Mathematics 242 – Principles of Analysis Midterm Exam 3 May 3, 2013

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 possible Extra Credit points. Possibly useful information:

∑_{i=1}^{n} = \frac{n(n + 1)}{2},  ∑_{i=1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}.

I. A) (25) Show directly, using upper and lower sums, that the function f(x) = x^{2} is

integrable on [0, 1] and determine the value of ∫_{0}^{1} x^{2} dx. B) (5) Compute ∫_{0}^{1} x^{2} dx using the Fundamental Theorem of Calculus to check your

work.

II. Let f(x) = \frac{sin(x)}{x} for x ≠ 0.

A) (5) How should a value f(0) be defined to make f continuous at x = 0? Explain. (Note: If you do not know how to decide, you may “buy” the answer for 5 points so that you can do the next part.)

B) (15) Using your value from part A for f(0), use the limit definition to show that the resulting function is differentiable at 0 and find the value f′(0). Say what method you are using to compute this limit.

III. A) (10) State the Mean Value Theorem. B) (10) Determine

lub \{\frac{ln(x)}{x} | x ∈ (e, +∞)\}

and justify your answer completely using facts we have developed in this course.

IV. True-False. Say whether each of the following statements is true or false. For true statements, give short proofs or reasons; for false ones give reasons or counterexamples.

A) (10) f(x) = ln(x) is uniformly continuous on (1, +∞).

B) (10) If A ⊂ R contains no nonempty interval (a, b), then A is finite or countably infinite.

C) (10) Let f be continuous on [a, b] and assume f(x) > 1 for all x ∈ [a, b]. It is possible for glb{f(x) | x ∈ [a, b]} to equal 1.

Extra Credit (10) Show that if f is differentiable on an open interval I and [a, b] ⊂ I where f′(a) < 0 and f′(b) > 0, then there must be some c ∈ (a, b) where f′(c) = 0.