MA 126
Review for Exam 2
March 2015
Here are some things you should know for the exam. You are also expected to know relevant precalculus and
Calculus I material, for example, the graphs of basic functions, how to use functional notation, the laws of
exponents and logarithms, principal values of sine and cosine, differentiation rules.
The exam is on Tuesday, March 17. Material covered: 7.1, 7.3, 8.1–8.4, 8.7, 9.1.
Transcendental Functions: Make sure you know the laws of exponents and logarithms! You should know
the basic integration formulas 1-20 in the table on page 431 in the text. (You were already supposed to
know most of these for exam 1, but some review is in order.) Know the definitions of the hyperbolic sine
and cosine functions in terms of exponentials, be able to sketch their graphs, and know their derivatives.
Techniques of Integration: Make sure you know how to use Change of Variables (or u-Substitution
Method) for both indefinite and definite integrals. Practice recognizing when an integral can be done with
just a substitution, or when a substitution is needed to put it into a form for which one of the techniques
from chapter 8 applies. Re-read my advice on the review sheet for exam 1.
Know how to state and apply the integration by parts formula for definite and indefinite integrals. Review
examples and homework for practice in making good choices for u and v. You do not need to memorize any
reduction formulas, but you should be able to use a reduction formula if it is given.
Be able to evaluate trigonometric integrals of the types found in section 8.2. You do not need to memorize
the formulas for products of sines and cosines.
Know the three trigonometric substitutions and be able to use them. Learn each substitution as a pair: for
√
example, x = a sin θ, a2 − x2 = a cos θ.
P (x)
Know the method of partial fraction decomposition for integrating rational functions Q(x)
. First, make sure
the degree of P (x) is less than the degree of Q(x). Factor Q(x), making sure the quadratic factors are really
irreducible. Write the correct form of the partial fraction decomposition, and solve for the undetermined
coefficients. Getting the form right is crucial, because with the wrong form you will generally not get a
solution. Finally, integrate the resulting expression.
Learn to express both types of improper integral correctly as a limit of proper integrals. Check the integrand
to see where vertical asymptotes occur (when is the denominator zero?) After integrating you must take a
limit, so review the types of limits that occur if you need to.
Sequences: Be able to apply Theorems 1–4 in section 9.1, together with basic limits from Theorem 5, to
find limits of sequences. Be prepared to state the Monotonic Sequence Theorem.
Read the problems carefully. Make sure you understand what is being asked. Reading and problemsolving skills require practice, so read all of the examples in the text and work a lot of exercises.
Document your work. You will get partial credit for any correct steps you show me, even if your final
answer is wrong. If you write the steps clearly you are much less likely to makes mistakes.