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:

$f(x)=(-3x+15)^{3/2}$

$f'(x)=\dfrac32(-3x+15)^{1/2}(-3)=-\dfrac92(-3x+15)^{1/2}$

$f''(x)=-\dfrac94(-3x+15)^{-1/2}(-3)=\dfrac{27}4(-3x+15)^{-1/2}$

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

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

$f(2)=9^{3/2}=27$

$f'(2)=-\dfrac929^{1/2}=-\dfrac{27}2$

$f''(2)=\dfrac{27}4\dfrac1{9^{1/2}}=\dfrac94$

$f'''(2)=\dfrac{81}8\dfrac1{9^{3/2}}=\dfrac38$

Then the Taylor polynomial is

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

$P_3(x)=27-\dfrac{27}2(x-2)+\dfrac98(x-2)^2+\dfrac1{16}(x-2)^3$