Q: A medical researcher models the concentration of a drug in the bloodstream over time by the sequence $a_n = 0.8^n$. What is the long-term behavior of the drug concentration?

It approaches $0$ as $n \to \infty$.

Q: Which of the following series can be tested for convergence using the Ratio Test, and what is the result? $\sum_{n=1}^\infty \frac{3^n}{n!}$

The Ratio Test shows the series converges.

Q: A chemical reaction is modeled by the alternating series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}$. Which statement is true?

The series converges absolutely.

Q: Which of the following is the interval of convergence for the power series $\sum_{n=1}^\infty \frac{(x-2)^n}{n}$?

$|x-2| < 1$ and diverges at $x=1$ and $x=3$

Q: Find the Maclaurin series for $f(x) = e^{2x}$.

$\sum_{n=0}^\infty \frac{2^n x^n}{n!}$

Q: Which of the following is a correct parametrization of the circle of radius $3$ centered at $(2, -1)$?

$x = 2 + 3 \cos t, \ y = -1 + 3 \sin t$

Q: Find the area inside one loop of the rose curve $r = \sin 2\theta$.

$\frac{1}{2} \int_0^{\pi/2} (\sin 2\theta)^2 d\theta$

Question 1

Which of the following sequences converges, and what is its limit? $a_n = \frac{2n+1}{n+3}$

Accepted Answer

The sequence converges to $2$.

Question 2

Which of the following series is absolutely convergent?

Accepted Answer

$\sum_{n=1}^\infty \frac{1}{n^2}$

Question 3

A medical researcher models the concentration of a drug in the bloodstream over time by the sequence $a_n = 0.8^n$. What is the long-term behavior of the drug concentration?

Accepted Answer

It approaches $0$ as $n 	o \infty$.

Question 4

Which of the following is the correct application of the Integral Test for the convergence of $\sum_{n=2}^\infty \frac{1}{n \ln n}$?

Accepted Answer

The series diverges because $\int_2^\infty \frac{1}{n \ln n} \, dn$ diverges.

Question 5

Suppose $0 < a_n \leq b_n$ for all $n$, and $\sum b_n$ converges. What can you conclude about $\sum a_n$ by the Direct Comparison Test?

Accepted Answer

$\sum a_n$ converges.

Question 6

Which of the following series can be tested for convergence using the Ratio Test, and what is the result? $\sum_{n=1}^\infty \frac{3^n}{n!}$

Accepted Answer

The Ratio Test shows the series converges.

Question 7

A chemical reaction is modeled by the alternating series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}$. Which statement is true?

Accepted Answer

The series converges absolutely.

Question 8

Which of the following is the interval of convergence for the power series $\sum_{n=1}^\infty \frac{(x-2)^n}{n}$?

Accepted Answer

$|x-2| < 1$ and diverges at $x=1$ and $x=3$

Question 9

Find the Maclaurin series for $f(x) = e^{2x}$.

Accepted Answer

$\sum_{n=0}^\infty \frac{2^n x^n}{n!}$

Question 10

A particle moves along a path given by $x(t) = t^2$, $y(t) = t^3$. What is the speed of the particle at $t=2$?

Accepted Answer

$\sqrt{16 + 81} = \sqrt{97}$

Question 11

Which of the following is a correct parametrization of the circle of radius $3$ centered at $(2, -1)$?

Accepted Answer

$x = 2 + 3 \cos t, \ y = -1 + 3 \sin t$

Question 12

Find the area inside one loop of the rose curve $r = \sin 2	heta$.

Accepted Answer

$\frac{1}{2} \int_0^{\pi/2} (\sin 2	heta)^2 d	heta$

Question 13

Which of the following statements about the convergence of the series $\sum_{n=1}^\infty \frac{1}{n^p}$ is true?

Accepted Answer

It converges for $p > 1$.

Question 14

A curve is given in polar coordinates by $r = 2 + 2\cos 	heta$. What is the Cartesian equation of the curve?

Accepted Answer

$x^2 + y^2 - 2x = 0$

Question 15

Evaluate the length of the curve $x = t^2$, $y = t^3$ from $t=0$ to $t=1$.

Accepted Answer

$\int_0^1 \sqrt{(2t)^2 + (3t^2)^2} dt$

Question 16

Which of the following is the Taylor series for $\ln(1+x)$ centered at $x=0$?

Accepted Answer

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n$

Question 17

A patient’s heart rate is modeled by the parametric equations $x(t) = 3\cos t$, $y(t) = 4\sin t$. What is the total distance traveled by a point on this curve as $t$ goes from $0$ to $2\pi$?

Accepted Answer

$2\pi \cdot 5$

Question 18

Which of the following is the correct formula for the area enclosed by a polar curve $r = f(	heta)$ from $	heta = a$ to $	heta = b$?

Accepted Answer

$\frac{1}{2} \int_a^b [f(	heta)]^2 d	heta$

Question 19

Suppose $\sum_{n=1}^\infty a_n$ converges absolutely. Which of the following must also be true?

Accepted Answer

$\sum_{n=1}^\infty |a_n|$ converges.

Calculus II: Series, Parametric Equations, and Polar Coordinates Study Guide

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