Q: A company’s revenue is given by $$R(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the marginal revenue when x = 20$?

$$R'(20) = 100 - 1 \times 20 = 80$$

Q: A business’s cost function is $$C(x) = 200 + 5x + 0.1x^2$. What is the marginal cost when x = 50$?

$$C'(50) = 5 + 0.2 \times 50 = 15$$

Q: A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the rate of change of the population at t = 10$?

$$P'(10) = 1000 \times 0.05 \times e$$^{0.5}$$ = 50e$$^{0.5}$$

Q: A company’s profit function is $$P(x) = 120x - 0.4x^2$ - 200. $$What is the marginal profit when $$x = 50$$?

$$P'(50) = 120 - 0.8 \times 50 = 80$$

Q: A company’s revenue function is $$R(x) = 80x - x^2$. What is the maximum revenue, and for how many units is it achieved?

Maximum at $$x = 40$$, $$R(40) = 80 \times 40 - 40^2$ = 3200 - 1600 = 1600$$

Q: A company’s cost function is $$C(x) = 500 + 10x + 0.5x^2$. What is the average cost per unit when x = 20$?

$$\frac{C(20)}{20} = \frac{500 + 10 \times 20 + 0.5 \times 400}{20} = \frac{500 + 200 + 200}{20} = \frac{900}{20} = 45$$

Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream over time with the function $$C(t) = 50e$$^{-0.2t}$$, where C$ is in mg/L and t$ is in hours. What is the instantaneous rate of change of concentration at t = 3$ hours?

Accepted Answer

$$-10e^{-0.6} mg/L/hr$$

Question 2

Which of the following is the correct derivative of $$f(x) = x^3$ - 4x^2$ + 7x - 5$$?

Accepted Answer

$$f'(x) = 3x^2$ - 8x + 7$$

Question 3

A company’s revenue is given by $$R(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the marginal revenue when x = 20$?

Accepted Answer

$$R'(20) = 100 - 1 	imes 20 = 80$$

Question 4

Which of the following is the correct factorization of $$x^2 - 9$?

Accepted Answer

$$(x - 3)(x + 3)$$

Question 5

If $$f(x) = \frac{2x}{x^2$ + 1}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2(x^2$ + 1) - 2x \cdot 2x}{(x^2$ + 1)^2$} = \frac{2x^2$ + 2 - 4x^2$}{(x^2$ + 1)^2$} = \frac{2 - 2x^2$}{(x^2$ + 1)^2$}$$

Question 6

A business’s cost function is $$C(x) = 200 + 5x + 0.1x^2$. What is the marginal cost when x = 50$?

Accepted Answer

$$C'(50) = 5 + 0.2 	imes 50 = 15$$

Question 7

Which of the following is the correct derivative of $$f(x) = \sqrt{x}$$?

Accepted Answer

$$f'(x) = \frac{1}{2\sqrt{x}}$$

Question 8

A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the rate of change of the population at t = 10$?

Accepted Answer

$$P'(10) = 1000 	imes 0.05 	imes e$$^{0.5}$$ = 50e$$^{0.5}$$

Question 9

Which of the following is the correct factorization of $$x^3 - 8$?

Accepted Answer

$$(x - 2)(x^2$ + 2x + 4)$$

Question 10

A company’s profit function is $$P(x) = 120x - 0.4x^2$ - 200. $$What is the marginal profit when $$x = 50$$?

Accepted Answer

$$P'(50) = 120 - 0.8 	imes 50 = 80$$

Question 11

If $$f(x) = x$$^{-2}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = -2x$$^{-3}$$

Question 12

A company’s demand function is $$p(x) = 60 - 0.5x. $$What is the elasticity of demand at $$x = 40$$?

Accepted Answer

$$E = \left| \frac{p'(x) \cdot x}{p(x)} \right| = \left| \frac{-0.5 \cdot 40}{60 - 0.5 \cdot 40} \right| = \left| \frac{-20}{40} \right| = 0.5$$

Question 13

Which of the following is the correct derivative of $$f(x) = \ln(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 14

A company’s revenue function is $$R(x) = 80x - x^2$. What is the maximum revenue, and for how many units is it achieved?

Accepted Answer

Maximum at $$x = 40$$, $$R(40) = 80 	imes 40 - 40^2$ = 3200 - 1600 = 1600$$

Question 15

If $$f(x) = x^2$ + 4x + 4$$, which of the following is the correct factorization?

Accepted Answer

$$(x + 2)^2$

Question 16

A company’s cost function is $$C(x) = 500 + 10x + 0.5x^2$. What is the average cost per unit when x = 20$?

Accepted Answer

$$\frac{C(20)}{20} = \frac{500 + 10 	imes 20 + 0.5 	imes 400}{20} = \frac{500 + 200 + 200}{20} = \frac{900}{20} = 45$$

Question 17

Which of the following is the correct derivative of $$f(x) = e$$^{3x}$$?

Accepted Answer

$$f'(x) = 3e$$^{3x}$$

Question 18

A company’s profit function is $$P(x) = 200x - 2x^2$. What is the break-even point (where profit is zero)?

Accepted Answer

$$x = 0$$ and $$x = 100$$

Business Calculus Test 2 Review: Study Guide (Sections 2.1–2.3, 2.5–2.7, 3.1, 3.2, Factoring)

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