# Taylor's formula

1. Sep 7, 2014

### Curious09

What is taylor formula and how it is used in calculators?

2. Sep 8, 2014

### HallsofIvy

Staff Emeritus
Taylor's formulas says that if f(x) is n+ 1 times differentiable in some neighborhood of $x= x_0$, then f(x) can be approximated by
$$T(x)= f(x_0)+ f'(x_0)(x- x_0)+ \frac{f''(x_0)}{2}(x- x_0)^2+ \frac{f'''(x_0)}{6}(x- x_0)^3+ \cdot\cdot\cdot+ \frac{f^{(n)}(x_0)}{n!}(x- x_0)^n$$
where "$f^{(n)}(x_0)$" indicates the nth derivative evaluated at $x= x_0$.

Further, the error, |f(x)- T(x)|, will be less than
$$\frac{f^{(n+1)}(x_0)}{(n+1)!}|x- x_0|^{n+1}$$.

I'm surprised you did not just look it up with Google or on Wikipedia. You will get a lot more information.

As for "how is it used in calculators"- it isn't. Calculators and Computers use a much more advanced numerical procedure called "CORDIC" to do calculations of trig functions, exponentials, etc.
http://en.wikipedia.org/wiki/CORDIC

3. Sep 10, 2014

### Curious09

But i am unable to understand how they calculate maximum number of values which it can accomodate before giving an error more than some specific value.

4. Sep 10, 2014

### fisicist

"maximum number of values"? You mean, in what environment of a specific argument the Taylor polynomial supplies a "sufficiently good" approximation? There is a formula for the error term, it was already posted above. It answers your question (if I got it right) more or less directly.
Please state your questions clearer and show a little bit more initiative.

5. Sep 12, 2014

### FactChecker

I would be surprised if the Taylor series was often used in calculators. The Taylor series has a lot of good theoretical properties and it is the first method of approximation you should learn. But it is usually not the most efficient way to approximate a given function. If it is used, @HallsOfIvy has posted the information.