MA 1023 LAB 1 C'98

IMPROPER INTEGRALS, SEQUENCES AND GEOMETRIC SERIES

The following maple commands may be helpful to you in doing this lab. I shall use the function

$$f(x) = \frac{1}{2^x}$$

as an example.

1.

To define a function, use

$$f := x \rightarrow \frac{1}{2^x};$$

2.

To integrate a function, use

int(f(x),x=1..infinity);

3.

To find the sum of a finite series

$$\sum_{n = 1}^{10}\,\frac{1}{2^n}$$

, use

sum(f(i),i=1..n) ;

where f(x) is define above;

4.

Recall that an infinite series is defined as the limit of the sequence of partial sums. To evaluate

$$\sum_{n = 1}^{\infty}\,\frac{1}{2^n}$$

, we can define the sequence of partial sums as a function of n by

$$Sn := n \rightarrow sum(f(i),i=1..n);$$

and then take the limit of Sn by using the limit command

limit(Sn(n),n=infinity);

Exercises

1.

Recall from Calculus II how to compute the volume of solid of revolution. Using

$$Vp := p \rightarrow .....$$

in Maple, define a function that represents the volume of the solid of revolution obtained by revolving the curve

$$y =\frac{1}{x^p}$$

about the x-axis, between x=1 and $x=\infty$. For what values of p will this volume be finite ? (Your answer should be an interval which may be unbounded.)

2.

Repeat the above problem but this time find the surface area of the solid of revolution. The formula for surface area is given on page 400 of the text. Is your answer the same as above ? Is it possible for a solid to have finite volume but infinite surface area? Is it possible for a solid to have finite surface area but infinite volume? (The last question may not have an easy answer. Make a conjecture and give some examples to support your case.)

3.

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.

4.

I ran across the following problem while teaching Calculus III A term which I don't know how to solve:

$$\sum_{k=1}^{\infty} \frac{\vert cos k^3\vert}{\sqrt{k}}\;\;\;\;\;\;\;\; \mbox{(p. 521, no 34)}$$

Will you help me out by forming the sequence of partial sums and finding the limit by using finite sums? Now repeat the problem without the absolute value in the infinite series. Does the series converge ? (Your Lab Instructors know some secret command that will allow you to see the graph of a sequence.)

DUE:LAB TIME ON JANUARY 21 OR 22