Assignment 10

Taylor Polynomials Homework:

7.7.{1,2,3,4,7,8,9,12,14, 17}

More problems:

1.  Differentiate the (entire) Taylor series for sin(x).  What do you notice?

2.  Simplify:  Pi - (Pi^3)/3! + Pi^5/5! - Pi^7/7! + ...

3.  Simplify:  1 + .2 + (.2^2) + (.2^3) + (.2^4) + ...

4.  We saw in class that e^(0.06) has a very structured form.  In this problem we'll examine e^(0.04).

• Calculate the first four terms (from constant term to cubic term) of the Taylor series for e^x.
• By hand, use these terms to estimate e^0.04. 
• What complication do you notice in the decimal expansion of the cubic term?

5.  Using a calculator or computer program of your choice, make a plot of sin(x), together with:

• The function obtained by taking the first term of its Taylor series.
• The function obtained by taking the first two terms of its Taylor series (i.e., linear and cubic).
• The function obtained by taking the first three terms of its Taylor series (i.e., linear, cubic, and quintic).

Explain what you notice.  Choose a region including [-2Pi, 2Pi]

6.  Repeat problem 5 using the function 1/(1-x).  On the region [-3,3], plot

• The function obtained by taking the first two terms of its Taylor series.
• The function obtained by taking the first four terms of its Taylor series.
• The function obtained by taking the first six terms of its Taylor series.

You should see similar patterns as in problem 5, but with one important difference--In what way does this Taylor series seem less adequate than the one for sin(x)?  If you can't see the pattern, use more terms!