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MATH 680 Analysis I ? Due: Wednesday, March 15th ? Winter 2000 Instructions : Do any ve problems. 1. Let f be Riemann integrable on [a; b]: Assume the result that if the limit n X b , a lim f a+k n n!1 exists, then its value is the integral lim n!1 2. n X n = ; 2 + n2 k 4 k=1 lim n!1 n X k=1 k=1 Z b a f (x) dx: Deduce the following limits: p p 1 = ln(1 + 2): n +k 2 2 Assume the result of dierentiating under the integral sign given as ! d Z b g (t; x) dt = Z b @ g (t; x) dt dx @x a a where g (t; x) is continuous for (t; x) 2 [a; b] [c; d] and continuously dierentiable in x for x 2 [c; d]. (This is not dicult to prove!) Dene Z x 2 2 Z 1 ,x2 (t2 +1) f (x) = e,t dt ; g(x) = e t2 + 1 dt: 0 0 (a) Show that g 0 (x) + f 0 (x) = 0 for all x and deduce that g (x) + f (x) = =4: (b) Use part (a) to prove that 3. lim x!1 Z x 0 p e,t2 dt = 21 : (a) Let f be a positive continuous function in [a; b]: Let M denote the maximum value of f on [a; b]: Suppose f (c) = M; where c 2 (a; b): Show that there exists a > 0 so that f (x) > 21 M; for all x 2 (c , ; c + ): (b) Let f be a positive continuous function in [a; b]: Let M denote the maximum value of f on [a; b]: Show that !1=n Zb n lim [f (x)] dx = M: n!1 a 4. Show that 2 is irrational. Let f (x) = xn (1 , x)n =n!. (a) Show that 0 < f (x) < 1=n!, for 0 < x < 1. 1 (b) Each k-th derivative f (k) (0) and f (k) (1) is an integer. Now assume that 2 = a=b; where a and b are positive integers, and let n X F (x) = bn (,1)k f (2k) (x) 2n,2k : k=0 Show the following parts: (c) F (0) and F (1) are both integers. (d) 2 an f (x) sin(x) = d (F 0 (x) sin(x) , F (x) cos(x)). dx Z1 (e) F (1) + F (0) = an f (x) sin(x) dx. 0 (f) Use (a) in (e) to deduce that 0 < F (1) + F (0) < 1 for n suciently large. This contradicts (c) and shows that 2 (and hence ) is irrational. 5. (a) Let f be continuous on [a; b]: If c 2 (a; b) and f (c) > 0, show that there exists a > 0 so that f (x) > 0 for all x 2 (c , ; c + ): (b) Left f be continuous on [a; b] and let f (x) = 0 whenever x is rational. Prove that f (x) = 0 for every x 2 [a; b]: 6. Let f be dened on all of the real line R: and assume that f is continuous at x = 0: Further, suppose also that for all x; y 2 R; that f satises the functional equation f (x + y) = f (x) + f (y ): Prove that there exists a constant c so that f (x) = cx; for all x: Hint: Show that f (n) = nf (1); for all positive integers n; by using induction. Next, determine the values of f (1=2); f (1=3); f (1=4); : : :: Generalize this method to determine f (1=m): After this, determine the value of f (r) for any rational number r: Finally, use continuity of f to determine f (x) for any real x: 7. (a) Suppose that f is dened on a subset S of Rn : If f is uniformly continuous on S and if S is bounded, prove that f is bounded on S: (b) Prove that f (x) = x2 is NOT uniformly continuous on R: 8. (a) Suppose that f : (X; dX ) ! (Y; dY ) is a uniformly continuous function between two metric spaces. Show that f preserves Cauchy sequences; that is, if fxn g is a Cauchy sequence in X , then ff (xn )g must be a Cauchy sequence in Y: (b) Show that the result of (a) is false if f is simply assumed to be continuous. 9. (a) Show: if A is a dense subset of a metric space S and S is a dense subset of a metric space T; then A is a dense subset of the metric space T: (b) Consider Q to be the metric space of rational numbers with metric d(x; y ) = jx , y j: Find an example of a closed and bounded subset of Q which is NOT compact! 2 10. Let fan g be a Cauchy sequence in a metric space (X; dX ): (a) Show that fan g must be a bounded sequence. (b) Show that fan g has a convergent subsequence with limit L, then the whole sequence itself must converge to L: 11. Consider the function x ); x 6= 0, : f (x) = x sin( 0; x=0 (a) If = 2 and = 1, show that f is dierentiable at x = 0, while if = 1 and = 1, f is not dierentiable at x = 0. (b) With = 2 and = 1, show that f 0 is not continuous at x = 0. 12. Let f; f 0 and f 00 be all continuous. (a) Show that f (x + h) , f (x , h) = 2hf (0) + O(h3 ): (b) Show that f (x + h) , f (x , h) = f 0 (x): lim h!0 2h (c) Give an example where the limit lim h!0 g(x+h)2,hg(x,h) exists for a function g (x) but g fails to be dierentiable at x. 13. Let f and g be two real-valued bounded functions on [a; b]. (a) Assume that f + g is Riemann-integrable, EITHER prove that both f and g must be Riemannintegrable OR give an example where neither f nor g are Riemann-integrable. (b) Assume that f g is Riemann-integrable, EITHER prove that both f and g must be Riemannintegrable OR give an example where neither f nor g are Riemann-integrable. (c) Assume that f 2 is Riemann-integrable, EITHER prove that f must be Riemann-integrable OR give an example where f is not Riemann-integrable. 14. Assume that f; f 0 ; f 00 are all continuous on the whole real line. For notational convenience, let F be an antiderivative of f so F 0 = f . Assume h > 0. By using appropriate Taylor expansions, show that F (h) , F (0) = 21 (f (0) + f (h)) + O(h3 ): 3