Q: Which of the following series diverges by the Ratio Test?

$$ \sum_{n=1}^{\infty} n! x^n $$ for any $$ x \neq 0 $$

Question 1

Which of the following integrals can be most efficiently evaluated using the method of integration by parts?

Accepted Answer

$$ \int x e^x \, dx $$

Question 2

A pharmaceutical company needs to determine the total amount of a drug in the bloodstream over time, given by the rate function $$ R(t) = 5e$$^{-0.2t}$$  mg/$$hour. Which integral represents the total amount of drug in the bloodstream from $$ t = 0  to$$ $$ t = 8 $$ hours?

Accepted Answer

$$ \$$int_0$$^8 5e$$^{-0.2t}$$ \, dt $$

Question 3

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

Accepted Answer

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

Question 4

A tank is being filled with water at a rate that varies with time, $$ r(t) = 3t^2$ $$ liters per minute. How much water enters the tank in the first 4 minutes?

Accepted Answer

$$ \$$int_0$$^4 3t^2$ \, dt = 64 $$ liters

Question 5

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

Accepted Answer

$$ \int_a^b \sqrt{1 + [f'(x)]^2} \, dx $$

Question 6

Evaluate the improper integral $$ \$$int_1$$^{\infty} \frac{1}{x^2$} \, dx . $$Which of the following is correct?

Accepted Answer

$$ 1 $$

Question 7

Which of the following is the Maclaurin series for $$ \sin x $$?

Accepted Answer

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

Question 8

A biologist models the population of a bacteria culture with the differential equation $$ \frac{dP}{dt} = kP $$, where $$ k \u003e 0 . $$Which of the following is the general solution?

Accepted Answer

$$ P(t) = Ce$$^{kt}$$ $$

Question 9

Which of the following integrals would be best solved using trigonometric substitution?

Accepted Answer

$$ \int \sqrt{9 - x^2$} \, dx $$

Question 10

A chemical reaction produces a byproduct at a rate $$ r(t) = 4 \cos t $$ grams per hour. What is the total byproduct produced from $$ t = 0  to$$ $$ t = \frac{\pi}{2} $$ hours?

Accepted Answer

$$ \$$int_0$$^{\frac{\pi}{2}} 4 \cos t \, dt = 4 $$ grams

Question 11

Which of the following series diverges by the Ratio Test?

Accepted Answer

$$ \sum_{n=1}^{\infty} n! x^n $$ for any $$ x 
eq 0 $$

Question 12

Which of the following is the correct setup for the volume of the solid generated by revolving the region bounded by $$ y = x^2$ $$ and $$ y = 4 $$ about the $$ x -$$axis?

Accepted Answer

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

Question 13

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

Accepted Answer

$$ \frac{A}{x+1} + \frac{B}{x-2} $$

Question 14

A cable weighing 2 pounds per foot hangs vertically from the top of a 50-foot building. How much work is required to lift the entire cable to the top of the building?

Accepted Answer

$$ \$$int_0$$^{50} 2x \, dx = 2500  ft-lb$$

Question 15

Which of the following is the correct formula for the $$ n th $$Taylor polynomial of $$ f(x) $$ centered at $$ x = a $$?

Accepted Answer

$$ T_n(x) = \sum_{k=0}^n \frac{f$$^{(k)}$$(a)}{k!} (x-a)^k $$

Question 16

Which of the following integrals is improper?

Accepted Answer

$$ \$$int_0$$^1 \frac{1}{\sqrt{x}} \, dx $$

Question 17

A patient receives a medication at a rate $$ r(t) = 10e$$^{-0.5t}$$  mg/$$hour. What is the total amount of medication administered in the first 2 hours?

Accepted Answer

$$ \$$int_0$$^2 10e$$^{-0.5t}$$ \, dt = 10(1 - e$$^{-1}$$)  mg$$

Question 18

Which of the following is the correct setup for the area between the curves $$ y = x^2$ $$ and $$ y = x $$ from $$ x = 0  to$$ $$ x = 1 $$?

Accepted Answer

$$ \$$int_0$$^1 (x - x^2$) \, dx $$

Calculus II (Math 182) Syllabus and Course Structure – University of Nevada, Reno

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