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

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

Question 1

Suppose a company’s revenue $R(x)$ in thousands of dollars is given by $R(x) = 50x - 0.5x^2$, where $x$ is the number of units sold (in thousands). What is the marginal revenue when $x = 20$?

Accepted Answer

$R'(20) = 30$

Question 2

Which of the following best describes the meaning of the derivative $f'(a)$ in a business context?

Accepted Answer

The instantaneous rate of change of $f$ with respect to $x$ at $x = a$

Question 3

A manufacturer’s cost function is $C(x) = 1000 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

Accepted Answer

$C'(50) = 30$

Question 4

If $f(x) = 3x^2 - 4x + 7$, what is the equation of the tangent line to $f$ at $x = 2$?

Accepted Answer

$y = 8x - 5$

Question 5

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. For what value of $x$ is the profit maximized?

Accepted Answer

$x = 10$

Question 6

Given $f(x) = \sqrt{x}$, what is $f'(x)$?

Accepted Answer

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

Question 7

If the demand function for a product is $p = 100 - 2x$, where $p$ is the price and $x$ is the quantity sold, what is the revenue function $R(x)$?

Accepted Answer

$R(x) = 100x - 2x^2$

Question 8

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

Accepted Answer

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

Question 9

A company’s cost function is $C(x) = 500 + 15x$. What is the average cost per unit when producing 50 units?

Accepted Answer

$C_{avg} = 25$

Question 10

If $f(x) = x^3 - 6x^2 + 9x + 5$, at what value of $x$ does $f(x)$ have a local minimum?

Accepted Answer

$x = 3$

Question 11

Suppose the marginal cost for producing $x$ units is $MC(x) = 4x + 10$. What is the total cost function $C(x)$ if the fixed cost is $200$?

Accepted Answer

$C(x) = 2x^2 + 10x + 200$

Question 12

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

Accepted Answer

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

Question 13

A company’s revenue function is $R(x) = 120x - x^2$. What is the maximum revenue, and at what quantity $x$ does it occur?

Accepted Answer

Maximum revenue is $3600$ at $x = 60$

Question 14

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

Accepted Answer

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

Question 15

A business finds that the cost to produce $x$ units is $C(x) = 2000 + 30x$. If each unit sells for $p = 50$, what is the break-even quantity?

Accepted Answer

$x = 100$

Business Calculus Exam Coverage Overview

Study Guide - Practice Questions

Study Guide - Flashcards