Q: A tank in the shape of a cylinder with radius 3 meters and height 5 meters is filled with water. How much work is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 \ \mathrm{kg/m^3$}$$ and $$g = 9.8 \ \mathrm{m/s^2$}.)$$

$$W = 1000 \times 9.8 \times \pi \times 3^2$ \times 5^2$

Q: Find the area enclosed by one loop of the polar curve $$r = 2 \sin \theta.$$

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

Question 1

Evaluate the definite integral representing the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$. Which of the following is the correct value?

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx = \frac{4}{3}$$

Question 2

Which of the following improper integrals converges?

Accepted Answer

$$\$$int_1$$^\infty \frac{1}{x^2$} \, dx$$

Question 3

A tank in the shape of a cylinder with radius 3 meters and height 5 meters is filled with water. How much work is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 \ \mathrm{kg/m^3$}$$ and $$g = 9.8 \ \mathrm{m/s^2$}.)$$

Accepted Answer

$$W = 1000 	imes 9.8 	imes \pi 	imes 3^2$ 	imes 5^2$

Question 4

Which of the following series converges?

Accepted Answer

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

Question 5

Find the sum of the infinite geometric series $$\sum_{n=0}^\infty 5 \left(\frac{1}{3}\right)^n.$$

Accepted Answer

$$\frac{15}{2}$$

Question 6

Determine the radius of convergence for the power series $$\sum_{n=1}^\infty \frac{x^n}{n}.$$

Accepted Answer

$$R = 1$$

Question 7

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

Accepted Answer

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

Question 8

Given the parametric equations $$x = t^2$, y$$ = $$t^3$$, find $$\frac{dy}{dx} in $$terms of $$t.$$

Accepted Answer

$$\frac{dy}{dx} = \frac{3t^2$}{2t}$$

Question 9

Find the area enclosed by one loop of the polar curve $$r = 2 \sin 	heta.$$

Accepted Answer

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

Question 10

Evaluate the limit of the sequence $$a_n = \frac{2n+1}{n-1} as$$ $$n 	o \infty.$$

Accepted Answer

$$2$$

Question 11

Which test is most appropriate to determine the convergence of the series $$\sum_{n=1}^\infty \frac{n!}{2^n}$$?

Accepted Answer

Ratio Test

Question 12

A medication is administered intravenously at a rate proportional to $$e^{-0.2t} mg/$$hour, where $$t is in $$hours. What is the total amount administered from $$t=0 to$$ $$t=5$$ hours?

Accepted Answer

$$\$$int_0$$^5 e$$^{-0.2t}$$ \, dt = \frac{1}{0.2}(1 - e$$^{-1}$$)$$

Question 13

Which of the following is the Taylor polynomial of degree 2 for $$\ln(1+x)$$ centered at $$x=0$$?

Accepted Answer

$$x - \frac{x^2$}{2}$$

Question 14

A solid is formed by rotating the region bounded by $$y = \sqrt{x}$$, $$x = 0$$, and $$y = 0$$ about the $$x-$$axis from $$x=0 to$$ $$x=4. $$Which integral gives its volume?

Accepted Answer

$$V = \pi \$$int_0$$^4 (\sqrt{x})^2$ \, dx$$

Question 15

Which of the following series diverges by the Alternating Series Test?

Accepted Answer

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

Question 16

Find the interval of convergence for the power series $$\sum_{n=1}^\infty \frac{(x-2)^n}{n}.$$

Accepted Answer

$$1 \u003c x \u003c 3$$

Question 17

Given the polar curve $$r = 1 + \cos 	heta$$, what is the area enclosed by one loop?

Accepted Answer

$$\frac{1}{2} \$$int_0$$^{2\pi} (1 + \cos \$$theta)^2$$ \, d	heta$$

Question 18

A spring with spring constant $$k = 200 \ \mathrm{N/m} is $$stretched from its natural length by 0.5 meters. How much work is done?

Accepted Answer

$$W = \frac{1}{2} 	imes 200 	imes (0.5)^2$

Question 19

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

Accepted Answer

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

Question 20

Suppose a population of bacteria grows at a rate proportional to its current size, and doubles every 3 hours. If the initial population is 1000, what is the population after 9 hours?

Accepted Answer

$$1000 	imes 2^3$ = 8000$$

Calculus II Final Exam Review: Key Topics and Concepts

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