Question 1

A water tank has a cross-sectional area that varies with height according to $$A(h) = 2h^2$. If the tank is filled to a height of 5$ meters, what is the volume of water in the tank?

Accepted Answer

$$\$$int_0$$^5 2h^2$ \, dh = \frac{250}{3}$$ cubic meters

Question 2

Which of the following integrals represents the volume of a solid obtained by rotating the region bounded by $$y = x^2$ and y = 4$ about the x$-axis using the method of cylindrical shells?

Accepted Answer

$$\$$int_0$$^2 2\pi x (4 - x^2$) \, dx$$

Question 3

A cable weighing $$2 kg/m is $$hanging from a building $$30$$ meters tall. How much work is required to lift the entire cable to the top of the building?

Accepted Answer

$$\$$int_0$$^{30} 2g x \, dx = 2g \cdot \frac{30^2$}{2}$$ Joules

Question 4

Which of the following is the correct formula for the arc length of the curve $$y = \ln x$$ from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$\$$int_1$$^3 \sqrt{1 + \left(\frac{1}{x}\$$right)^2$$} \, dx$$

Question 5

Evaluate the improper integral $$\$$int_1$$^{\infty} \frac{1}{x^2$} \, dx$$ and determine its convergence.

Accepted Answer

The integral converges to $$1$$

Question 6

Which of the following is the correct setup for finding the area enclosed by the curve $$r = 2 + 2\cos 	heta in $$polar coordinates from $$	heta = 0 to$$ $$	heta = 2\pi$$?

Accepted Answer

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

Question 7

Which of the following series converges by the Ratio Test?

Accepted Answer

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

Question 8

A pharmaceutical company uses the Taylor series to approximate $$e^x$$ for small $$x in $$drug concentration calculations. What is the third-degree Taylor polynomial for $$e^x$$ centered at $$x=0$$?

Accepted Answer

$$1 + x + \frac{x^2$}{2} + \frac{x^3$}{6}$$

Question 9

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

Accepted Answer

$$\frac{5}{x-1} - \frac{2}{x+2}$$

Question 10

A spring with force constant $$k is $$stretched from $$x = 0 to$$ $$x = 0.5$$ meters. What is the work done, according to Hooke's Law?

Accepted Answer

Question 11

Which of the following is the correct integral for the surface area of revolution when $$y = x^2$ is revolved about the x$-axis from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 2\pi x^2$ \sqrt{1 + (2x)^2$} \, dx$$

Question 12

Which of the following is the correct binomial series expansion for $$(1 + x$$)^{1/2}$$ up to the x^2$ term?

Accepted Answer

$$1 + \frac{1}{2}x - \frac{1}{8}x^2$

Question 13

Which of the following statements about the convergence of the alternating series $$\sum_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n} is $$correct?

Accepted Answer

The series converges conditionally

Question 14

A curve is parametrized by $$x = t^2$, y$$ = $$t^3. $$What is $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

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

Question 15

Which of the following is the correct formula for the length of the curve $$r = 3\cos 	heta$$ from $$	heta = 0 to$$ $$	heta = \frac{\pi}{2} in $$polar coordinates?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} \sqrt{(3\cos \$$theta)^2$$ + \left(\frac{d}{d	heta}(3\cos 	heta)\$$right)^2$$} \, d	heta$$

Question 16

Which of the following is the correct integral for the work required to pump water out of a cylindrical tank of radius $$2$$ meters and height $$5$$ meters, filled to the top, if the water is to be pumped to a height $$1$$ meter above the tank?

Accepted Answer

$$\$$int_0$$^5 \pi (2)^2$ \cdot 1000 \cdot (6 - y) \, dy$$

Question 17

Which of the following is the correct Maclaurin series for $$\sin x up to $$the $$x^3$$ term?

Accepted Answer

$$x - \frac{x^3$}{6}$$

Question 18

Which of the following is the correct setup for the integral test to determine the convergence of $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$?

Accepted Answer

Evaluate $$\$$int_1$$^{\infty} \frac{1}{x^p} \, dx$$ and check if it converges for $$p \u003e 1$$

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