$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Question 1

Which of the following best describes the concept of a function in business calculus?

Accepted Answer

A function is a rule that assigns each input exactly one output.

Question 2

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

Accepted Answer

$f(4) = 27$

Question 3

A company’s profit $P$ as a function of units sold $x$ is given by $P(x) = 100x - 0.5x^2$. What is the marginal profit when $x = 20$?

Accepted Answer

$P'(20) = 90$

Question 4

Which of the following is the correct definition of the derivative of a function $f(x)$ at a point $x=a$?

Accepted Answer

$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 5

If the revenue function for a product is $R(x) = 500 \ln(x+1)$, what is the marginal revenue when $x = 9$?

Accepted Answer

$R'(9) = 500/10 = 50$

Question 6

Which of the following statements about limits is TRUE?

Accepted Answer

A limit describes the behavior of a function as the input approaches a specific value.

Question 7

A business’s cost function is $C(x) = 200 + 15x$. What is the average cost per unit when 10 units are produced?

Accepted Answer

$C_{avg} = 35$

Question 8

Which rule is used to find the derivative of the product of two functions $f(x)$ and $g(x)$?

Accepted Answer

Product Rule: $[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)$

Question 9

If $f(x) = e^{2x}$, what is $f'(x)$?

Accepted Answer

$f'(x) = 2e^{2x}$

Question 10

A company’s demand function is $p(x) = 100 - 2x$. What is the elasticity of demand when $x = 20$?

Accepted Answer

Elasticity $E = \frac{p}{x} \cdot \frac{dp}{dx} = \frac{60}{20} \cdot (-2) = -6$

Question 11

Which of the following is the correct application of the Chain Rule for $y = \sin(3x)$?

Accepted Answer

$y' = 3\cos(3x)$

Question 12

If the cost function is $C(x) = x^3 - 6x^2 + 9x + 15$, at what value of $x$ is the marginal cost minimized?

Accepted Answer

$x = 2$

Question 13

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

Accepted Answer

$R_{max} = 3600$

Question 14

Which of the following is the correct derivative of $f(x) = \ln(x^2 + 1)$?

Accepted Answer

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

Question 15

A manufacturer’s cost function is $C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when $x = 10$?

Accepted Answer

$C'(10) = 25$

Question 16

Which of the following is the correct formula for the derivative of a quotient $\frac{f(x)}{g(x)}$?

Accepted Answer

$\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Question 17

If $f(x) = x^4 - 8x^2 + 16$, find all critical points.

Accepted Answer

$x = -2, 0, 2$

Question 18

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the maximum profit?

Accepted Answer

$P_{max} = 300$

Question 19

Which of the following is the correct interpretation of the second derivative $f''(x)$ in a business context?

Accepted Answer

It measures the rate of change of the rate of change, indicating concavity and acceleration.

Business Calculus Syllabus and Course Overview

Study Guide - Practice Questions