Q: Using the CAPM formula $$E(R_i) = R_f + \beta_i[E(R_m) - R_f]$$, if $$R_f = 2\%$$, $$\beta_i = 1.5$$, and $$E(R_m) = 8\%$$, what is the expected return $$E(R_i)$$?

$$E(R_i) = 2\% + 1.5 \times (8\% - 2\%) = 11\%$$

Question 1

Which of the following sets of ordered pairs represents a function?

Accepted Answer

[(2, 5), (3, 7), (4, 9)]

Question 2

Given the linear function $$y = 7x - 15$$, what is the slope and what does it represent in a business context?

Accepted Answer

Slope is $$7$$; it represents the rate of change of $$y$$ with respect to $$x.$$

Question 3

A company’s cost function is $$C(q) = 12q + 400. $$What is the fixed cost?

Accepted Answer

$$400$$

Question 4

If the points $$(3, 8)$$ and $$(7, 20)$$ lie on a line, what is the equation of the line in the form $$y = mx + b$$?

Accepted Answer

$$y = 3x - 1$$

Question 5

Suppose the demand function is $$Q_d = 120 - 4p$$ and the supply function is $$Q_s = 30 + 2p. $$What is the equilibrium price?

Accepted Answer

$$p = 12$$

Question 6

A business sells a product for $$p = 25$$ and sells $$q = 80$$ units. What is the revenue?

Accepted Answer

$$R = 2,000$$

Question 7

If the cost function is $$C(q) = 10q + 250$$ and the revenue function is $$R(q) = 20q$$, what is the break-even quantity?

Accepted Answer

$$q = 20$$

Question 8

Which of the following best describes a proportional function?

Accepted Answer

$$y = mx$$ with $$b = 0$$

Question 9

Given two lines: $$y = 4x + 2$$ and $$y = 4x - 5$$, what is their relationship?

Accepted Answer

They are parallel.

Question 10

If the demand function is $$Q_d = 80 - 2p$$, what happens to quantity demanded as price increases?

Accepted Answer

Quantity demanded decreases.

Question 11

A linear regression model predicts wage as $$w = -0.90 + 0.54e$$, where $$e is $$years of education. What does the slope $$0.54$$ represent?

Accepted Answer

The increase in predicted wage for each additional year of education.

Question 12

If a company’s profit function is $$\pi(q) = 18q - (8q + 400)$$, what is the profit when $$q = 50$$?

Accepted Answer

$$\pi = 400$$

Question 13

Which formula is used to calculate the slope between two points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

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

Question 14

If a supply function is $$Q_s = 15 + 5p$$, what is the relationship between price and quantity supplied?

Accepted Answer

Positive relationship: as price increases, quantity supplied increases.

Question 15

Given $$y = mx + b$$, what does $$b$$ represent?

Accepted Answer

The value of $$y$$ when $$x = 0.$$

Question 16

A company’s cost function is $$C(q) = 6q + 150. If $$the marginal cost increases to $$8$$, what is the new cost function?

Accepted Answer

$$C(q) = 8q + 150$$

Question 17

If a line has a slope of $$0$$, which of the following best describes the function?

Accepted Answer

Constant function.

Question 18

Using the CAPM formula $$E(R_i) = R_f + \beta_i[E(R_m) - R_f]$$, if $$R_f = 2\%$$, $$\beta_i = 1.5$$, and $$E(R_m) = 8\%$$, what is the expected return $$E(R_i)$$?

Accepted Answer

$$E(R_i) = 2\% + 1.5 	imes (8\% - 2\%) = 11\%$$

Question 19

If a company’s revenue function is $$R(q) = 30q$$ and its cost function is $$C(q) = 18q + 360$$, what is the profit function $$\pi(q)$$?

Accepted Answer

$$\pi(q) = 30q - 18q - 360$$

Question 20

A company’s break-even quantity is given by $$q^{be}$$ = \frac{b}{p - m}$$. If fixed cost b = 500$, price p = 25$, and marginal cost m = 10$, what is the break-even quantity?

Accepted Answer

$$q^{be} = 33.33$

Business Calculus Study Guide: Linear Functions and Applications

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