Extra Credit, Fall Semester 2000

If you would like to obtain 60 points extra credit, work on the problems below. The rules for this extra credit are as follows:

  1. ONLY COMPLETE SOLUTIONS of part 1 below will receive 35 points extra credit. Not everything needs to be correct to receive the 35 points, but work must be done and shown on each and every little item of the problem. Precisely follow the instructions given below and do exactly what is asked for in the problem, using ONLY things that we have discussed in class during the semester. Again, only complete solutions will be eligible for the 35 points. In other words, you will obtain either 0 or 35 points. In addition, the work needs to be nicely legible, written down neatly on sheets of paper stapled together, and the deadline for turning in your work is Wednesday, November 29, 2000, 8:30 am. Solutions that don't satisfy these standards will not be considered.
  2. For those who obtain 35 points in part 1, they are eligible for another 10 points in part 2, but again, only if their solution to part 2 is complete.
  3. Everybody, regardless whether work was turned in or not, can receive another 15 points by participating in an in-class-quiz on Wednesday, November 29. The quiz will consist of certain items selected from the first part of the problem below. Again, only complete solutions of this quiz will obtain the 15 points.
So, here are the problems:
  1. For x>0, define L(x) as the integral from 1 to x over the function 1/t. Find L(1). Find all zeros of L. Where is L positive, negative? Draw a picture and indicate the areas that correspond to L(1/2) and L(2). Use Property 8 on page 333 to estimate L(2). Use the Midpoint Rule on page 330 (with n=10) to estimate L(2). Use the Fundamental Theorem of Calculus, Part I, on page 338 to find the derivative of L. Where is L increasing, decreasing? Where are the maxima and minima of L? Find the second derivative of L. Where is L concave upward, concave downward? Where are the inflection points of L? Let f(x)=L(ax) and find f'. Conclude that L(ax)=L(a)+L(x) for all a and x. Similarly show the formulas L(a/x)=L(a)-L(x) and L(x^a)=aL(x). Calculate L(2^n). What does this imply about the limit of L(x) as x tends to infinity? How about the limit of L(x) as x tends to 0, x>0? Now, estimate L(1/1000), L(1/2), L(40), and L(1000) using the Midpoint Rule (with n=10). Use all the information obtained to carefully draw a graph of L.
  2. Find the derivatives of L(cos x), sin(L(x)) and L(|x|). Find antiderivatives of x/(x^2+1) and tan(x). From the graph in the first part of this problem, roughly determine the number e that satisfies L(e)=1. Now find the integral of L(x)/x from 1 to e.