Math 131A Homework 5 Please turn this homework in to me at the start of lecture on March 10th Remember the quiz will on the same material so not doing this homework would be very silly The quiz for this homework is on March 16th 1 a Suppose f R R is differentiable and f 0 x 4 for all x R Prove that there is at most one a 2 such that f a a2 Help Use a proof by contradiction and the MVT b Suppose f and g are differentiable on R that f 0 g 0 and f 0 x g 0 x for all x R Prove that f x g x for x 0 Help Using the MVT somehow would be good practice even though it is not necessary a corollary of the MVT given in lecture notes suffices consider h x g x f x c Suppose f R R and that f x f y x y 2 for all x y R Prove f is a constant function Help Calculate the derivative of f 2 Suppose f R R is continuous and differentiable on R 0 and that limx 0 f 0 x exists and is finite Prove that f is differentiable at 0 and that f 0 is continuous at 0 Help You need to show that f 0 0 exists and that it is equal to limx 0 f 0 x Do NOT try and do weird interchanging of limits that is exactly the sort of thing we are not allowed to do in this class without justification Write down the quotient that appears in the definition of the derivative and use the MVT 3 Suppose that f R R is differentiable on R and that f 0 R R is bounded Prove that f is uniformly continuous Help Use the MVT 4 Suppose an n 1 is a sequence and that an 0 1 9 for each n N P ak Prove that the series k 1 10k is Cauchy What is the point of this question 5 Determine which of the following series converge Justify your answer You can use the fact that limk 1 k1 k exists and is greater than 1 P 1 a k 1 kk P k b k 1 2k P 2 cos k c k 1 3k P k d k 1 sin 9 P k e k 1 kk 1 P 6 a Suppose that ak is a convergent series of nonnegative numbers and p 1 Prove that P p ak converges Help You should use the comparison theorem and you may want to deal with early terms and later terms differently b Let an bePa sequence such that lim an 0 Prove that there is a subsequence ank k 1 such that k 1 ank converges Help You should use the comparison theorem Think of your favorite convergent series that should tell you how to construct the subsequence 7 For each of the following power series find the radius of convergence and determine the exact interval of convergence P 2 n a n 0 n x P 1 n b n 1 nn x P 2n n c n 1 n2 x P 4 2 1 n n n x d n 0 5 P n e n 0 x The last two are harder Either you need the definition of lim sup we haven t done this or it is also possible to calculate the radius of convergence directly using knowledge of geometric series 2 8 Random and fun interchange of quantifier question Let f R2 R be defined by f x y y x2 y 2x2 Consider the following sentences 0 0 r R 0 r f r cos r sin f 0 0 0 0 r R 0 r f r cos r sin f 0 0 One is true one is false Figure out which is which and demonstrate with one equation that the false sentence is false If you care show the true sentence is true 9 These questions won t be quizzed this week but you could use them as revision for the final They were inspired by Cody s great questions a Suppose r R R and s R R are differentiable functions Let r x if x Q f x s s if x R Q and suppose a R i Prove f is continuous at a if and only if r a s a ii Prove f is differentiable at a if and only if r a s a and r0 a s0 a b Let f be a real valued function Prove that f is uniformly continuous if and only if for each pair of sequences xn and yn in the domain of f with limn xn yn 0 we have limn f xn f yn 0 Said in quanitifiers show f is uniformly continuous if and only if xn yn n N xn yn dom f and lim xn yn 0 n lim f xn f yn 0 n 3 Math 131A Homework 5 Solutions 1 a Suppose f is differentiable on R and that f a a2 and f b b2 where 2 a b The mean value theorem gives an x0 a b with f 0 x0 f b f a b2 a2 a b 2 2 4 b a b a We have proved the contrapositive b Since f 0 x g 0 x for all x R we have g f 0 x g 0 x f 0 x 0 for all x R A corollary of the mean value theorem says that g f is increasing and so for all x 0 we have g x f x g f x g f 0 g 0 f 0 0 i e g x f x c Fix y R For all x with x 6 y we have 0 f x f y x y x y By continuity of and polynomials we have lim x y x y lim x y y y 0 x y and so by definition of f 0 y continuity of and the squeeze lemma we have f 0 y lim x y f y f x f y f x lim 0 x y y x y x Thus f 0 y 0 for all y R and so f is constant 2 Suppose f R R is continuous and differentiable on R 0 and that limx 0 f 0 x exists and is finite First we will demonstrate that limx 0 f x f 0 x 0 exists and is equal to limy 0 f 0 y Let xn be a sequence in 0 converging to 0 For each n f is continuous on 0 xn and differentiable on 0 xn and so the mean value theorem provides us with a yn 0 xn such that f xn f 0 f 0 yn xn 0 Since 0 yn xn yn is a sequence in R 0 and by the squeeze lemma yn converges to 0 Thus by definition of limy 0 f 0 y we have …