**MAT 320 Introduction to Analysis**

December 19, 1996 Final Examination

A-track

THIS TEST IS ``OPEN BOOK.'' TOTAL SCORE = 100.

**1.** (10 points)
Prove that the function
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
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
by applying Newton's method to find a root of .
Show that if
is any number greater than , then the
numbers calculated by iterating Newton's
method give a sequence converging to . Proceed as
follows.
*NOTE: the various parts of this problem may be worked
independently: assume that earlier parts are proven.*

**a.** Show that if then

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

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

**6.** (10 points) Suppose 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
is bounded in absolute value by *M*.

Prove that can be approximated by

with an error less than in absolute value.

March 26 1997