What do you notice about the error as the interval distance between x and a gets

Question

What do you notice about the error as the interval distance between x and a gets smaller? What happens as n increases? We will explore this formula in the exercises below. Part A Find the first four terms of the Taylor Series expansion for y=ex about x=0. Part B Plot the function f(x)=ex 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)=ex ? Where is the approximation the "best"? Part 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? Part 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. Part D 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? Part E 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

y=e^x 1st 4 terms= 1+x+x^2/2+x^3/6 1st 6 terms= 1+x+x^2/2+x^3/6+x^4/24+x^5/120 y=sinx 1st 5 terms= x-x^3/6+x^5/120-x^7/5040+x^9/362880 y=cosx 1st 5 terms= 1-x^2/2+x^4/24-x^6/720+x^8/40320 y= 1/(x-1) 1st 4 terms= -1+(x-2)-(x-2)^2+(x-2)^3

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