Mathematics 242 – Principles of Analysis Makeup Midterm Exam 3 May 10, 2011

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 possible Extra Credit points (see question IV below). Possibly useful information:

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

I. Both parts of this question refer to the function f : R → R defined by f(x) = x^{2} − 4x.

A) (20) Consider the regular partitions P_{n} of the interval [0, 3] and show directly, using

the upper and lower sums, that f is integrable on [0, 3]. B) (15) Explain why the hypothesis of the Mean Value Theorem is satisfied for f on the

interval [1, 5] and find the number c mentioned in the conclusion.

II. (15) Let

f(x) = \{ \begin{matrix}x^{a} sin(1/x) & if x ≠ 0 \\ 0 & if x = 0 \end{matrix}

For which a ≥ 0 does lim_{x→0^{+}} f(x) exist? Is f(x) differentiable at x = 0 when a = 2/3? Explain.

III. (20) Let f be continuous on [a, b]. Show that the function F (x) = ∫_{a}^{x} f(t) dt is differentiable and F′(x) = f(x).

IV. True-False. Say whether each of the following statements is true or false. For true statements, give short proofs; for false ones give reasons or counterexamples. Do any three parts. If you submit solutions for all four, then I will consider the other one for Extra Credit.

A) (10) Let f(x) = sin(4x) − 3 cos(2x). There exists some c ∈ (0, π/4) such that f(c) = −1.

B) (10) f(x) = \frac{1}{x} is uniformly continuous on (0, 1).

C) (10) If A ⊂ R contains no nonempty interval (a, b), then A is at most countably infinite.

D) (10) Let f be continuous on [a, b] and assume f(x) < 0 for all x ∈ [a, b]. Then there exists a constant k < 0 such that f(x) ≤ k < 0 for all x ∈ [a, b].