What is Taylor Series

In mathematics, sometimes it is really difficult to evaluate some functions. In such cases, we use the approximation formulas where the function is expressed as a series. There are two such approximation formulas:

  1. Taylor series formula
  2. Maclaurin series formula

Taylor series formula helps us in writing a function as a series (or sum) of terms involving the derivatives of the function.

This formula helps us in finding the approximate value of the function.

Taylor series formula

f( x+h )=f( x ) + f'(x)h + f”{(x) h2}/2! + f”'{(x) h3}/3! + ⋯

or

taylor series formula
Taylor Series Formula
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1 Example Solved

Given f(4)=125, f’(4)=74, f’’(4)=30, f’’’(4)=6. Find f(6)=?

Sol: Here h = 2, and all other derivates are zero.

F(6) = 125 + 74(2) + 30(2)2 / 2 + 6(2)3 / 3 + 0 + 0 + 0

F(6) = 125 + 148 + 60 + 8

F(6) = 341 Ans.

2 Example Solved

Given The Ordinary differential dy/dx = 3×2 – x2y, y(2)=5. Find 2nd Order Polynomial for y as a function of x & x=2.

Sol.

Let the 2nd order polynomial be a0+a1x + a2x2

Using taylor series, f( x+h )=f( x ) + f'(x)h + f”{(x) h2}/2! + f”'{(x) h3}/3! + ⋯

Given, y(2) = 5 and x =2.

Therefore,

y’(2) = 3x2 – x2y

3(2)2 – (2)2y

12 – 4y

Y = 12/4

Y=3, put this value of y

Y’(2)=3(2)2 – (2)-8

Y’(2) = -8

Now, y’’(2) = (d/dx)(3x2 – x2y)

6x – (d/dx)(x2y)

Use product Rule of differentiation, xy

product rule differentiation
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6x – ( 2xy – x2 d/dx(y))

6(2) – 2(2)(5) – 22(-8) = 24

Therefore, y(2+h)= 5 + (-8)h + 24h2

Y(2+h) = 5 – 8h + 12h2

Y(x) = 5-8(x-2)+12(x-2)2

Y(x)=12x2-56x+69   (Answer.)

Suggested Methods:

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