Question

Find the degree 3 Taylor polynomial P3(x) of function f(x)=(-3x+15)^(3/2) at a=2

Answer 1

Take up to the third-order derivative:

`f(x)=(-3x+15)^(3/2)`

`f'(x)=\frac32(-3x+15)^(1/2)(-3)=-\frac92(-3x+15)^(1/2)`

`f''(x)=-\frac94(-3x+15)^(-1/2)(-3)=\frac{27}4(-3x+15)^(-1/2)`

`f'''(x)=-\frac{27}8(-3x+15)^(-3/2)(-3)=\frac{81}8(-3x+15)^(-3/2)`

Evaluate each derivative at
`x=a=2`:

`f(2)=9^(3/2)=27`

`f'(2)=-\frac929^(1/2)=-\frac{27}2`

`f''(2)=\frac{27}4\frac1{9^(1/2)}=\frac94`

`f'''(2)=\frac{81}8\frac1{9^(3/2)}=\frac38`

Then the Taylor polynomial is

`P_3(x)=f(2)+f'(2)(x-2)+\frac{f''(2)}2(x-2)^2+\frac{f'''(2)}6(x-2)^3`

`P_3(x)=27-\frac{27}2(x-2)+\frac98(x-2)^2+\frac1{16}(x-2)^3`