Question 1

A company’s revenue $R(x)$ from selling $x$ units of a product is given by $R(x) = 50x$. The cost $C(x)$ to produce $x$ units is $C(x) = 30x + 200$. What is the profit function $P(x)$, and what is the profit when 40 units are sold?

Accepted Answer

$P(x) = 20x - 200$, Profit = $600$

Question 2

Which of the following best describes the domain of the function $f(x) = \dfrac{1}{x-3}$?

Accepted Answer

$x 
eq 3$

Question 3

A biologist models the population of bacteria in a petri dish by $P(t) = 500e^{0.2t}$, where $t$ is in hours. What is the population after 5 hours?

Accepted Answer

$P(5) = 500e^{1} \approx 1,359$

Question 4

Given the quadratic function $f(x) = 2x^2 - 8x + 6$, what are the coordinates of the vertex?

Accepted Answer

$(2, -2)$

Question 5

A car depreciates in value according to the exponential function $V(t) = 20,000e^{-0.15t}$, where $t$ is the number of years since purchase. What is the value of the car after 3 years?

Accepted Answer

$V(3) = 20,000e^{-0.45} \approx 12,740$

Question 6

Which of the following is the inverse of the function $f(x) = 3x + 7$?

Accepted Answer

$f^{-1}(x) = \dfrac{x-7}{3}$

Question 7

A hospital monitors the concentration of a drug in a patient’s bloodstream using $C(t) = 100e^{-0.4t}$, where $t$ is in hours. After how many hours will the concentration drop to 25 mg/L?

Accepted Answer

$t = \dfrac{\ln(0.25)}{-0.4} \approx 3.47$

Question 8

Which of the following is the correct formula for the average cost function $A(x)$ if the total cost function is $C(x) = 5x + 200$?

Accepted Answer

$A(x) = \dfrac{5x + 200}{x}$

Question 9

Given two points $(1, 4)$ and $(5, 12)$, what is the equation of the line passing through them?

Accepted Answer

$y = 2x + 2$

Question 10

A scientist uses the Richter scale, $M = \log_{10}(A/A_0)$, to measure earthquake intensity. If $A = 10,000A_0$, what is the magnitude $M$?

Accepted Answer

$M = 4$

Question 11

Which of the following is the correct composition $(f \circ g)(x)$ if $f(x) = x^2$ and $g(x) = x + 1$?

Accepted Answer

$(f \circ g)(x) = (x + 1)^2$

Question 12

A quadratic equation $x^2 - 6x + 8 = 0$ models the height of a ball thrown upward. What are the times when the ball is at ground level?

Accepted Answer

$x = 2$ and $x = 4$

Question 13

Which of the following is a piecewise function that models a taxi fare: $F(x) = \begin{cases} 3, & 0 < x \leq 1 \ 3 + 2(x-1), & x > 1 \end{cases}$, where $x$ is the number of miles traveled?

Accepted Answer

A fare of $3$ for the first mile, then $2$ per mile after that

Question 14

If $f(x) = \sqrt{x-2}$, what is the domain of $f$?

Accepted Answer

$x \geq 2$

Question 15

A chemical reaction decays according to $N(t) = 100e^{-0.3t}$. How long until only 20 units remain?

Accepted Answer

$t = \dfrac{\ln(0.2)}{-0.3} \approx 5.36$

Question 16

Which of the following is the correct formula for the slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$?

Accepted Answer

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Question 17

A logistic function models the spread of a virus: $P(t) = \dfrac{1000}{1 + 9e^{-0.5t}}$. What is the initial population at $t = 0$?

Accepted Answer

$P(0) = 100$

Question 18

Which of the following is the correct quadratic formula for solving $ax^2 + bx + c = 0$?

Accepted Answer

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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