Mathematics 133 -- Intensive Calculus for Science 1
Exam 3 -- Things to Know
November 19, 2001
General Information
The third full-period exam of the semester will be given
in class on Friday, November 30. There will be seven or
eight questions (maybe grouped together as parts of larger
questions).
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 29. 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 Sections 3.3,3.4,3.5,3.6,3.7,
4.1, 4.3, and 4.5. However, as you have probably realized, this
course is really cumulative. The material on the meaning of
derivatives from Chapter 2 is the basis for what we did in
Chapter 4. Also, the basic derivative rules from Sections 3.1 and 3.2
will come up frequently in computing derivatives. 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.
And the basic algebra review we did the first week of the course
is always relevant -- you'll need to be able to factor, apply rules for
exponents, manipulate fractions, etc.
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.
The following are the new topics to be tested on this exam:
- The product rule for derivatives: d/dx(f(x)g(x)) = f(x)g'(x) + f'(x)g(x)
- The quotient rule for derivatives:
d/dx(f(x)/g(x)) = (g(x)f'(x) - f(x)g'(x))/(g(x))2
- The chain rule for derivatives: d/dx(f(g(x)) = f'(g(x))g'(x)
- Derivatives of trigonometric functions: d/dx(sin(x)) = cos(x),
d/dx(cos(x)) = -sin(x)
- Derivatives of inverse functions via the chain rule;
d/dx(ln(x)) = 1/x, d/dx(arcsin(x)) = 1/sqrt(1 - x2),
d/dx(arctan(x)) = 1/(1 + x2)
- Implicit differentiation
- Know how to use these derivative rules singly and in combination. You should be
able to find the derivative of any function such as those in problems 1-54 in the
Review Exercises for Chapter 3 (see page 159 in text).
- Critical points; the First Derivative Test for Local Maxima and Local
Minima; The Second Derivative Test for Local Maxima and Local Minima.
(Be able to give precise statements of what these say, in addition to
knowing how to use them.) Good review problems: Review Exercises for Chapter
4 (page 212): 7-19.
- Optimization: Finding the overall maximum and/or minimum of a continuous
function on an interval a <= x <= b.
- Applied Optimization problems like Section 4.5/4-17. Important Note:
Because of the short time remaining in the semester,
there will not be a problem set covering these applied optimization problems,
but there will be a group discussion on them Tuesday, November 20. You will need
try a number of these problems on your own as well.
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
the ``Check Your
Understanding'' Problems from Chapter 3 (p. 162 in text)
and Chapter 4 (p. 216)
- Try the sample exam last after you have done the rest of
your review.