Finding Taylor Polynomial Approximations of Functions

Hannah Hill

6 min read

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Study Guide Overview

This study guide covers Taylor Polynomial Approximations for AP Calculus BC. It explains the Taylor Approximations Theorem, including the formula and its components. The guide focuses on applying the theorem to find Maclaurin series (Taylor series centered at x=0) and Taylor series centered at other values. It provides practice problems and solutions for finding Taylor and Maclaurin polynomials of various functions.

#10.11 Finding Taylor Polynomial Approximations of Functions

Welcome to AP Calc 10.11! In this lesson, you’ll learn how to approximate a function over at a point.

#📈 Taylor Approximations Theorem

This theorem states that for a function $f(x)$ , it’s Taylor series approximation at $x=a$ is…

$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}\cdot(x-a)^n$

This can be rewritten as…

$f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n$

where $f^{(n)}(a)$ is the $n^{\text{th}}$ deriviative of the function and $f^{(0)}(a)=f(x)$ . The $n^{\text{th}}$ -order Taylor polynomial is the $n^{\text{th}}$ partial sum of the infinite series.

Taylor series centered at $x=0$ are common and are called Maclaurin series.

#🧱 Breaking Down the Theorem

Taylor series look very daunting when you first approach them. Let’s define each portion and build a table that will help you tackle problems of this type!

 $\begin{array}{ |c|c|c|c|c|c| } \hline n & n! ...$

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Question 1 of 10

What is the general formula for the Taylor series approximation of a function $f(x)$ at $x=a$ ?

$\sum_{n=0}^\infty f^{(n)}(a) \cdot (x-a)^n$

$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x)^n$

$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x-a)^n$

$\sum_{n=1}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x-a)^n$