Intensive Calculus 1 -- Exam 2

Mathematics 133 -- Intensive Calculus for Science 1

Exam 2 -- Things to Know

October 28, 2005

General Information

The second full-period exam of the semester will be given in class on Friday, November 4. There will be nine or ten questions (maybe grouped together as parts of larger questions) similar to problems from the quizzes so far. I will provide one of the department calculators for your use.

We will review for the exam in class on Wednesday, November 2 or Thursday, November 3. Rosie Arcuri will also run a ``last minute question'' session on Thursday evening.

Material To Know

The exam will cover the material since the first exam: Sections 1.7 and 1.8, Chapter 2, and Sections 3.1-3.4. Note: Many of the properties of linear, polynomial, exponential, polynomial and rational 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.

Limits and Continuity -- know how to compute one-sided and two-sided limits, what the property of continuity means
Average and instantaneous speed -- see the problems from Section 2.1 on Problem Set IV.
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 2.3 problems 13-16)
How to sketch y = f'(x), given the graph y = f(x) (See section 2.3/23 - 30)
The derivative as rate of change, the units of dy/dx, interpreting statements about dy/dx (like Section 2.4/1-4,11-14)
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 2.6/19-21.)
2. The interpretation of the second derivative as acceleration.
First ``shortcut rules'' for computing derivatives (Sections 3.1-3.4)
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)
5. The derivative product rule: d/dx(f(x)g(x)) = f'(x)g(x)+f(x)g'(x)
6. The derivative quotient rule: d/dx(f(x)/g(x)) = (g(x)f'(x)-f(x)g'(x))/(g(x))²
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 just to teach the particular problems assigned, but to practice using the ideas we have discussed in class and come to a full understanding of the mathematics. Don't focus too much on the particular types of questions you have seen before -- be prepared for questions that might look slightly different.
Go 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 - 19 from the ``Check Your Understanding'' Problems from Chapter 2 (p. 106 in text).
Try the sample exam last after you have done the rest of your review.