MATH 104 : Final Exam

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1 MATH 104 : Final Exam 10 May, 2017 Name: You have 3 hours to answer the questions. You are allowed one page (front and back) worth of notes. The page should not be larger than a standard US letter size. Use of electronic items and calculators is not permitted during the exam. If you need to use the restroom, please deposit your phones with the proctor first. Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Notation. R, Q and N denote the sets of real, rational and natural numbers respectively. Question Points Score Total: 80 1

2 1. In each of the following questions mark out ALL the true statements. If none of the statements is true, then indicate so. An incorrect answer is either a false statement marked as true or a true statement not marked at all. Each incorrect answer will cost one point. (a) (3 points) Let Q be endowed with the usual Euclidean metric (that is, d Q (p, q) = p q ) and let N be endowed with the discrete metric (that is, d N (m, n) = 1 if m n and d(n, n) = 0). 1. [1, 2) is a closed subset of (Q, d Q ). 2. [1, 2) is a compact subset of (Q, d Q ). 3. {0} is an open set in (N, d N ). 4. An interior point of any subset E Q is also a limit point of E. (b) (3 points) 1. The polynomial p(x) = x 3 3x + 1 has three real roots. 2. The point x = 1 is a local maximum for p(x) = x 3 3x The polynomial p(x) = x 3 + 3x + 1 has exactly three real roots. 4. There is no local max or local min for the polynomial p(x) = x 3 + 3x + 1. (c) (3 points) 1. If f 2 is Riemann integrable on [a, b], then so is f. 2. If f 2 is Riemann integrable on [a, b], and f 0, then f is also integrable. 3. If f 3 is Riemann integrable on [a, b], then so is f. 4. If f is Riemann integrable on [a, b] then F (x) = is a continuous function on [a, b]. x a f(t) dt 2

3 (d) (3 points) For a real valued function f, consider the equality lim + h) f(a h)] = 0. h 0 +[f(a 1. The function f(x) = { 1 x, x 0 x, x < 0, satisfies the above equality at a = A function f that is continuous at x = a satisfies the above equality. 3. If a function f satisfies the above equality, then it is continuous at x = a. 4. Let f(x) = { cos (π/x), x 0 0, x = 0. Then the equality above holds with a = 0, but neither f(0+) or f(0 ) exist. (e) (3 points) 1. f(x) = e x sin x 2 is uniformly continuous on (0, 1). 2. f(x) = 1 x is uniformly continuous on (0, 1). 3. f(x) = 1 x is uniformly continuous on [1, ). 4. If f is a continuous, bounded function on (0, 1), then lim t 0 + f(t) exists. (f) (3 points) 1. ( )) n=1 (1 ( 1)n cos 1 n is absolutely convergent. 2. n=1 ( 1)n 1 n(ln n) 2 3. ( 1) n n=1 n 1/n is conditionally convergent. is conditionally convergent. 4. If a 2 n converges then a n converges. 3

4 2. (a) (4 points) State the Cauchy criteria for uniform convergence of a sequence of functions f n : [0, 1] R. (b) (6 points) Let {f n } be a sequence of real valued functions defined on [0, 1] such that f n+1 (t) f n (t) < 1 2 n for all n = 0, 1, 2,, and for all t [0, 1]. Show that {f n } converges uniformly. 4

5 3. (a) (4 points) State the version of the fundamental theorem of calculus useful for the problem below. (b) (6 points) Compute 1 lim h 0 h h 0 e t cos t dt. 5

6 4. (a) (6 points) Let γ : [0, 1] R 3 be a continuous function (such functions are called curves) such that γ(0) = (0, 0, 0) and γ(1) = (1, 1, 1). Show that the image of the curve intersects S 2 = {(x, y, z) R 3 x 2 + y 2 + z 2 = 1} at some point. Hint. Intermediate value theorem. 6

7 (b) (6 points) Let K R n be a closed subset, and let q be a point in R n \ K. Show that there exists a point p 0 K at the shortest distance from q. That is, show that there is a p 0 K such that p 0 q = inf p q. p K Recall that for any vector v = (v 1,, v n ) R n, v = v v2 n. Hint. Consider a minimizing sequence. Note. If you provide a proof for the case when K is compact, you will be awarded 3 points. 7

8 5. (a) (8 points) Find all the points x R at which the series converges. f(x) = n=1 (2x 1)n 1 n 2 n 8

9 (b) (6 points) Verify that [0, 1] lies in the interval of convergence, and compute 1 0 f(x) dx. 9

10 6. Consider the sequence of functions (a) (4 points) Show that f n (x) = for all x, y [0, 1] and each n = 1, 2,. x 0 cos nt 1 + t 2 dt. f n (x) f n (y) x y (b) (4 points) Use Ascoli-Arzela to show that there is a subsequence which converges uniformly to a continuous function on [0, 1]. 10

11 (c) (4 points) Show that f n (x) = sin(nx) n(1 + x 2 ) + 2 x t sin nt n 0 (1 + t 2 ) 2 dt. (d) (4 points) Use the above identity to show that in fact f n 0 uniformly on [0, 1]. Note. The proof of part(d) works with 1/(1 + t 2 ) replaced by any f(t) C 1 [0, 1], and is a special case of the famous Riemann-Lebesgue lemma which is foundational in Fourier analysis. 11

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d.

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d. Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books