Problem 1. Basics on Derivatives.
1. Be able to state the definition of the derivative
2. Be able to prove that a differentiable function is continuous (Theorem 5.2)
3. Be able to prove the product rule for derivatives (any proof you like).
4. Be able to prove the chain rule for derivatives (any proof you like).
Problem 2. Mean Value Theorems
1. Be able to state and prove the Interior Extrumum Theorem, which is what
I would call Theorem 5.8
2. Be able to state and prove Rolle’s Theorem: Suppose that f is continuous
on [a, b] and differentiable on (a, b) and that f (a) = f (b) Then there is a
point c ∈ (a, b) so that f 0 (c) = 0.
3. Be able to state and prove the Mean Value Theorem, Theorem 5.10.
4. Be able to state and prove the Cauchy Mean Value Theorem, Theorem 5.9
5. Be able to prove the statements in Theorem 5.11.
6. Be able to state and prove the Intermediate Value Theorem for Derivatives,
Theorem 5.12.
7. Be able to state Taylor’s Theorem.
8. Be able to prove Theorem 5.19.
1
Problem 3. Defining the integral and basic properties.
If f is a bounded function on [a, b] and α is an increasing function on [a, b],
be able to define the following concepts.
1. A partition of [a, b].
2. One partition refines another partition.
3. The upper and lower sums U (f, P, α) and L(f, P, α).
4. The upper an lower integals of f with respect to α on [a, b].
5. f is integrable with respect to α on [a, b].
6. The integral of f with respect to α on [a, b].
Know an example of a functions f and α so that f is not integrable with
respect to α.
Be able to prove the basic properties of the integral given in Theorem 6.12.
(Of course, I wouldn’t ask them all!)
Problem 4. Theorems about integrals.
1. Be able to prove Theorem 6.13b.
2. Be able to prove the Fundamental Theorem of Calculus, Theorem 6.21
3. Be able to prove Theorem 6.25.
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EXAM
Review Topics for Exam 1
Math 5319, Spring 2010
February 23, 2010
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 4 problems. There are 0 points
total.
Good luck!