5.6 Taylor Polynomials

We began this chapter by finding tangent lines for a graph by matching the graph's height and slope. Now that we know how to calculate bend, we can use height, slope, and bend together to find parabolas that hug our graph.

Definition.

Consider the graph of a function \( z \depends x \). The Taylor polynomial is the function \( z_T \depends x_T \) described by the equation

\[ \begin{align} z_T - z &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult (x_T - x) + \operatorname{\overset{\mathit{x}}{\bend}} z \mult (x_T - x)^2 \end{align} \]

where

\[ \begin{align} &x : \Number, & &z : \Number, & &\operatorname{\overset{\mathit{x}}{\slope}} z : \Number, & &\operatorname{\overset{\mathit{x}}{\bend}} z : \Number \end{align} \]

are the constants found at the center.

Let's find a Taylor polynomial!

Example.

Consider the square root function

\[ \begin{align} z &\depends x \\ z &= \sqrt[2]{x}. \end{align} \]

Let's find the Taylor polynomial centered at \( x = 1 \). We've calculated the slope and bend.

\[ \begin{align} \operatorname{\overset{\mathit{x}}{\slope}} z &= \frac{1}{2 \mult \sqrt[2]{x}} \\ \operatorname{\overset{\mathit{x}}{\bend}} z &= \frac{-1}{8 x \mult \sqrt[2]{x}} \end{align} \]

So at the center \( x = 1 \) our function has height, slope, and bend:

\[ \begin{align} &z = 1, & &\operatorname{\overset{\mathit{x}}{\slope}} z = \frac{1}{2}, & &\operatorname{\overset{\mathit{x}}{\bend}} z = \frac{- 1}{8}. \end{align} \]

We can use these constants to write down the Taylor polynomial.

\[ \begin{align} z_T - 1 &= \frac{1}{2} \mult (x_T - 1) - \frac{1}{8} \mult (x_T - 1)^2 \end{align} \]

Here are the graphs of the functions from the example. The square root function is on the left, and its Taylor polynomial is on the right.

The graph of the Taylor polynomial is the best fit parabola. The Taylor polynomial, \( z_T \depends x_T \), is defined so that its height, slope, and bend match the function \( z \depends x \) at the center.