Q: A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 100x - 2x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 10$?

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

Q: A manufacturer finds that the cost $$C(x) to $$produce $$x$$ units is $$C(x) = 500 + 20x + x^2$. What is the average rate of change of cost as production increases from x = 10$ to x = 15$?

$$\frac{C(15) - C(10)}{15 - 10} = \frac{500 + 20 \times 15 + 15^2$ - (500 + 20 \times 10 + 10^2$)}{5} = \frac{500 + 300 + 225 - (500 + 200 + 100)}{5} = \frac{1025 - 800}{5} = 45$$

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

$$R(60) = 120 \times 60 - 60^2$ = 7200 - 3600 = 3600$$

Q: Which of the following is the correct formula for the area under the curve $$f(x) = 3x$$ from $$x = 0 to$$ $$x = 4$$?

$$\int_{0}^{4} 3x \, dx = \left. \frac{3}{2}x^2$ \right|_{0}^{4} = \frac{3}{2} \times 16 = 24$$

Question 1

Which of the following best describes the primary purpose of calculus in business and economics applications?

Accepted Answer

To analyze rates of change and optimize outcomes using mathematical models.

Question 2

Given the function $$f(x) = 2x^3$ - 5x^2$ + 4x - 7$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x^2$ - 10x + 4$$

Question 3

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 100x - 2x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 10$?

Accepted Answer

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

Question 4

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

If $$f(x) = u(x) \cdot v(x)$$, then $$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$$

Question 5

Suppose the demand for a product is given by $$p = 50 - 2q$$, where $$p is $$price and $$q is $$quantity. What is the slope of the demand curve?

Accepted Answer

$$-2$$

Question 6

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 7

A manufacturer finds that the cost $$C(x) to $$produce $$x$$ units is $$C(x) = 500 + 20x + x^2$. What is the average rate of change of cost as production increases from x = 10$ to x = 15$?

Accepted Answer

$$\frac{C(15) - C(10)}{15 - 10} = \frac{500 + 20 	imes 15 + 15^2$ - (500 + 20 	imes 10 + 10^2$)}{5} = \frac{500 + 300 + 225 - (500 + 200 + 100)}{5} = \frac{1025 - 800}{5} = 45$$

Question 8

Which of the following statements about the limit $$\lim_{x 	o 0} \frac{\sin x}{x} is $$true?

Accepted Answer

The limit is $$1.$$

Question 9

A business wants to maximize its revenue, given by $$R(x) = x(100 - x)$$, where $$x is $$the number of units sold. What value of $$x$$ maximizes revenue?

Accepted Answer

$$x = 50$$

Question 10

Which of the following is the correct formula for the derivative of $$f(x) = e$$^{3x}$$?

Accepted Answer

$$f'(x) = 3e$$^{3x}$$

Question 11

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the derivative of h(x) = f(x$$) \cdot g(x)$$ using the product rule?

Accepted Answer

$$h'(x) = 2x \sqrt{x} + x^2$ \cdot \frac{1}{2\sqrt{x}}$$

Question 12

A company’s cost function is $$C(x) = 1000 + 5x. $$What is the marginal cost?

Accepted Answer

$$5$$

Question 13

Which of the following is the correct application of the chain rule for differentiation?

Accepted Answer

If $$f(x) = g(h(x))$$, then $$f'(x) = g'(h(x)) \cdot h'(x)$$

Question 14

Suppose $$f(x) = x^3$ - 6x^2$ + 9x + 15. $$Find all critical points of $$f(x).$$

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 15

Which of the following best describes the use of the second derivative in business calculus?

Accepted Answer

It determines the concavity of a function and helps identify relative maxima and minima.

Question 16

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

Accepted Answer

$$R(60) = 120 	imes 60 - 60^2$ = 7200 - 3600 = 3600$$

Question 17

Which of the following is the correct formula for the area under the curve $$f(x) = 3x$$ from $$x = 0 to$$ $$x = 4$$?

Accepted Answer

$$\int_{0}^{4} 3x \, dx = \left. \frac{3}{2}x^2$ ight|_{0}^{4} = \frac{3}{2} 	imes 16 = 24$$

Question 18

If $$f(x) = x^2$ + 4x + 3$$, what is the indefinite integral $$\int f(x) \, dx$$?

Accepted Answer

$$\int f(x) \, dx = \frac{1}{3}x^3$ + 2x^2$ + 3x + C$$

Introductory Calculus for Business & Social Sciences – Syllabus Study Guide

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