# Estimation with Taylor Polynomials and Error Bounds

Recall that if $f$ is a function that is $n$ times differentiable on an open interval containing $c$ then the Taylor polynomial of $f$ at $c$ is:

(1)Furthermore, if $c = 0$ then the polynomial above is sometimes called the Maclaurin polynomial of $f$.

We recently saw on the Taylor's Theorem and The Lagrange Remainder page that the Lagrange Remainder $E_n(x) = f(x) - P_n(x)$ can be obtained for some $\xi$ between $c$ and $x$ with the following formula:

(2)We will now look at some examples of estimating sums with Taylor polynomials and bounding the error in these approximations.

## Example 1

**Use the second order Taylor polynomial $P_2(x)$ for $f(x) = \sqrt{x}$ centered about $64$ to approximate the value of $\sqrt{61}$. Bound the error of this approximation.**

We have that $f'(x) = \frac{1}{2\sqrt{x}}$ and $f''(x) = \frac{-1}{4(\sqrt{x})^3}$. Therefore the second order Taylor polynomial $P_2(x)$ for $f(x) = \sqrt(x)$ centered about $64$ is:

(3)Plugging in $x = 61$ gives us:

(4)Now let's bound the error. For some $\xi$ between $61$ and $64$ we have that:

(5)We compute the third derivative of $\sqrt{x}$ as $f^{(3)}(x) = \frac{3}{8x^{5/2}}$. Note that $\max_{[61, 64]} \mid f^{(3)} (x) \mid = \frac{3}{8(\sqrt{61})^5}$

Therefore we have that:

(6)