Compute $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$; if $$L \u003c 1$$ the series converges.

Question 1

Which of the following is the correct antiderivative of $$f(x) = \frac{4}{1 + x^2$}$$?

Accepted Answer

$$2 \arctan(x) + C$$

Question 2

Evaluate the integral $$\int \frac{e$$^{x}$$}{\sqrt{x}} \, dx.$$

Accepted Answer

$$2 \sqrt{x} e$$^{x}$$ + C$$

Question 3

Which substitution is most appropriate for evaluating $$\int \frac{5x^2$}{x^3$ - 1} \, dx$$?

Accepted Answer

Let $$u = x^3$ - 1$$

Question 4

Find the value of the improper integral $$\int_{1}^{\infty} \frac{1}{x^3$} \, dx.$$

Accepted Answer

$$\frac{1}{2}$$

Question 5

Which of the following is the correct formula for the arc length of $$y = x^2$ from x = 1$ to x = 2$?

Accepted Answer

$$\int_{1}^{2} \sqrt{1 + (2x)^2$} \, dx$$

Question 6

Determine if the series $$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$$ converges or diverges.

Accepted Answer

It converges by telescoping series.

Question 7

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

Accepted Answer

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

Question 8

Evaluate $$\int 	an(x) \, dx.$$

Accepted Answer

$$-\ln|\cos(x)| + C$$

Question 9

A tank is being filled with water at a rate given by $$R(t) = 3t^2$ liters per minute. How much water is added between t = 0$ and t = 2$ minutes?

Accepted Answer

$$8$$ liters

Question 10

Which of the following is the correct process for the Ratio Test for series convergence?

Accepted Answer

Compute $$L = \lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight|$$; if $$L \u003c 1$$ the series converges.

Question 11

Find the surface area generated by rotating $$y = x^2$ from x = 1$ to x = 2$ about the y$-axis.

Accepted Answer

$$2\pi \int_{1}^{2} x \sqrt{1 + (2x)^2$} \, dx$$

Question 12

Which of the following integrals can be evaluated using trigonometric substitution $$x = 2\sin	heta$$?

Accepted Answer

$$\int \frac{1}{\sqrt{4 - x^2$}} \, dx$$

Question 13

Convert the polar point $$\left(-4, \frac{2\pi}{3}\right) to $$rectangular coordinates.

Accepted Answer

$$\left(2, -2\sqrt{3}\right)$$

Question 14

Which of the following series is absolutely convergent?

Accepted Answer

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

Question 15

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

Accepted Answer

$$1 \u003c x \u003c 5$$

Question 16

Which of the following is the correct parametric equation for a circle centered at $$(2, -3)$$ with radius $$7$$?

Accepted Answer

$$x = 2 + 7\cos(t), \ y = -3 + 7\sin(t)$$

Question 17

Evaluate $$\int \sec(x) \, dx.$$

Accepted Answer

$$\ln|\sec(x) + 	an(x)| + C$$

Question 18

Which of the following is the correct process for the $$p-$$series test?

Accepted Answer

If $$\sum \frac{1}{n^p}$$, the series converges if $$p \u003e 1$$ and diverges if $$p \leq 1.$$

Calculus II Final Review: Integration, Series, Parametric and Polar Equations

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