Work Sheet 1
For one of the following write a paragraph or two explaining your method of solution
and in what sense it solves the problem. Of course, give your solution too!!
- A dart is thrown randomly at a dart board
(
English Competition Board) What is the probability of hitting the red
. Note, you can print the picture of the board and measure the ring sizes from the
picture. Since you want probability, actual measurements are not important, only
relative measurements -- so you can use the picture!
- The problem is that nobody wants a nuclear dump in their backyard. But Minnesota
needs one, so to be fair, the state legislature (in their collective wisdom) has decided to randomly choose a location
somewhere in Minnesota and build the dump there. However, they
also instruct the mathematicians at St. Olaf to "determine the probability that the site
chosen will be within 100 miles of a major metropolitan area." Estimate this probability.
Problem Set 1
- State and prove both versions of the FTC.
- Prove that every differentiable function is continuous.
- Find a derivative which is not continuous. (Inherent here is that you
not only give the function and find its derivative, but prove
the derivative is discontinuous at some point.)
- Find a derivative which is not continuous at 100 points.
- Find a derivative which is not continuous at infinitely many points.
- Let C10={.d1d2d3... : di= 0 or 9 for each i}. Determine the "geometry"
of C10.
ie locate portions of R which contain points from C and portions devoid of
points of C.
- Find and record a proof that every continuous function defined on a compact (closed
and bounded) interval is Riemann integrable.
- Complete the details of the arguments of Example 1.3 on page 8 proving that f(x)= x1/2 is integrable on [0,1] with integral 2/3.
- Complete the details of the arguments of the two counter examples given on pages 10 and 11.
- In Example 1, what IS the integral of each function fn and why is the integral what you say it is? Too, what IS the limit function, f, and why is f not Riemann integrable?
- In Example 2, why does the sequence {fn} conveerge pointwise to 0? Too, why is this convergence NOT uniform?
