MAT 320 Introduction to Analysis

December 19, 1996 Final Examination



1. (10 points) Prove that the function tex2html_wrap_inline87 is continuous, but not uniformly continuous, on the open interval (0,1).2. (10 points) Consider a continuous function f defined on [0,1] and with values in [0,1], i.e. such that tex2html_wrap_inline97 for every x in [0,1]. Show that f has a fixed point, i.e. that there is a point x in [0,1] such that f(x)=x. Hint: Consider the function g(x)=f(x)-x.

3. (10 points) Consider the function


Prove carefully and directly that f is Riemann integrable on [0,3]. ``Directly'' means show that f satisfies the definition of Riemann integrability; DO NOT use Theorem 3.5.2!

4. (20 points) Consider the function


a. Show that this function is differentiable at x=0.

b. Show that f is continuously differentiable, i.e. that f' is a continuous function.

5. (30 points) Consider the problem of calculating tex2html_wrap_inline129 by applying Newton's method to find a root of tex2html_wrap_inline131 . Show that if tex2html_wrap_inline133 is any number greater than tex2html_wrap_inline129 , then the numbers tex2html_wrap_inline137 calculated by iterating Newton's method give a sequence converging to tex2html_wrap_inline129 . Proceed as follows. NOTE: the various parts of this problem may be worked independently: assume that earlier parts are proven.

a. Show that if tex2html_wrap_inline141 then


(Hint: for the first inequality, draw the graph of f and argue using its convexity.)

b. Conclude that the sequence tex2html_wrap_inline147 has a limit as tex2html_wrap_inline149 .
c. Show that the limit must be tex2html_wrap_inline129 .
d. What happens if the initial guess tex2html_wrap_inline133 lies strictly between 0 and tex2html_wrap_inline129 ?

6. (10 points) Suppose tex2html_wrap_inline159 is a sequence of bounded functions which converges uniformly to a limit function f. Prove that f is bounded.

7. (10 points) Suppose the function f is (n+1) times continuously differentiable on [0,1], and that its (n+1)-st derivative tex2html_wrap_inline173 is bounded in absolute value by M.

Prove that tex2html_wrap_inline177 can be approximated by


with an error less than tex2html_wrap_inline181 in absolute value.

Tony Phillips
March 26 1997