a) Find the first four terms of the Taylor Series expansion for y = e^x about x

Question

a) Find the first four terms of the Taylor Series expansion for y = e^x about x = 0.

b) Plot the function f(x) = e^x along with the Taylor approximation you found in part A on the same set of axes. Use x = -5..5 for your x-axis bounds.  What do you notice about the two curves? How well does the Taylor series expansion approximate the exact curve of f(x) = e^x?   Where is the approximation the "best"?

c) Repeat parts A and B, but this time, expand the Taylor series out for the first 6 terms. Which is better and why?  How could you measure the "error" of the approximation at the point x = 2?

d) Use the remainder formula to compare the error for the Taylor approximation found in Part A and the Taylor approximation found in part C.

e) Find the first five terms of the Taylor Series expansion for y = sin(x) about x = 0.  How does this compare/contrast to the series expansion for cosine?

f) Find the first four terms of the Taylor Series expansion for y = 1/(x-1) about x = 2.  Plot the Taylor approximation, along with the original function on the same set of axes.  Where is the approximation the "best"?

Explanation / Answer

A)The easiest way is to repeatedly differentiate the function and evaluate at x=0 to get the Taylor coefficients.

Another way, not really easier, is to take the Taylor series of e^x up to x^4 and the Taylor series of up to x^4 and multiply those partial series. That will give you the Taylor series of the product, up to x^4:

(1 + x + x^2/2 + x^3/6 + x^4/24) (1 - x^2/2 + x^4/24) = 1 +x - x^3/3 - x^4/6 + higher order terms.

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