Mathematics 242 – Principles of Analysis Final Examination December 15, 2004

Directions

Do all work in the blue exam booklet. There are 200 possible regular points and 10 possible Extra Credit points.

I. Let A = {cos(x) : x ∈ [0, 3π/4]} and let B = {x : 1 < x^{2} < 4}.

A) (10) What is the set A ∪ B? B) (10) What is the least upper bound of the set C = {|x − 2| : x ∈ A}?

II.

A) (10) State the ε, N definition for convergence of a sequence. B) (20) Identify L = lim_{n→∞} x_{n} for the sequence

x_{n} = \frac{3n^{2} + n}{n^{2} + 1}

and prove using the definition that lim_{n→∞} x_{n} = L.

III. Let x_{n} = sin(n) and y_{n} = cos(n). In this problem you may use without proof any facts about sequences, subsequences, etc. that we proved in class. A) (10) Show that there exists an index sequence of positive integers n_{k}, k ∈ N, such

that x_{n_{k}} = sin(n_{k}) converges. B) (10) Does there exist an index sequence n_{k} of positive integers such that both x_{n_{k}} and

y_{n_{k}} converge? Prove your assertion.

IV. A) (10) Give the ε, δ definition for the statement lim_{x→c} f(x) = L. B) (20) Identify the limit

L = lim_{x→0} x^{2} cos(1/x)

and prove using the definition that lim_{x→0} x^{2} cos(1/x) = L.

V. All parts of this problem refer to the function

f(x) = \frac{32x}{x^{4} + 48}

A) (5) What are f(0) and f(2) for this function? B) (10) Show using the Intermediate Value Theorem that for each k with 0 < k < 1, the

equation f(x) = k has at least two solutions x ∈ R, with x > 0.

C) (10) Show, using the Mean Value Theorem, that if f is differentiable on an interval I = (a, b) and f′(x) ≠ 0 for all x ∈ I, then for each k the equation f(x) = k has at most one solution with x ∈ I. (Hint: Prove the contrapositive.)

D) (5) Show using part C that there are exactly two solutions of the equation f(x) = k from part B for each k with 0 < k < 1.

VI. A) (20) In this part you may use without proof the summation formulas:

∑_{i=1}^{n} 1 = n ∑_{i=1}^{n} i = \frac{n^{2} + n}{2} ∑_{i=1}^{n} i^{2} = \frac{2n^{3} + 3n^{2} + n}{6}.

Show that f(x) = x^{2} + x − 1 is integrable on [a, b] = [0, 3] by considering upper and lower sums for f and determine the value of ∫_{0}^{3} x^{2} + x − 1 dx.

B) (20) State and prove “part 1” of the Fundamental Theorem of Calculus.

VII. True - False. For each true statement, give a short proof or reason. For each false statement give a reason or a counterexample. A) (10) Let ∑_{n=1}^{∞} a_{n} be an infinite series with positive terms. If the partial sums s_{N} are

bounded above by some B for all N, then ∑_{n=1}^{∞} a_{n} converges. B) (10) The infinite series

∑_{n=1}^{∞} \frac{(−1)^{n}}{n^{3}}

converges conditionally. C) (10) The function

f(x) = \{ \begin{matrix}x & if x is rational \\ −x^{2} & if x is irrational \end{matrix}

is continuous at x = 0.

Extra Credit (10) Let

f(x) = \{ \begin{matrix}cos(2x) & if x < 0 \\ ax^{2} + bx + c & if x ≥ 0 \end{matrix}

There is exactly one set of constants a, b, c for which f′(0) and f′′(0) both exist. Find them.

Have a peaceful and joyous holiday season!