AP Calculus BC – Series, Parametric and Polar

Determine whether each series converges or diverges. Name the test used.(a) $\sum_{n=1}^{\infty} \frac{n}{3n+1}$(b) $\sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}}$(c) $\sum_{n=1}^{\infty} \frac{2^n}{n!}$

Use the limit comparison test to determine whether $\sum_{n=1}^{\infty} \frac{3n + 1}{n^3 + 2}$ converges.

(a) Find the sum of the geometric series $\sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^n$.(b) Determine whether $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges absolutely, converges conditionally, or diverges.

Find the interval of convergence of the power series $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}$. Check the endpoints.

(a) Write the first four nonzero terms of the Maclaurin series for $e^x$.(b) Use your answer to write the first four nonzero terms of the Maclaurin series for $x e^{x^2}$.(c) Hence find $\frac{d^5}{dx^5}\left(x e^{x^2}\right)$ at $x = 0$.

Let $f(x) = \sin x$.(a) Write the third-degree Taylor polynomial $P_3(x)$ for $f$ about $x = 0$.(b) Use the Lagrange error bound to show that $|f(0.5) – P_3(0.5)| < 0.0003$.(c) Use $P_3$ to approximate $\int_0^{0.5} \frac{\sin x}{x}\,dx$.

The Taylor series for a function $g$ about $x = 1$ is $\sum_{n=0}^{\infty} \frac{(-1)^n (x-1)^n}{2^n (n+1)}$.(a) Find $g(1)$ and $g'(1)$.(b) Find the radius of convergence.(c) The series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$ is approximated by its first four terms. Use the alternating series error bound to find the maximum possible error.

A particle moves in the plane with position $x(t) = t^2 – 3t$, $y(t) = 2t^3$ for $t \ge 0$.(a) Find the velocity vector at $t = 1$ and the speed of the particle at $t = 1$.(b) Find $\frac{dy}{dx}$ at $t = 1$.(c) Find the time when the particle is moving vertically.

The curve $C$ is given by $x = t^2$, $y = t^3$ for $0 \le t \le 2$.(a) Find $\frac{d^2y}{dx^2}$ in terms of $t$.(b) Find the length of $C$.

The polar curve $r = 2 + 2\cos\theta$ (a cardioid) is drawn for $0 \le \theta \le 2\pi$.(a) Find the area enclosed by the curve.(b) Find $\frac{dy}{dx}$ at $\theta = \frac{\pi}{2}$.

The polar curves $r = 3$ and $r = 2 + 2\cos\theta$ are drawn.(a) Find the values of $\theta$ in $[0, 2\pi)$ where the curves intersect.(b) Set up and evaluate an integral for the area inside $r = 2 + 2\cos\theta$ and outside $r = 3$.(c) For the cardioid, find the rate of change of $r$ with respect to $\theta$ at $\theta = \frac{\pi}{3}$ and interpret its sign.

Let $h(x) = \frac{1}{1 + x^2}$.(a) Write the Maclaurin series for $h(x)$ and state its interval of convergence.(b) By integrating your series, write the first four nonzero terms of the Maclaurin series for $\arctan x$.(c) Use the first three nonzero terms to approximate $\arctan(0.2)$ and use the alternating series bound to estimate the error.