Question

Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series. 

 

Answer 1

Let `f(x)=2x^3+3x^2-8x+7`

Here a=2 

`f(x)=2x^3+3x^2-8x+7`                   `f(2)=19`

`f'(x)=6x^2+6x-8`                            ` f'(2)=28`

`f'(x)=12x+6`                                    ` f''(2)=30`

`f'''(x)=f'''(2)=12`

Taylor’s series is : 

`f(x)=f(x)=f(a)+(x-a)f'(a)+(x-a)^2/(2!) f''(a)+........` 

`2x^3+3x^2-8x+7=19+(x-2)28+(x-2)^2/(2!) 30+(x-a)^3/(3!) 12`

`2x^3+3x^2-8x+7=19+28(x-2)+15(x-2)^2+2(x-2)^3`