**MAT 320 MIDTERM #2**

*B-track *

**NOVEMBER 26, 1996**

*This is an 80-minute test. Work all questions. *

**1.** *(20 points)* Let *f* be a continuous function
defined on the closed interval [*a,b*]. Prove that |*f*|
is bounded, i.e. that there exists a number *M* such that
for all .

**2.** *(20 points)*.
The function is monotonic increasing
on . Find *n* so that a left-hand sum
with *n* equal subdivisions is within .01 of .

**3.** *(20 points)* Prove that

is differentiable at *x*=0. What is the value of its
derivative there?

**4.** *(20 points)* Prove that

is a convergent integral, i.e. that the integrals from 1 to *c*
tend to a finite limit *L* as .

**5.** *(20 points)* Suppose *f* is continuously
differentiable, and that for all *x*. If
*f*(1)=1, how large can *f*(2) possibly be?

Mon Dec 2 18:52:53 EST 1996