Q: A business calculates the average rate of change in sales over a month using the function $$S(t) = 200 + 10t$$, where $$t is $$the number of weeks. What is the average rate of change from week 2 to week 6?

$$rac{S(6) - S(2)}{6 - 2} = rac{260 - 220}{4} = 10$$

Question 1

Which of the following best describes the purpose of Business Calculus as outlined in the course catalog?

Accepted Answer

To introduce students to calculus with an emphasis on conceptual understanding and applications to business, including differentiation, curve sketching, optimization, integration, and partial derivatives.

Question 2

A student is asked to explain the concept of a tangent line to a curve at a point. Which of the following best describes the instantaneous rate of change at that point?

Accepted Answer

It is the slope of the tangent line to the curve at that specific point, given by $$f'(a)$$ where $$a is $$the point of tangency.

Question 3

Suppose a company’s profit $$P(x) in $$thousands of dollars is modeled 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

$$P'(x) = 100 - 10x$$, so $$P'(4) = 100 - 10(4) = 60$$

Question 4

Which of the following is NOT permitted during exams according to the course policies?

Accepted Answer

Use of a graphing calculator.

Question 5

A business wants to maximize its revenue $$R(x) = 50x - x^2$, where x$ is the number of items sold. What is the number of items that maximizes revenue?

Accepted Answer

$$x = 25$$

Question 6

If $$f(x) = 3x^2$ + 2x - 5$$, what is the derivative $$f'(x)$$ using correct mathematical notation?

Accepted Answer

$$f'(x) = 6x + 2$$

Question 7

A company’s cost function is $$C(x) = 200 + 15x$$, where $$x is $$the number of units produced. What does the constant 15 represent in this context?

Accepted Answer

The marginal cost, or the cost to produce one additional unit.

Question 8

Which of the following best demonstrates the use of correct mathematical notation in written solutions, as required by the course learning outcomes?

Accepted Answer

Writing $$f(x) = x^2$ and f$$'(x) = 2x$$ when finding the derivative.

Question 9

A business calculates the average rate of change in sales over a month using the function $$S(t) = 200 + 10t$$, where $$t is $$the number of weeks. What is the average rate of change from week 2 to week 6?

Accepted Answer

$$rac{S(6) - S(2)}{6 - 2} = rac{260 - 220}{4} = 10$$

Question 10

Which of the following is a real-world application of optimization in business calculus?

Accepted Answer

Finding the maximum profit a company can earn by adjusting production levels.

Question 11

A company’s revenue function is $$R(x) = 80x - 2x^2$. What is the maximum revenue, and at what value of x$ does it occur?

Accepted Answer

Maximum revenue is $$R(20) = 80(20) - 2(400) = 1600 - 800 = 800 at$$ $$x = 20$$

Question 12

If $$f(x) = x^3$ - 4x + 7$$, what is $$f''(x)$$, the second derivative?

Accepted Answer

$$f''(x) = 6x$$

Question 13

A business wants to minimize its cost function $$C(x) = x^2$ - 12x + 36. $$What is the minimum cost and at what value of $$x$$ does it occur?

Accepted Answer

Minimum cost is $$C(6) = 36 - 72 + 36 = 0 at$$ $$x = 6$$

Question 14

Which of the following is an example of using partial derivatives in a business context?

Accepted Answer

Calculating how profit changes with respect to price while holding advertising constant.

Question 15

A company’s demand function is $$D(p) = 500 - 20p$$, where $$p is $$the price per unit. What is the price that will result in zero demand?

Accepted Answer

$$p = 25$$

Question 16

Which of the following best describes the role of academic integrity in the context of Business Calculus assignments?

Accepted Answer

Students must submit their own work and avoid plagiarism, misrepresentation, and unauthorized collaboration.

Question 17

A business’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 $$P(10) = -200 + 400 - 100 = 100 at$$ $$x = 10$$

Question 18

Which of the following is a correct application of the chain rule in differentiation?

Accepted Answer

If $$f(x) = (3x + 2)^4$, then f$$'(x) = 4(3x + $$2)^3$$ 	imes 3$$

Business Calculus Syllabus and Course Overview

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