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Question
Expand `2x^3+7x^2+x-1` in powers of x - 2
Sum
Solution
Let `f(x) =2x^3+7x^2+x-1`
Here a = 2
| `f(x) =2x^3+7x^2+x-1` | 𝒇(𝟐)=𝟒𝟓 |
| `f'(x) =6x^2+14x+1` | 𝒇′(𝟐)=𝟓𝟑 |
| `f''(x) =12x+14` | 𝒇′′(𝟐)=𝟑𝟖 |
𝒇′′′(𝒙)=𝒇′′′(𝟐)=𝟏𝟐
Taylor’s series is :
`f(x)=f(a)+(x-a)f'(a)+(x-a)^2/(2!)f''(a)+....`
`2x^3+7x^2+x-1=45+(x-2)53+(x-2)^2/(2!)38+(x-a)^3/(3!)12`
`2x^3+7x^2+x-1=45+53(x-2)+19(x-2)^2+2(x-2)^3`
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