Q: Which of the following is a necessary condition for the convergence of the alternating series $$\sum_{n=1}^{\infty} (-1$$)^{n+1}$$ a_n$$?

$$a_n \to 0 as$$ $$n \to \infty$$

Question 1

Suppose you want to calculate the volume of a solid formed by rotating the region bounded by $$y = x^2$ and y = 4$ about the x-axis. Which integral correctly represents the volume using the washer method?

Accepted Answer

$$\int_{-2}^{2} \pi (4^2$ - (x^2$)^2$$) \, dx$

Question 2

Which of the following is the correct application of the Fundamental Theorem of Calculus for $$F(x) = \int_{1}^{x} \sin(t^2$) \, dt$$?

Accepted Answer

$$F'(x) = \sin(x^2$)$$

Question 3

Evaluate the improper integral $$\int_{1}^{\infty} \frac{1}{x^2$} \, dx.$$

Accepted Answer

$$1$$

Question 4

Which substitution would you use to evaluate $$\int \sqrt{4 - x^2$} \, dx$$?

Accepted Answer

$$x = 2\sin 	heta$$

Question 5

A medication is absorbed into the bloodstream at a rate modeled by $$R(t) = 5e$$^{-0.5t}$$ mg/hour. What is the total amount absorbed in the first 4 hours?

Accepted Answer

$$\int_{0}^{4} 5e$$^{-0.5t}$$ \, dt$$

Question 6

Which of the following series converges by the Ratio Test?

Accepted Answer

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

Question 7

Find the area between the curves $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$.

Accepted Answer

$$\int_{0}^{1} (x - x^2$) \, dx$$

Question 8

Which of the following is the Maclaurin series for $$e^x$$?

Accepted Answer

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

Question 9

Evaluate $$\int x e$$^{x^2}$$ \, dx.$$

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 10

Which of the following integrals can be evaluated using partial fractions?

Accepted Answer

$$\int \frac{3x+2}{x^2$ - x} \, dx$$

Question 11

Which of the following is a necessary condition for the convergence of the alternating series $$\sum_{n=1}^{\infty} (-1$$)^{n+1}$$ a_n$$?

Accepted Answer

$$a_n 	o 0 as$$ $$n 	o \infty$$

Question 12

A tank is being filled with water at a rate $$R(t) = 10 - t$$ liters per minute for $$0 \leq t \leq 10. $$How much water is added in the first 5 minutes?

Accepted Answer

$$\int_{0}^{5} (10 - t) \, dt$$

Question 13

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u \, dv = uv - \int v \, du$$

Question 14

Which of the following series diverges?

Accepted Answer

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

Question 15

Find the sum of the geometric series $$\sum_{n=0}^{\infty} ar^n$$ for $$|r| \u003c 1.$$

Accepted Answer

$$\frac{a}{1 - r}$$

Question 16

Which test would you use to determine the convergence of $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ for $$p \u003e 0$$?

Accepted Answer

The p-series test

Question 17

Which of the following is the Taylor series for $$\sin x$$ centered at $$x=0$$?

Accepted Answer

$$\sum_{n=0}^{\infty} \frac{(-1)^n x$$^{2n+1}$$}{(2n+1)!}$$

MATH 205: Differential & Integral Calculus II – Course Syllabus and Topic Overview

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