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: 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 $$0 \u003c a_n \leq b_n$$ 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

$$|x-2| \u003c 1$$ 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\$$theta)^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 $$p \u003e 1.$$

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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