Math 409 - Test 2 Review
General Information
Test 1 (Friday, 8/5/05) will have 6 to 8 questions, some with
multiple parts. The format will be similar to Test 1 and the quizzes
you've taken. You should expect to have to state definitions - and
some theorems -, work problems similar to ones on the homework and
quizzes, and be able to prove theorems. Although you will not be
directly tested over material up to Test 1, you will need to known it
well enough to answer the questions from the material covered since
then. Test 2 will cover the following material:
- Chapter 4
- 4.1-4.3, and 4.4 through the Inverse Function Theorem (4.27).
- Chapter 5
- 5.1-5.4, and 5.6 through Jensen's Inequality (5.64).
- Chapter 6
- 6.1, 6.2, and 6.3 through the Ratio Test (6.24) and Remark 6.25.
- Chapter 7
- 7.1-7.3
Definitions and Statements of Theorems
- Derivative of f at x = a
- f is differentiable on I
- f is continuously differentiable on I
- Partition of [a,b], norm of a partition, refinement of a
partition
- Upper Riemann sum U(f,P) and lower
Riemann sum L(f,P) of a bounded function
over a partition P of the interval [a,b].
- Upper, lower, and Riemann integral on [a,b], and existence for an
integrable function
- Integrable function over a closed, bounded interval
- Improperly integrable function over an interval I
- Comparison Theorem for Integrals
- Convex function
- Jensen's Inequality
- Series, partial sums
- Absolute and conditional convergence of a series
- Limit superior, limit inferior for sequences
- Pointwise convergence of a sequence or series of functions
- Uniform convergence of a sequence or series of functions
- Weierstrass M-Test
Proofs
Be able to state and prove the following:
- Rolle's Theorem
- Inverse function Theorem
- First Mean Value Theorem for Integrals
- Fundamental Theorem of Calculus
- Integral Test
Problems
Be familiar enough with the following items to be able to use them to
solve problems similar to ones on quizzes and homework: various ways
of calculating derivatives, limits (L'Hospital's rule), integrals;
simple Riemann sums; calculation of improper integrals; geometric
series, harmonic series, telescoping series; tests for convergence of
series; conditions for interchange of sum and derivative and sum and
integral, power series.
Updated 8/3/05 (fjn).