Question

A function f has derivatives of all orders for all real numbers, x. Assume f(2) = −3, f ′(2) = 5, f ″(2) = 3 and the third derivative of f at x = 2 is −8. Write the 3rd degree Taylor polynomial for f about x = 2 and use it to approximate f(1.5).

Answer 1

The Taylor polynomial would be

$T_3(x)=f(2)+f'(2)(x-2)+\dfrac{f''(2)}2(x-2)^2+\dfrac{f'''(2)}6(x-2)^3$

$T_3(x)=-3+5(x-2)+\dfrac32(x-2)^2-\dfrac43(x-2)^3$

Then

$f(1.5)\approx T_3(1.5)=-\dfrac{119}{24}\approx-4.96$