Taylor Series Approximation | AP Calculus BC | ShowMeClass

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Taylor Series Approximation

AP Calculus BC · Infinite Sequences & Series

Key Objectives

Understand Taylor polynomial tracking.

Visualize Convergence radius visually.

Observe error bound decreasing inside convergence interval.

Maclaurin Series (c=0)

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k

By matching derivatives at a single point (x=0), polynomial $P_n$ curves itself to hug the true function $f(x)$.

Maclaurin SeriesConvergencePolynomials

Degree: n = 1

$f(x)$ (True)
$P_n(x)$ (Approx)

Target Function: f(x)

Degree (n): 1

Expanded Polynomial

x