Mathematics 101 Midterm 2 topics

The test will cover Chapter 6 of the textbook - see the topics below. While you will not be specifically tested on the earlier material (Chapter 5), that material is in many ways essential to everything we have been doing since then, eg. inverse trig substitutions (Chapter 6) will be useless to you if you can't evaluate the resulting trig integrals using the methods of Chapter 5.

Section 6.1:
- Integration for parts: indefinite and definite integrals. (You do not need to remember the "reduction formulas" at the end of the section, but it is useful to read this paragraph and try one or more of the exercises 31-34, just as it would be useful to do any other exercise involving integration by parts.)
- Exercises: 1-30.
Section 6.2:
- Inverse trigonometric substitutions: sine, tangent, secant.
- Completing the square (this should already be familiar to you).
- Skip "other inverse substitutions", page 361, and "the tan(t/2) substitution", page 362. These methods can be useful sometimes, but they often lead to complicated calculations and we did not have the time to cover them.
- Exercises: 1-28, 40-46.
Section 6.3:
- Integrating rational functions: reduction to the case P(x)/Q(x) with the degree of P less than the degree of Q.
- Partial fractions: setting up a partial fraction decomposition (remember that the cases of quadratic and repeated factors are a little bit different from distinct linear factors!); two methods of finding the coefficients.
- Exercises: 1-34.
Section 6.4: we skipped it.
Section 6.5:
- Improper integrals: infinite intervals, unbounded functions
- Evaluating improper integrals as limits of proper integrals
- Comparison test
- Exercises: in 1-29 you should calculate the integral as a limit of proper integrals; in 30-41 use the comparison test. (Note: if you are specifically asked to use the comparison test, your best bet is to do just that. The integral in question may be either very difficult or impossible to evaluate explicitly.)
Section 6.6:
- Basic approximate integration formulas: Trapezoid Rule, Midpoint Rule
- Error estimates: given n, how large can the error be? How large does n need to be in order to achieve a prescribed accuracy?
- Exercises: 1-11. Some of the computational exercises can get a bit tedious, even if you use a calculator. You probably do not need to do too many of them.
- For the trapezoid, midpoint, and Simpson's methods, you should memorize the formulas for T_n, M_n, S_n, and the error estimates.
Section 6.7:
- Simpson's Rule: computational formula and error estimate.
- Exercises: 1-10.
Section 6.8:
- Other aspects of numerical integration: improper integrals (skip Taylor's formula and Romberg integration).
- Exercises: 1-9.