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 tex2html_wrap_inline51 for all tex2html_wrap_inline53 .

2. (20 points). The function tex2html_wrap_inline55 is monotonic increasing on tex2html_wrap_inline57 . Find n so that a left-hand sum with n equal subdivisions is within .01 of tex2html_wrap_inline65 .

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 tex2html_wrap_inline79 .

5. (20 points) Suppose f is continuously differentiable, and that tex2html_wrap_inline83 for all x. If f(1)=1, how large can f(2) possibly be?

Tony Phillips
Mon Dec 2 18:52:53 EST 1996