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?

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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