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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