Question 1

Which of the following best describes the domain of a linear function $$f(x) = mx + b$$?

Accepted Answer

The set of all real numbers $$\mathbb{R}$$

Question 2

Given the supply and demand functions for sugar: $$p = S(q) = 1.4q - 0.6$$ and $$p = D(q) = -2q + 3.2$$, what is the equilibrium quantity $$q_0$$?

Accepted Answer

$$q_0 = 1.118$

Question 3

For the cost function $$C(x) = 0.091x + 1.32$$, what is the fixed cost?

Accepted Answer

$$1.32$$

Question 4

If the price per unit is $$p$$ and the number of units sold is $$x$$, which formula correctly represents the revenue function?

Accepted Answer

$$R(x) = px$$

Question 5

Suppose the cost to produce 100 cups of coffee is $$11.02$$ and 400 cups is $$40.12. $$Assuming a linear cost function, what is the marginal cost per cup?

Accepted Answer

$$0.091$$

Question 6

Given the demand function $$p = D(q) = -2q + 3.2$$, what happens to the price $$p as $$the quantity $$q$$ increases?

Accepted Answer

Price decreases

Question 7

If the break-even quantity for a product is $$80$$ and the fixed cost is $$400$$, what is the slope (marginal profit) of the profit function if revenue and cost are linear?

Accepted Answer

$$m = 5$$

Question 8

Which statement best describes the Law of Supply?

Accepted Answer

If the price of an item increases, the supply of the item increases.

Question 9

A company’s profit function is $$P(x) = R(x) - C(x). If$$ $$R(x) = 10x$$ and $$C(x) = 7x + 400$$, what is the break-even quantity?

Accepted Answer

$$x = 400/3$$

Question 10

Which formula gives the slope $$m of $$the least-squares regression line for data points $$(x_1$, y_1$), ..., (x_n, y_n)$$?

Accepted Answer

$$m = \frac{\sum x_i y_i - (\sum x_i)(\sum y_i)/n}{\sum x_i^2$ - (\sum x_i)^2$/n}$$

Question 11

If the correlation coefficient $$r$$ for a set of data is close to $$1$$, what does this indicate?

Accepted Answer

There is a strong positive linear relationship between the variables.

Question 12

Given the cost function $$C(x) = mx + b$$, which parameter represents the marginal cost?

Accepted Answer

$$m$$

Question 13

A company sells a product at $$p = 5$$ per unit. The cost function is $$C(x) = 3x + 200. $$What is the profit when $$x = 100$$ units are sold?

Accepted Answer

Question 14

If the least-squares regression line for consumer credit is $$y = 116.84x + 812.28$$, what is the predicted consumer credit in the year 2030 ($$x = 30$$)?

Accepted Answer

$$y = 6117.5$$

Question 15

Which of the following is true about the equilibrium price in a market?

Accepted Answer

It is the price at which supply equals demand.

Question 16

If the supply function is $$p = S(q) = 1.4q - 0.6$$ and the demand function is $$p = D(q) = -2q + 3.2$$, what is the equilibrium price $$p_0$$?

Accepted Answer

$$p_0 = 0.96$

Question 17

A restaurant’s cost function for coffee is $$C(x) = 0.091x + 1.32. $$What is the cost to produce 1000 cups?

Accepted Answer

$$C(1000) = 91 + 1.32 = 92.32$$

Question 18

If the correlation coefficient $$r$$ for a set of data is $$0$$, what does this mean?

Accepted Answer

There is no linear correlation between the variables.

Business Calculus: Linear Functions, Supply & Demand, Marginal Analysis, and Regression

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