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(Mean Value Property. Friday, April 7.) Solve Exercise 4 and 5, Rudin, Chapter 5

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(Mean Value Property. Friday, April 7.) Solve Exercise 4 and 5, Rudin, Chapter 5 Math 140B { Homework 2. Due Friday, April 14. Problem 1. (Mean value property. Friday, April 7.) Solve Exercise 4 and 5, Rudin, Chapter 5. Problem 2. (Mean value property. Friday, April 7.) A function f : R ! R is said to be Lipschitz if there exists a constant M > 0 such that for all x; y 2 R we have jf(x) − f(y)j ≤ Mjx − yj: (i) Show that any Lipschitz function is uniformly continuous over R. (ii) Show that if f is differentiable and f 0 is bounded, then f is Lipschitz. In partic- ular, conclude that f(x) = sin x is uniformly continuous over R. (iii) Give an example of a differentiable function which is not Lipschitz. You may try f(x) = x2: (iv) Give an example of a function which is Lipschitz but not differentiable. You may try f(x) = jxj: Problem 3. (Mean value property. Friday, April 7.) For any continuous function f :[−1; 1] ! R such that f is three times differentiable on (−1; 1), and f(−1) = 0; f(0) = 0; f(1) = 1; f 0(0) = 0; 000 there must exist x0 2 (−1; 1) such that f (x0) = 3: This is a modification of Rudin, Problem 17, Chapter 5. 1 3 2 Hint: You may want to consider the auxiliary function g(x) = f(x) − 2 (x + x ): Compute g(−1); g(0); g(1); g0(0). Then apply the mean value property a number of times to the function g and conclude. Problem 4. (Rolle's theorem for vector-valued functions. Friday, April 7.) Let ~ k ~ f : I ! R ; f = (f1; : : : ; fk) be a vector-valued function defined over an interval I. We say that f~ is differentiable if each fi is differentiable and we set ~0 0 0 f (x) = (f1(x); : : : ; fk(x)): 2 Show that Rolle's theorem fails for vector-valued functions. Specifically, let f~ : R ! R be the differentiable function f~(x) = (cos x; sin x): Show that f~(0) = f~(2π); but that there is no c 2 (0; 2π) such that f~0(c) = 0. 1 2 Remark: The mean value theorem can be salvaged if we replace equality by inequality. In other words, there exists c 2 (a; b) such that jjf~(b) − f~(a)jj ≤ jjf~0(c)jj(b − a): As usual, jj~xjj denotes the length of a vector. This is proven in Rudin, Theorem 5.19, and you can read the proof if you wish. Problem 5. (L'Hopital rule. Monday, April 10.) Solve Exercises 9 and 11, Rudin, Chapter 5. Problem 6. (Taylor's theorem. Wednesday, April 12.) Use Taylor's theorem for the π function f(x) = cos x to prove that for all x 2 [0; 2 ] we have x2 x2 x4 1 − ≤ cos x ≤ 1 − + : 2! 2! 4! Extra credit (not required). (Taylor series. Real analytic functions. Wednesday, April 12.) Let f : R ! R be infinitely many times differentiable. We say the function f is of class C! (or that f is a real analytic function) if for all a 2 R, the function f equals its Taylor expansion f 00(a) f (n)(a) f(x) = f(a) + f 0(a)(x − a) + (x − a)2 + ::: + (x − a)n + :::: 2! n! for all x in a sufficiently small neighborhood of a. In calculus, we sometimes incorrectly learn that every function which is infinitely many times differentiable equals its Taylor series, but this is not the case as we will show below. Consider the function ( exp 1 if x < 0 f(x) = x : 0 if x ≥ 0 First calculate the first derivative of f: (i.1) Show that f 0(0) = 0, directly from the definition. (i.2) Write down a formula for f 0(x) which is valid for all values of x. Now calculate the second derivative. (ii.1) Consider the formula you found in (i.2). First, from this formula, show that f 00(0) = 0 by using the definition of the derivative. (ii.2) Next, differentiate f 0 and write down a formula for f 00(x) which is valid for all x. Now onto higher order derivatives. 3 (iii) Do the same for third derivatives. You should be able to prove first that f 000(0) = 0 from the definition. Write down an expression for the third derivative. Do you notice any patterns in the general formula? (iv) Continuing in this fashion, show that f (n)(0) = 0 for all n. You will need to use induction on n and formulate a hypothesis about the form of f (n)(x) for all n, which you will then need to prove by induction. (v) Explain that (iv) implies that f is not real analytic i.e. it does not equal its Taylor series at a = 0 in any neighborhood of a..
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