A 16-Year-Old Asks: How Can I Apply Taylor Series to Delta-F?

[richardlhp]

Hi! I am a 16 year old trying to figure out the application of taylor series. I understand most of its uses when applied to functions like e^x, sinx, cosx, but in a mechanics book, i am required to find delta-F, a finite change in a function F. Ostensibly, this appears to be a step that needs the application of taylor series, so can anyone help me to explain slowly and clearly how taylor series can be applied to delta-F? (sorry i do not know how to use symbols and stuff)

 

[LCKurtz]

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

If you lop off the higher order terms you have the approximation

$$f(x) \doteq f(a) + \frac{f'(a)}{1!}(x-a)$$

or

$$f(x) - f(a) \doteq f'(a)(x-a)$$

If x = b you have

$$f(b) - f(a) \doteq f'(a)(b-a)$$

In the delta-y notation you might write this as

$$\Delta y \doteq f'(x)\Delta x$$

Is that what you are getting at?

 

[richardlhp]

Yeah! Thanks a lot. Hence for a more accurate value, one can keep the higher order terms?

 

[LCKurtz]

Yeah! Thanks a lot. Hence for a more accurate value, one can keep the higher order terms?

Yes, exactly, as long x is within the radius of convergence from a. Many functions have good approximation with just a few terms for x near a but not every function's Taylor series converges to it.

 

[catface4]

You can find more information here:

http://tinyurl.com/taylorseries