter 2 Review Problems: p. 146-7/1,2,5 (Note: 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!), 7 - 12
• Be able to differentiate any functions like p. 237/1 - 30 given by formulas. (The derivative computation might come as part of a larger problem.)
• Chapter 5 Review Problems: p. 281/3 - 6
• Chapter 9 Review Problems: p. 471/1,2,3,5,6,7,8,

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| (absolute value function).
  Which of them would eventually look the same? Which would look locally linear? For those that would look linear, find the equations of the lines you would see 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 at a general x.

Sample Exam

I. (20) The gross domestic product (GDP) of a country has grown as in the following table

t = time, in years	0	10	20	30	40	50
G(t) = GDP	500	610	720	831	940	1051

Is a linear function or an exponential function a better "fit" for these data points? Explain how you are deciding which is better and find a formula for a function that approximates the values in this table.

II. (10) What is the exact mathematical definition of the derivative of a function f(x) at x = a? Explain how this relates to the idea of the rate of change of f at a.

III. A population of bacteria grows in a medium that can support no more than P₁ of bacteria. At time t = 0, the number of bacteria present is P₀ < P₁. The population grows more and more rapidly at first, but then the rate of growth decreases and the population approaches P₁, without exceeding it.

• A. (15) Sketch a graph of the population of bacteria as a function of time that fits this description.
• B. (15) Sketch the graph of the derivative of the population function.

IV. Find the derivative functions of each of the following functions defined by formulas:

• A. (5) f(x) = 3x⁴ + 5 sqrt(x) - 3 cuberoot(x)
• B. (7.5) g(x) = sin(x) /(1 + cos(x))
• C. (7.5) h(x) = arctan(2^x)

V.

• A. (15) Find the Taylor polynomial of degree 4 for f(x)=e^sin(x) with a = 0.
• B. (5) Use your polynomial to approximate f(.2).