# Taylor polynomial Question

The function $f(x)$ is approximated near $x=0$, by the 3rd dgeree Taylor polynomial

$T_3(x)=4-3x+\frac{1}{5}x^2+4x^3$. Give the values of $f(0)$, $f'(0)$, $f''(0)$, $f'''(0)$

To find this, do I take the antiderivatives starting at the 3rd degree Taylor polynomial, and go back? I'm not sure how to approach this problem.

• Do you recall how to get a Taylor polynomial from a function? – Simply Beautiful Art Oct 19 '16 at 20:39
• Write down the Taylor formula and compare the terms. – Yves Daoust Oct 19 '16 at 20:42

We have

$$T_3(x)=f(0)+xf'(0)+\frac{x^2}{2} f''(0)+\frac{x^3}{3!}f'''(0)$$

so by identification of the coefficients of $x^0,x,x^2$ and $x^3$, we get

$f(0)=4$

$f'(0)=-3$

$f''(0)=\frac{2}{5}$

and

$f'''(0)=24$.

$f(0)=T_3(0)=4$

$f'(0)=T_3'(0)=-3$

$f''(0)=T_3''(0)={2\over 5}$

$f'''(0)=T_3'''(0)=24$

We can't say nothing for highter degree derivatives.

• Hm, similar can be done to approximately find $f(1),f'(1),f''(1)$ etc. with this idea. :) Intuitive answer. – Simply Beautiful Art Oct 19 '16 at 22:13