# Thread: Error in Taylor Polynomial

1. ## Error in Taylor Polynomial

Completely lost, all help appreciated

1) Bound the error in the approximation

sin(x) ≈ x

for -pi/4 ≤ x ≤ pi/4

--------------------------------------

Let p(x) be the Taylor polynomial of degree n of the function
f(x) = log(1-x) about a = 0.

How large should n be to have

| f(x)-p(x) | ≤ 10^(-4) for -1/2 ≤ x ≤ 1/2 ?

For -1 ≤ x ≤ 1/2

2. $\displaystyle \sin(x)\;=\;x - \frac{1}{6}x^{3} + OtherStuff$

The alternating sign of the terms is VERY helpful.

What is the size of the first term omitted on the desired interval?

3. The error is less than the first term not used in the approximation

|-x^3/3!| < pi^3/(4^3*3)= pi^3/384.

If they ask, "What is the taylor polynomial of degree 3?"

Do you make sure that you have x^3 and no higher degree? Or do you keep 3 terms or? How do you know what's degree 3, etc. Thanks.

4. Originally Posted by amor_vincit_omnia
The error is less than the first term not used in the approximation

|-x^3/3!| < pi^3/(4^3*3)= pi^3/384.
That's the idea, but are you sure that's where the maximum value occurs, AT $\displaystyle \frac{\pi}{4}$? Also, 3! = 6, not 3. 3! = 3*2*1

If they ask, "What is the taylor polynomial of degree 3?"
Quit when you have a term in x^3 or of higher degree. For example, I dare you to find a degree 3 polynomial for f(x) = cos(x). Please don't count the number of terms while looking for degree.