Quick Question on Taylor Expansions

[StephenD420]

Hello all,

I am a senior physics undergraduate student. I have wondered about the Taylor Expansion for a few years now and just have never bothered to ask. But I will now:

I know the Taylor Expansion goes like:

f(a) + \frac{f'(a)}{1!}*(x-a) + \frac{f''(a)}{2!}*(x-a)^{2} + \frac{f'''(a)}{3!}*(x-a)^{3} + ...

which is the same as \sum \frac{f^{n}(a)}{n!}*(x-a)^{n}

but how do you know when you use this to approximate a formula? Any problem that my professors have given they have explicitly said to use a Taylor Expansion, but I know there has to be a rule of thumb when to use the Taylor Expansion to approximate a formula.

Any ideas?
Thanks much.
Stephen

 

[Chestermiller]

You are asking about typical practical applications of Taylor series expansions. Here are two examples:

1. Suppose you need to evaluate the definite integral of a function over a fairly narrow range of integration limits, and the integral of the function is not conveniently tabulated. You can still get an accurate estimate of the integral by expanding in a Taylor series with respect to some point between the two limits of integration, and integrating several terms in the resulting Taylor series.

2. Taylor series expansions are used extensively in numerical analysis to provide accurate finite difference formulas for the derivatives of various orders of a function.