Maclaurin Series Approximation – AP Calculus BC

Learning Objectives

Visualize how adding polynomial terms ($n$-th degree) forces an approximation to "wrap" around a transcendental curve.

Observe the error divergence as $x$ moves away from the center of approximation ($x=0$).

Recognize the exact Maclaurin expansions for $\sin(x)$, $\cos(x)$, and $e^x$.

General Maclaurin Definition

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

A Maclaurin Series is just a Taylor Series strictly centered at $c = 0$.

Tags

Calculus BCSeriesTaylorMaclaurinApproximation

Transcendental Function vs Polynomial Expansion

Target $f(x)$$\sin(x)$

Polynomial Degree ($n$)1

Current Approximation: $P_n(x)$

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Target Function

Polynomial Degree ($n$) 1

Notice: The polynomial exactly matches the target at $x=0$. As you increase $n$, the "closeness" spreads outwards. For $\sin$, $\cos$, and $e^x$, the radius of convergence is $\infty$, so an infinite polynomial will perfectly trace the entire line.

Quick Quiz

If you set the degree $n=5$ for $f(x)=\sin(x)$, what does the final term look like?