Intensive Calculus 1 -- Exam 2

Mathematics 133 -- Intensive Calculus for Science 1

Exam 2 -- Things to Know

October 26, 2001

General Information

The second full-period exam of the semester will be given in class on Friday, November 2. There will be seven or eight questions (maybe grouped together as parts of larger questions) similar to problems from the quizzes so far. I will let you use a graphing calculator on this exam, but calculators like the TI 89 or 92 with symbolic manipulation are not allowed on this or any other exam.

We will review for the exam in class on Thursday, November 1. If there is interest I can also see if Kathryn Blaisdell is available for an evening ``last minute question'' session on Thursday.

Material To Know

The exam will cover the material from Section 1.6, the sections we covered in Chapter 2, and Sections 3.1 and 3.2. Note: Many of the properties of linear, polynomial, and exponential functions you will need to have at your fingertips come from the material from Chapter 1, though. And the basic algebra review we did the first week of the course is always relevant -- you'll need to be able to do things like that accurately, quickly, and without having to think about the steps too much. You might want to devote part of your review time to those topics if you feel like you need it.

You should know the following material.

Polynomial functions: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀. Know:
1. How to find x-axis intercepts by factoring
2. How the sign of a_n determines the overall shape of the graph
3. How to determine the smallest possible n and/or a possible equation from a graph (like Section 1.6/5,10-13)
Rational functions: f(x) = p(x)/q(x), where p(x), q(x) are polynomials. Know:
1. How to determine x-axis intercepts, vertical asymtotes, horizontal asymptotes from the equation (like Section 1.6/6)
2. How to determine a possible equation for a graph of a rational function. I would concentrate on relatively simple cases here -- for instance where p(x), q(x) both have degree 2 or less. (See for instance, Section 1.6/31, 32, Review problems for Chapter 1/16
Average and instantaneous speed -- see the problems from Section 2.1 on Problem Set 4.
Limits -- see the problems from Section 2.2 from Problem Set 4.
How to spot points on a graph where f'(x) = 0, where f'(x) > 0 (intervals where the function f is increasing), and where f'(x) < 0 (intervals where f is decreasing).
How to compute a simple derivative (say a degree 2 polynomial function, or f(x) = 1/x, etc.) from the limit definition of f'. (See section 2.4 problems 13-16)
How to sketch y = f'(x), given the graph y = f(x) (See section 2.4/23 - 30)
The derivative as rate of change, the units of dy/dx, interpreting statements about dy/dx (like Section 2.5/1-4,11-14)
The second derivative f''(x), its relation to concavity Know
1. How to identify where f''(x) > 0 (intervals where the graph y = f(x) is concave up, or f'(x) is increasing), and where f''(x) < 0 (intervals where y = f(x) is concave down, or f'(x) is decreasing). (See Section 2.6/1-4.)
2. The interpretation of the second derivative as acceleration.
First ``shortcut rules'' for computing derivatives (Sections 3.1 and 3.2)
1. The derivative power rule: d/dx(xⁿ) = n x^n-1.
2. The derivative sum rule: d/dx(f(x) + g(x)) = f'(x) + g'(x)
3. The derivative constant multiple rule: d/dx(cf(x)) = cf'(x)
4. The derivative exponential rule: d/dx(a^x) = a^x ln(a)
Know how to use these singly and in combination. Also be able to do some preliminary algebra to put functions into a form where these methods apply, if necessary -- for instance splitting up quotients into sums of constants times powers, or multiplying out factored expressions.
Applications of these rules to rate of change problems, questions about whether a function is increasing or decreasing, concave up or concave down.

Suggestions on how to study

The goal of the homework problems is not to learn how do the particular problems assigned, but to practice using the ideas we have discussed in class and come to a full understanding of the ideas behind the problems by thinking through them. In addition to going over the quizzes and problem sets, make sure you look over the class notes and the discussions of why things are true in the text.
Along the same lines, spend some time with 1 - 20 from the ``Check Your Understanding'' Problems from Chapter 2 (p. 103 in text), especially if you weren't satisfied with your score on the lab and/or Quiz 5 (the one from Friday, October 19).
Try the sample exam last after you have done the rest of your review.