AP Calculus BC - Advanced Topics & Theorems

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Determine the Interval of Convergence for a Power Series

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To find the interval of convergence, apply the Ratio Test: $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges absolutely. Solve for $x$ such that $L < 1$. You **must** check the endpoints individually by substituting them into the original series, as the Ratio Test is inconclusive when $L=1$.

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Lagrange Error Bound (Alternating Series Remainder)

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For a function expanded in a Taylor polynomial about x=c, the Lagrange error bound estimates the remainder by \[|R_n(x)| \le \frac{M}{(n+1)!}|x-c|^{n+1},\] where \(M\) is a maximum value of \(|f^{(n+1)}(z)|\) on the interval between \(c\) and \(x\).

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Arc Length for Parametric Curves

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For a curve defined by parametric equations $x(t)$ and $y(t)$ where $t_1 \leq t \leq t_2$, the arc length $L$ is $L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$. Ensure the derivatives are squared and summed before taking the square root.

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Distance Traveled by a Particle in Motion (Parametric)

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Distance is the integral of speed (the magnitude of velocity), not velocity. For $x(t)$ and $y(t)$, distance is $\int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. Contrast this with displacement, which is simply $\int_{t_1}^{t_2} v(t) \, dt$ or $\sqrt{(\Delta x)^2 + (\Delta y)^2}$.

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Taylor Series Expansion about x = a

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The Taylor series for a function $f(x)$ centered at $x=a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$. If $a=0$, it is specifically called a Maclaurin series. Memorize common series like $e^x$, $\sin x$, and $\cos x$ (centered at 0) to derive related series.