Question 1

Which of the following integrals can be most effectively solved using the method of trigonometric substitution?

Accepted Answer

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

Question 2

A pharmaceutical company needs to determine the total amount of a drug in the bloodstream over a 12-hour period, given the rate of absorption $$ R(t) = 5e$$^{-0.2t}$$  mg/hr. $$Which integral represents the total amount absorbed?

Accepted Answer

$$ \$$int_0$$^{12} 5e$$^{-0.2t}$$ \, dt $$

Question 3

Evaluate the improper integral $$ \$$int_1$$^{\infty} \frac{1}{x^2$} \, dx $$ and determine if it converges or diverges.

Accepted Answer

It converges to $$1$$

Question 4

Which of the following series is a geometric series, and what is its sum if it converges?

Accepted Answer

$$ \sum_{n=0}^{\infty} 3 \left( \frac{1}{2} \right)^n $$; sum is $$6$$

Question 5

A biologist models the population of a bacteria culture by $$ P(t) = 100e$$^{0.3t}$$ . $$What is the average population between $$ t = 0 $$ and $$ t = 5 $$ hours?

Accepted Answer

$$ \frac{1}{5} \$$int_0$$^5 100e$$^{0.3t}$$ \, dt $$

Question 6

Which of the following statements about the convergence of the series $$ \sum_{n=1}^{\infty} \frac{1}{n^p}  is $$true?

Accepted Answer

It converges if and only if $$ p \u003e 1 $$

Question 7

A water tank has a cross-sectional area $$ A(y) = 4y  at $$height $$ y $$ meters. What is the volume of the tank from $$ y = 0  to$$ $$ y = 3 $$ meters?

Accepted Answer

$$ \$$int_0$$^3 4y \, dy $$

Question 8

Which of the following integrals represents the area between the curves $$ y = x^2$ $$ and $$ y = 2x $$ from $$ x = 0  to$$ $$ x = 2 $$?

Accepted Answer

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

Question 9

Evaluate $$ \int \frac{dx}{x^2$ + 4} .$$

Accepted Answer

$$ \frac{1}{2} \arctan \left( \frac{x}{2} \right) + C $$

Question 10

A patient receives a continuous IV drip at a rate of $$ r(t) = 2 + \sin t  mg/hr $$for $$ 0 \leq t \leq \pi . $$What is the total amount of medication administered?

Accepted Answer

$$ \$$int_0$$^{\pi} (2 + \sin t) \, dt $$

Question 11

Which of the following is the correct setup for finding the length of the curve $$ y = \ln x $$ from $$ x = 1  to$$ $$ x = e $$?

Accepted Answer

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

Question 12

Which of the following is an application of the definite integral in a real-world context?

Accepted Answer

Calculating the total distance traveled by a car given its velocity function over time

Question 13

Which test would you use to determine the convergence of the series $$ \sum_{n=1}^{\infty} \frac{n!}{2^n} $$?

Accepted Answer

Ratio Test

Question 14

A chemical reaction's rate is modeled by $$ r(t) = 10e$$^{-0.5t}$$ $$ mol/L/hr. What is the total amount of product formed from $$ t = 0  to$$ $$ t = 4 $$ hours?

Accepted Answer

$$ \$$int_0$$^4 10e$$^{-0.5t}$$ \, dt $$

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

A patient’s heart rate is modeled by $$ H(t) = 70 + 10\sin(\pi t/12) $$ beats per minute, where $$ t  is in $$hours. What is the average heart rate over a 24-hour period?

Accepted Answer

$$ \frac{1}{24} \$$int_0$$^{24} [70 + 10\sin(\frac{\pi t}{12})] \, dt $$

Question 18

Which of the following is the correct setup for the volume of the solid obtained by rotating the region bounded by $$ y = x^2$ $$ and $$ y = 0 $$ from $$ x = 0  to$$ $$ x = 1 $$ about the x-axis?

Accepted Answer

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

Question 19

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

Accepted Answer

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

Honors Calculus 2: Course Structure, Content, and Study Guide

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