Q. 5 - Calculus

Question

Let f be a twice-differentiable function at a point $x_0$. Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_2\left(x\right)$ might be a good approximation for f close to $x_0$.

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{second} \mathrm{Taylor} \mathrm{polynomial} \mathrm{has} \mathrm{the} \mathrm{same} \mathrm{value}, \mathrm{same} \mathrm{slope} \mathrm{for} \\ \mathrm{its} \mathrm{tangent} \mathrm{line}, \mathrm{and} \mathrm{the} \mathrm{same} \mathrm{concavity} \mathrm{as} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{the} \mathrm{point} {\mathrm{x}}_0. \end{gathered}$$

Step 1. Given information is:

$\mathrm{f} \mathrm{is} \mathrm{a} \mathrm{twice}-\mathrm{differentiable} \mathrm{function} \mathrm{at} \mathrm{a} \mathrm{point} {\mathrm{x}}_0$

Step 2. Result

$$\begin{gathered} \mathrm{If} \mathrm{f} \mathrm{is} \mathrm{a} \mathrm{twice} \mathrm{differentiable} \mathrm{function} \mathrm{at} \mathrm{a} \mathrm{point} {\mathrm{x}}_0, \mathrm{then} \mathrm{the} \mathrm{second} \mathrm{Taylor} \\ \mathrm{polynomial} {\mathrm{P}}_2\left(\mathrm{x}\right) \mathrm{might} \mathrm{be} \mathrm{a} \mathrm{good} \mathrm{approximation} \mathrm{for} \mathrm{f} \mathrm{close} \mathrm{to} {\mathrm{x}}_0. \\ \mathrm{This} \mathrm{is} \mathrm{because} \mathrm{the} \mathrm{second} \mathrm{Taylor} \mathrm{polynomial} \mathrm{has} \mathrm{the} \mathrm{same} \mathrm{value}, \mathrm{same} \\ \mathrm{slope} \mathrm{for} \mathrm{its} \mathrm{tangent} \mathrm{line}, \mathrm{and} \mathrm{the} \mathrm{same} \mathrm{concavity} \mathrm{as} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{the} \mathrm{point} {\mathrm{x}}_0. \end{gathered}$$