Maple Project 1 MTH 142 Spring 2001

Department of Mathematics, University of Rhode Island

PROBLEM 1. (About of one of the versions of the Fundamental Theorem of Calculus.)

Define in Maple the functions

\begin{displaymath}g(t) = e^{\sin(t)} \quad \mbox{and} \quad G(x) = \int_0^x g(t) dt\end{displaymath}

and calculate G(1) in decimal form.

\begin{displaymath}h(t) = \frac{G(t+0.1) - G(t)}{0.1}\end{displaymath}

Plot $h(t)$ and $g(t)$ (not $G(t)$) on the same set of coordinate axes for $0\leq t \leq 10$. Also produce a plot for $0\leq t \leq 1$. Comment on what you see.

PROBLEM 2. (About the behavior of the error when calculating integrals numerically.)

Use Maple to define the function $f(t) = 4 \sqrt{1-t^2}$. Then verify that $\int_0^1 f(t) dt = \pi$.
Use Maple to define the ``error'' functions (where $n$ is a positive integer)

E_{left}(n) = LEFT(n) - \pi \quad \mbox{and} \quad
E_{trap}(n) = TRAP(n)-\pi

Here $LEFT(n)$ and $TRAP(n)$ are the left Riemann sum and the Trapezoid approximations to $\int_0^1 f(t) dt$. Calculate $E_{left}(10)$ and $E_{trap}(10)$ in decimal form.
Plot $E_{left}(n)$ and $E_{trap}(n)$ versus $n$ in the same set of coordinate axes. Use $2 \leq n \leq 20$ ($n$ is an integer). (Use the pointplot command) Explain the plot.

PROBLEM 3. (About the $p$-test for improper integrals.)

Define the functions

\begin{displaymath}r(t) =
\frac{1}{\cot(t^{-1/2})+\sqrt{t}\ (\sin(\pi t))^2}
\quad \mbox{and} \quad s(t) = \frac{K}{t^p}

Set $K=1$ and $p=1$, and plot $r(t)$ and $s(t)$ in the same set of coordinate axes for $1 \leq t \leq 10$. Verify that the inequality $0 \leq s(t) \leq r(t)$ for $1 \leq t$ is not satisfied.
Find (if possible) suitable values of $K>0$ and $p<1$ so that $0 \leq s(t) \leq r(t)$ for $1 \leq t$. Support your answer with appropriate plots. (You may use use $1 \leq t \leq 10$.)
What do your plots suggest about $\int_1^\infty r(t) dt$?

COMMENTS and additional information

Your name, date, class and section should be at the top of your paper. The final project should have only one author. You may discuss the project with your classmates, but what you turn in should contain your own original answers. Plagiarism is considered a serious offence.
Before each Maple computation, you should insert an explanation of what you are about to do. Neatness and good English will be taken into account. It is not sufficient to have it right; you should communicate it well.
Maple should be used in all calculations and plots.
For additional information on Plotting, solving equations and calculating integrals in maple: see the maple worksheet `` Introduction to Maple in Calculus II''(intro142.mws) , located in
MAPLE HELP will be available in some of the labs. The schedule and location will be announced at the web site .
To submit this project, you may use the Mathematics Department's electronic submission system, available at ( you must register as soon as possible ).


> restart;                 # good to have this at the top of worksheet; 
> with(student);           # adds leftsum, rightsum, trapezoid functions (and others)
> with(plots);             # adds extra functionality for plots
> f:=x->x^2;               # define a function
> f:=x->evalf(x^2);        # define a function, force it to give decimal result
> plot([f(x),g(x)],x=0..5);# plot two functions 
> leftsum(f(t),t=0..1,6);  # left Riemann sum of f(t) with 6 subintervals
> p1:=pointplot({seq([n,g(n)],n=1..10)}):
                           # plot points (1,g(1)), (2,g(2)),...,(10,g(10)), 
                           # and store the plot under the name p1
> display([p1,p2]);        # show two plots called p1 and p2 in one set of coordinate axes
> Pi                       # the number 3.1415...

Orlando Merino