Thread: Need help understanding how to find Taylor polynomials

1. Need help understanding how to find Taylor polynomials

I, for the life of me, cannot figure out how Taylor polynomials/series work. Frankly, the material we were given to work with isn't very explanatory, and doesn't have examples that tell us what to do.

The questions are below. Using online programs, I was able to determine the final answers to the following. However, I'm unable to determine HOW to get to these answers. Also, the material does not contain anything relating to Big-Oh notation, so don't include that.

Find the Taylor polynomial $P_4$ for the function $f$

1) $f(x)=x-\cos x$
$Ans.=-1+x+\frac{x^2}{2}-\frac{x^4}{24}$

2) $f(x)=\sqrt{1+x}$
$Ans.=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{x^4}{128}$

3) $f(x)=\ln\cos x$
$Ans.=-\frac{x^2}{2}-\frac{x^4}{12}$

4) $f(x)=\sec x$
$Ans.=1+\frac{x^2}{2}+\frac{5x^4}{24}$

Any help in determining how to reach these answers would be appreciated, so I might have a chance at understanding how the process works.

2. Wikipedia has a good article on Taylor series here.

In particular your example appears to be a specific case of the Taylor series called the Maclaurin series. Have you had any experience with these at all?

Let me know if the Wikipedia article helps.

3. Sorry had to change computers before, didn't have time for a fuller explanation.

Here's the formula for the Maclaurin series.

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$

Hope this helps.

4. Originally Posted by craig
Sorry had to change computers before, didn't have time for a fuller explanation.

Here's the formula for the Maclaurin series.

$f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$

Hope this helps.
I hope this isn't interpreted as being too nitpicky, but it is more correct to write

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$

with $=$ instead of $\approx$ due to convergence; we use $\approx$ when the infinite series is truncated. I write this only for clarity's sake.

5. Originally Posted by undefined
I hope this isn't interpreted as being too nitpicky, but it is more correct to write

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$