Mathematics 242 – Principles of Analysis Midterm Exam 3 December 3, 2004

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).

I. A) (10) Give the ϵ - δ definition for the statement lim_{x→c} f(x) = L. B) (10) Show using the definition that

lim_{x→2} x^{2} − 4x + 2 = −2.

C) (5) Using any applicable method, determine the limit

L = lim_{x→0} \frac{x^{2}}{sin(2x^{2})}.

II. A) (10) State the Intermediate Value Theorem. B) (15) Let f(x) = a_{1} + a_{2} sin(x), where a_{1}, a_{2} are constants with 0 < a_{1} < a_{2}. Show

that there exists some c ∈ [−π/2, π/2] such that f(c) = 0.

III. (20) Show that every monotone increasing function on a closed interval [a, b] is integrable.

IV. True-False. Say whether each of the following statements is true or false, and give a complete, correct reason for your assertion. Do any three parts. If you submit solutions for all four, then I will consider the other one for Extra Credit.

All parts refer to the function f defined by:

f(x) = \{ \begin{matrix}−x − 2 & if x < 0 \\ 2 − x − x^{3} & if x ≥ 0 \end{matrix}

A) (10) f is differentiable at x = 0. B) (10) f integrable on [−2, 2]. C) (10) The conclusion of the Mean Value Theorem is true for f on the interval [−2, 2]. D) (10) The conclusion of the Extreme Value Theorem is true for f on the interval [−2, 1].

(Note: The conclusion of theorem phrased as “if p, then q” is the statement q; the hypotheses are the statement(s) p.)