MA 1023 LAB 1 A'97

IMPROPER INTEGRALS and INFINITE SERIES

Before solving the exercises from today's lab, you should copy in your home-directory a MAPLE file by using the following command:

cp ~/vernescu/Mlab1.mws ~;

The file contains some examples with sequences, visualization of sequences and series. This should complement the information and examples that you already have in the file Maple.mws that you copied last week.

You can then start the Maple engine by using the xmaple command. Click on FILE and OPEN Mlab1.mws. Read the examples and work some of your own. Once you feel confortable with the commands, you can try to solve the following exercises. You have to create your own MAPLE file and attach a printout to your report.

1. A ball has bounce coefficient 0 < r < 1 if when it is dropped from height h, it bounces back to a height of rh. Suppose that such a ball is dropped from an initial height a and subsequently bounces infinitely may times. Find the total up-and-down distance in all its bouncing. (Bonus:find the total amount of time the ball spends on bouncing given that r = 0.64 and it is dropped from an initial height of . Assume that .)
2. Let denote the graph of the function .

(i) Find the area of the region bounded between , the positive x-axis, x=1 and .

(ii) Find the volume of the solid of obtained by revolving the curve about the x-axis, between x=1 and .

(iii) Find the area of the surface generated by revolving about the x-axis between x=1 and .

(iv) Give the range of values of p > 0 for which your answers to the above 3 problems are finite. Is there any relationship between the three quantities. (The formula for area of surface of revolution may be found on p. 400 of the text.)

3. By now, it is well known that the harmonic series diverges. That is to say, the limit of s_n is infinite , where . But how fast does go to infinity? To answer this question, consider the sequence . Using maple, discover what properties this sequence have. Does it converge? If so, what does it converge to? Can you give me a rigorous reason why this sequence converges. Therefore, converges to infinity like .

DUE:LAB TIME ON SEPTEMBER 22 OR SEPTEMBER 23.