\(MATH\ 257\ Calculus\ III				Week\ of\ April\ 28, \ 
        2003\)
Lab 5 :  More Taylor Fun
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Last time, we used the Normal[Series[f[x], {x, a, n}]] command to make Mathematica generate the nth degree Taylor Polynomial for the function of f[x] centered at the point a.  

In this lab you'll practice doing this some more, this time with functions we'd never want to do using only pencil and paper...

For example, let's look at the function Sin[(e^x)].

First take a look at the graph of this function from -π to π:

Plot[Sin[(E^x)],  {x,  -π, π},  
      PlotRange -> {-5, 5}]

Rather unusual, huh?  Let's see what the 4th Taylor polynomial centered at a=0 looks like:

T4[x_] = 
      Normal[Series[Sin[(E^x)],  {x, 0,  4}]]

Blech!  That doesn't look like something I would have wanted to do by hand!  

Let's plot T4[x] together with Sin[(e^x)] to see how good a job T4[x] does approximating it.  I'll use the Thickness[] command to make the original function a thick line, and the RGBColor[] command to make T4[x] a different color:

Plot[{Sin[(E^x)], T4[x]}, {x,-π, π}, PlotRange ->  {-5, 5}, PlotStyle ->  {Thickness[ .01],  RGBColor[1, 0, 0]}]

Not too bad for only the 4th Taylor polynomial.  In the questions below you'll be asked to do better with higher-order Taylor polynomials.

Question 1 Find the 10th, 20th, and 30th order Taylor polynomials centered at the origin for Sin[(e^x)].  Give them the names T10[x], T20[x], and T30[x].

Question 2  Plot the original function Sin[(e^x)] along with the 10th, 20th, and 30th degree Taylor polynomials.  Make sure the original function is thick and the Taylor polynomials different colors.  Explain which colors are which polynomial!

Question 3  Give the approximate interval on which the Taylor polynomial of degree 30 approximates Sin[(e^x)].  What do you think the interval of convergence for the whole, infinite Taylor series is for this function?  (Hint: what it the interval of convergence for Sin[x] and (e^x) by themselves?)

Question 4 Find the 5th, 10th, and 15th degree Taylor polynomials for Tan[x] centered at the origin.

Question 5 Plot Tan[x] along with the 5th, 10th, and 15th degree Taylor polynomials found in #4 on the interval (-(π/2), (π/2)).  Make Tan[x] a thick graph and the polynomials different colors, and explain which colors correspond to which polynomial.

Question 6a Find the 9th order Taylor polynomial for f[x]=2x(e^x).

Question 6b Find the 9th order Taylor polynomial for f[x]=(e^x) and then multiply it by 2x.

Question 6c Did you get the same answers in the above two questions?  Explain why this happened.

Now that you're done, 
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		There should only be the title, your names, section number, and the questions and your answers
		(with any needed explanations, graphs, etc.).
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