Question 1

Which of the following integrals represents the area enclosed by one loop of the polar curve $$r = 2 \sin 	heta$$?

Accepted Answer

$$\displaystyle \frac{1}{2} \int_{0}^{\pi} (2 \sin \$$theta)^2$$ \, d	heta$$

Question 2

Evaluate the improper integral $$\displaystyle \int_{1}^{\infty} \frac{1}{x^2$} \, dx$$ and determine if it converges or diverges.

Accepted Answer

It converges to $$1.$$

Question 3

Which of the following is the correct partial fraction decomposition for $$\displaystyle \frac{3x+5}{(x-1)(x+2)}$$?

Accepted Answer

$$\displaystyle \frac{A}{x-1} + \frac{B}{x+2}$$, where $$A=2$$, $$B=1$$

Question 4

A tank contains 100 liters of water and salt is being added at a rate modeled by $$r(t) = 5e$$^{-0.1t}$$ grams per minute. What is the total amount of salt added to the tank after 10 minutes?

Accepted Answer

$$\displaystyle 50(1 - e$$^{-1}$$)$$ grams

Question 5

Which of the following series converges?

Accepted Answer

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

Question 6

Find the Maclaurin series for $$\sin x up to $$the $$x^5$$ term.

Accepted Answer

$$\displaystyle x - \frac{x^3$}{3!} + \frac{x^5$}{5!}$$

Question 7

Which of the following is an example of an improper integral?

Accepted Answer

$$\displaystyle \int_{0}^{\infty} e$$^{-x}$$ \, dx$$

Question 8

Evaluate $$\displaystyle \int x e$$^{x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$\displaystyle x e$$^{x}$$ - \int e$$^{x}$$ \, dx = x e$$^{x}$$ - e$$^{x}$$ + C$$

Question 9

A force $$F(x) = 10x (in $$Newtons) moves an object along the $$x-$$axis from $$x=0 to$$ $$x=5$$ meters. What is the work done by the force?

Accepted Answer

$$\displaystyle \int_{0}^{5} 10x \, dx = 125$$ Joules

Question 10

Which substitution is most appropriate for evaluating $$\displaystyle \int \frac{dx}{\sqrt{4 - x^2$}}$$?

Accepted Answer

$$x = 2 \sin 	heta$$

Question 11

Given the sequence $$a_n = \frac{1}{n}$$, what can be said about its convergence?

Accepted Answer

It converges to $$0.$$

Question 12

Which of the following is the correct formula for the length of the curve given by $$y = f(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\displaystyle \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \$$right)^2$$} \, dx$$

Question 13

Which test would you use to determine the convergence of the series $$\displaystyle \sum_{n=1}^{\infty} \frac{(-1$$)^{n}$$}{n}$$?

Accepted Answer

Alternating Series Test

Question 14

Find the volume of the solid generated by revolving the region bounded by $$y = x^2$ and y = 0$ from x=0$ to x=2$ about the x$-axis.

Accepted Answer

$$\displaystyle \pi \int_{0}^{2} (x^2$)^2$$ \, dx = \frac{16\pi}{5}$

Question 15

Which of the following is the Taylor series for $$e^x$$ centered at $$x=0$$?

Accepted Answer

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

Question 16

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

$$\displaystyle \frac{1}{2} \int_{a}^{b} [f(	heta)]^2 \, d	heta$$

Question 17

Which of the following integrals is improper due to an infinite discontinuity in the interval of integration?

Accepted Answer

$$\displaystyle \int_{-1}^{1} \frac{1}{x} \, dx$$

Question 18

Suppose $$\displaystyle \sum_{n=1}^{\infty} a_n$$ converges. What can you say about $$\displaystyle \lim_{n 	o \infty} a_n$$?

Accepted Answer

It must be $$0.$$

Question 19

Which of the following is the correct trigonometric substitution for evaluating $$\displaystyle \int \frac{dx}{x^2$ \sqrt{x^2$ - 9}}$$?

Accepted Answer

$$x = 3 \sec 	heta$$

Calculus II Syllabus and Course Structure Overview

Guida di studio - Domande di pratica

Study guide - Flashcard