ModelingProject2
Maple Project 2 MTH 142 Fall 2001

Department of Mathematics, University of Rhode Island






PROBLEM 1.

a) define in Maple the function $f(x) = \frac{1}{2+ 0.8 \sin (x)}$. Produce a plot of the function for $0 \leq x \leq 6$.

b) Obtain taylor polynomials $P_{1}(x)$, $P_{2}(x)$, $P_{3}(x)$ of $f(x)$ for $x$ near $a=3.2$.

c) Plot $f(x)$ together with $P_{1}(x)$, $P_{2}(x)$, $P_{3}(x)$.

d)The error when approximating f(x) by $P_{n}(x)$ at $x$ is defined as

\begin{displaymath}E_{n}(x) = f(x) - P_{n}(x) = \mbox{exact value - approximation}\end{displaymath}

Calculate $E_{1}(1.01)$, $E_{2}(1.01)$, and $E_{3}(1.01)$. Comment on the numbers you obtain.

e) Produce a plot of $f(x)$ and $P_{20}(x)$ with suitable ranges for the dependent and independent variables, so that it is possible to see what part of the plot of $f(x)$ is well matched by $P_{20}$. What is a guess for the interval of convergence of the Taylor series of $f(x)$ for $x$ near $a=3.2$?






PROBLEM 2.

a) It can be shown mathematically that

\begin{displaymath}S = \sum_{n=1}^{\infty} \frac{1}{n^{4}} =
\frac{\pi^{4}}{90}\end{displaymath}

Verify that Maple produces the same result.

b) Define the the partial sum function

\begin{displaymath}partialsum(N) = \sum_{n=1}^{N}\frac{1}{n^{4}}, \quad N=1,2,\ldots\end{displaymath}

Produce a pointplot of $partialsum(N)$, for $N=1,2,\ldots,20$.

c)The error when approximating $S$ by $partialsum(N)$ is defined as

\begin{displaymath}Err(N) = S - partialsum(N)\end{displaymath}

Calculate $Err(10)$, $Err(100)$, $Err(1000),Err(10,000)$ . Comment on the rate at which the error goes to zero as $N$ is increased.




COMMENTS and additional information

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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.
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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.
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Maple should be used in all calculations and plots.
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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 www.math.uri.edu/Center/workc2.html
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MAPLE HELP will be available in some of the labs. The schedule and location will be announced at the web site www.math.uri.edu .
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To submit this project, you may use the Mathematics Department's electronic submission system, available at www.math.uri.edu ( you must register as soon as possible ).


USEFUL MAPLE COMMANDS



> 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 
> 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...
> exp(2.5);                # exponential function evaluated at 2.5
> taypol:=n->convert( taylor(f(x),x=1,n+1),polynom);
                           # taypol(5) gives the Taylor polynomial of degree 5 of
			   # the function f(x) about x = 1;
> sub(x=2,p);              # substitute x=2 into p (here p could be a polynomial, for example)			   
> sum(1/n,n=1..infinity);  # infinite sum from n=1 to n=infinity of 1/n