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Each problem is worth the indicated number of points. For full credit
you must give an appropriate explanation of your reasoning.
This Problem Set is basically just a walk-through tour of the important parts of Section 12.10, which is the real focus of Chapter 12. The central idea is finding a general method for getting polynomial approximations to functions.
Start by reading page 785 and page 786 down to formula 7. Don't get hung up on details, just keep in mind that the general idea is to have the derivatives of our polynomials match the derivatives of the functions they're imitating.
[2 pts] 1. Without looking at Example 1 yet, try using formula 7 to write the MacLaurin series for f(x)=ex. Check yourself with the answer in Example 1. Notice that finding the radius of convergence for this power series is actually something we've already done (as part of problem 10 on the exam).
[2 pts] 2. Read Example 3. Mimic it to find the Taylor series for f(x) = ex at a=1. Write your answer both in sigma notation (the way the example does) and by listing out the first 5 terms.
Read Example 4 about finding the MacLaurin polynomial for sin x.
[2 pts] 3. Find the MacLaurin polynomial for sinh x (if you don't remember sinh x from somewhere in your past, all you need to know about it for now is that the derivative of sinh x is cosh x, the derivative of cosh x is sinh x, and your calculator should be able to tell you what sinh 0 and cosh 0 are).
Read Example 5, which shows how to use the Maclaurin polynomial for sin x to get the MacLaurin polynomail for cos x.
[2 pts] 4. Find the MacLaurin polynomial for cosh x (remember, the derivative of sinh x is cosh x).
Read Example 6, which gives another easy way to find new power series based on ones you already know.
[2 pts] 5. Find the MacLaurin series for the function f(x) = x sin x.
Read Example 7. Pay attention to the way they lay out the work in columns -- organizing things like that really does help avoid little mistakes.
[2 pts] 6. Represent f(x) = cos x as the sum of its Taylor series centered
at a=.
Skim page 792 through the end of Example 9 for ideas of some things these polynomials can do.
[2 pts] 7. Do Stewart 4th, §12.10 #4.
[2 pts] 8. Do Stewart 4th, §12.10 #6.
[2 pts] 5. Do Stewart 4th, §12.10 #10.
[2 pts] 6. Do Stewart 4th, §12.10 #14.