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?

Accepted Answer

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

Guida di studio - Domande di pratica

Study guide - Flashcard