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: $$F(x) = \begin{cases} 3, & 0  1 \end{cases}$$, 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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