Q: A business calculates its average cost per item as $AC(x) = \dfrac{C(x)}{x}$, where $C(x)$ is the total cost for producing $x$ items. If $C(x) = 100 + 5x$, what is $AC(10)$?

$AC(10) = \dfrac{100 + 5 \times 10}{10} = 10$

Q: A company’s revenue function is $R(x) = 120x - x^2$. What is the marginal revenue when $x = 10$?

$R'(10) = 120 - 2 \times 10 = 100$

Question 1

Which of the following best describes the primary objective of Business Calculus as outlined in the course syllabus?

Accepted Answer

To apply calculus concepts to solve real-world business and economic problems.

Question 2

Suppose a company’s profit $P(x)$, in thousands of dollars, is given by $P(x) = 100x - 5x^2$, where $x$ is the number of units sold (in thousands). What is the marginal profit when $x = 4$?

Accepted Answer

$P'(4) = 100 - 10(4) = 60$

Question 3

Which of the following is NOT a skill emphasized in the Business Calculus course according to the syllabus?

Accepted Answer

Proving advanced theorems in abstract algebra.

Question 4

A manufacturer’s cost function is $C(x) = 500 + 20x$, where $x$ is the number of items produced. What does the derivative $C'(x)$ represent in a business context?

Accepted Answer

The marginal cost, or the cost of producing one additional item.

Question 5

If the revenue function for a product is $R(x) = 50x - x^2$, what is the maximum revenue, and at what value of $x$ does it occur?

Accepted Answer

Maximum revenue is $625$ at $x = 25$

Question 6

Which of the following best describes the geometric significance of the first derivative of a function $f(x)$?

Accepted Answer

It represents the slope of the tangent line to the graph of $f(x)$ at a point.

Question 7

A business wants to minimize its cost function $C(x) = x^2 - 10x + 25$. What is the minimum cost and at what value of $x$ does it occur?

Question 8

Which of the following is a correct application of the chain rule for the function $f(x) = (3x^2 + 2x)^5$?

Accepted Answer

$f'(x) = 5(3x^2 + 2x)^4 \cdot (6x + 2)$

Question 9

A company’s demand function is $p(x) = 200 - 4x$, where $p$ is the price per unit and $x$ is the quantity sold. What is the revenue function $R(x)$?

Accepted Answer

$R(x) = 200x - 4x^2$

Question 10

Which of the following statements about the second derivative $f''(x)$ is TRUE?

Accepted Answer

It measures the rate of change of the slope of $f(x)$.

Question 11

A business calculates its average cost per item as $AC(x) = \dfrac{C(x)}{x}$, where $C(x)$ is the total cost for producing $x$ items. If $C(x) = 100 + 5x$, what is $AC(10)$?

Accepted Answer

$AC(10) = \dfrac{100 + 5 	imes 10}{10} = 10$

Question 12

Which of the following is an example of an applied problem that can be solved using calculus in a business context?

Accepted Answer

Finding the maximum profit for a company by analyzing its profit function.

Question 13

If $f(x) = x^3 - 6x^2 + 9x$, at what value of $x$ does $f(x)$ have a local minimum?

Accepted Answer

$x = 3$

Question 14

A company’s revenue function is $R(x) = 120x - x^2$. What is the marginal revenue when $x = 10$?

Accepted Answer

$R'(10) = 120 - 2 	imes 10 = 100$

Question 15

Which of the following is the correct definition of a function in the context of business calculus?

Accepted Answer

A rule that assigns each input exactly one output.

Question 16

If the cost function is $C(x) = 2x^2 + 3x + 100$, what is the marginal cost when $x = 5$?

Accepted Answer

$C'(5) = 4 	imes 5 + 3 = 23$

Question 17

Which of the following is a real-world example of using the second derivative in business?

Accepted Answer

Determining whether a company’s profit is increasing at an increasing or decreasing rate.

Question 18

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the maximum profit and at what value of $x$ does it occur?

Accepted Answer

Maximum profit is $-2(10)^2 + 40 	imes 10 - 100 = 100$ at $x = 10$

Business Calculus Syllabus and Course Overview

Study Guide - Practice Questions

Study Guide - Flashcards