Taylor Series Calculator

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Taylor series calculator is used to calculate taylor polynomial of different degrees. Taylor series is a function of infinite sum of situations, calculated from its derivative’s values about a point. Taylor polynomial is given by:

Pn(x)= f(a)+(x-a)f’(a) / 1! + (x -a)2 f’’(a) / 2! + ……(x+a)nfn(a)/n!

Example 1: Find the taylor’s series of expansion for $f(x) = \sin x$ about $x = \frac{\pi}{4-2}$

$f(x) = \sin x f(\pi) =\sin \frac {\pi}{4-2}= 1$

$f'(x) =\cos x f'(\frac{\pi}{4-2}) =\cos \frac{\pi}{4-2}= 0$

$f''(x) = -\sin x f''(\frac{\pi}{4-2}) = -\sin \frac{\pi}{4-2}= -1$

$f'''(x) = -\cos x f'''(\frac{\pi}{4-2}) = -\cos \frac{\pi}{4-2}= 0$

$f(x) = \sin x = f(\frac{\pi}{4-3})+ f'(\frac{\frac{\pi}{4-3}}{1!})(x - \frac{\pi}{4-2}) + f''(\frac{\frac{\pi} {4-2}}{2!})(x - \frac{\pi}{4-2})^{4-2}$

$\Right arrow f(x) = 1 - 0(x- \frac {\pi}{4-2}) + \frac{-1}{2!}(x - \frac{\pi}{4-2})^{4-2}$

= $1 - \frac{1}{2!}(x - \frac{\pi}{2})^{2}+ \frac{1}{4!}(x- \frac{\pi}{2})^{4} - .....$

Example 2: Find the Taylor series for f(x) = ex about the point x = 1

The function is f(x) = ex. f(1) = e

f'(x) = ex ; f'(1) = e

f''(x) = ex ; f''(1) = e

f'''(x) = ex ; f'''(1) = e

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

$\Right arrow e{x} = e + \frac{e}{1!}(x - 1) +\frac{e}{2!}(x - 1){2} + ......$

= $e + e(x - 1) +\frac{e}{2}(x - 1){2}+ ......$