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  3. AP Calculus AB/BC
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Finding Taylor Polynomial Approximations of Functions

Hannah Hill

Hannah Hill

6 min read

Next Topic - Lagrange Error Bound
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.

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Previous Topic - Alternating Series Error BoundNext Topic - Lagrange Error Bound

Question 1 of 10

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

∑n=0∞f(n)(a)⋅(x−a)n\sum_{n=0}^\infty f^{(n)}(a) \cdot (x-a)^n∑n=0∞​f(n)(a)⋅(x−a)n

∑n=0∞f(n)(a)n!⋅(x)n\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x)^n∑n=0∞​n!f(n)(a)​⋅(x)n

∑n=0∞f(n)(a)n!⋅(x−a)n\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x-a)^n∑n=0∞​n!f(n)(a)​⋅(x−a)n

∑n=1∞f(n)(a)n!⋅(x−a)n\sum_{n=1}^\infty \frac{f^{(n)}(a)}{n!} \cdot (x-a)^n∑n=1∞​n!f(n)(a)​⋅(x−a)n