CBSE Class 12-science Answered

Consider the differentiation definition,

df(a)/dx = (f(x) - f(a))/(x-a)

Rearranging, f(x) = f(a) + (x-a)df(a)/dx

This simple formula to obtain the function, using a linear function, can be extended to include higher derivatives and get as accurate as possible,

f(x ) = f(a) + (x-a)df(a)/dx + ((x-a)²/2!)d²f(a)/dx² + ((x-a)³/3!)d³f(a)/dx³ +((x-a)⁴/4!)d⁴f(a)/dx⁴ + ............

Note that we don't know what the function is, all we know is the value of function at a and it's derivatives. 

Such a series representation is called a Taylor series of the function, since a representation of the function can be obtained, around some point, a.

f(x) = ((x-a)ⁿ/n!)dⁿf(a)/dxⁿ       The summation can go over infinite terms.

If the point a = 0, then the series is called Maclaurin Series.

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