Question 1

A company’s profit function is given by $P(x) = 2x^2 - 12x + 20$, where $x$ is the number of units sold. What is the minimum profit, and at how many units sold does it occur?

Accepted Answer

Minimum profit is $-8$ at $x = 3$

Question 2

Which of the following is the correct derivative of $f(x) = x^3 + 2x^2 - 5x + 7$?

Accepted Answer

$f'(x) = 3x^2 + 4x - 5$

Question 3

A business’s revenue function is $R(x) = 100x - x^2$. What is the marginal revenue when $x = 20$?

Accepted Answer

$MR = 60$

Question 4

Given $f(x) = 3x^2 - 4x + 1$, use the definition of the derivative to find $f'(x)$.

Accepted Answer

$f'(x) = 6x - 4$

Question 5

If $f(x) = x^2$ and $g(x) = 2x + 1$, what is the derivative of $h(x) = f(x)g(x)$ using the product rule?

Accepted Answer

$h'(x) = 2x(2x + 1) + x^2(2)$

Question 6

A manufacturer’s cost function is $C(x) = 5x^3 - 2x^2 + 4x$. Find the rate of change of cost when $x = 2$.

Accepted Answer

$C'(2) = 56$

Question 7

Which of the following statements about the concavity of a function is correct?

Accepted Answer

If $f''(x) > 0$, the function is concave up at $x$

Question 8

Evaluate the definite integral $\int_0^2 (3x^2 - 2x + 1)dx$ to find the total quantity produced over 2 hours if the rate of production is $3x^2 - 2x + 1$ units per hour.

Accepted Answer

Total quantity is $8$

Question 9

A company wants to maximize its profit given by $P(x) = -2x^2 + 40x - 100$. What is the optimal number of units to produce?

Accepted Answer

$x = 10$

Question 10

Which of the following is the correct application of the quotient rule for $f(x) = \frac{x^2}{x+1}$?

Accepted Answer

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

Question 11

If $f(x) = \ln(x)$, what is $\frac{d}{dx}f(x)$?

Accepted Answer

$\frac{1}{x}$

Question 12

A business’s demand function is $p(x) = 50 - 2x$. What is the elasticity of demand at $x = 10$?

Accepted Answer

Elasticity is $-0.4$

Question 13

Which of the following is the correct antiderivative of $f(x) = 4x^3 - 2x$?

Accepted Answer

$F(x) = x^4 - x^2 + C$

Question 14

A company’s cost function is $C(x) = 10x + 500$. What is the average cost per unit when producing 50 units?

Accepted Answer

Average cost is $20$

Question 15

Which of the following is the correct application of the chain rule for $f(x) = (2x + 1)^5$?

Accepted Answer

$f'(x) = 5(2x + 1)^4 \cdot 2$

Question 16

Evaluate $\int_0^1 (6x^2 - 4x + 2)dx$ to find the total change over the interval.

Accepted Answer

Total change is $4/3$

Question 17

Which of the following is the correct formula for the area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$?

Accepted Answer

$\int_a^b [f(x) - g(x)]dx$

Question 18

If $f(x) = e^{2x}$, what is $\frac{d}{dx}f(x)$?

Accepted Answer

$2e^{2x}$

Question 19

A company’s revenue is given by $R(x) = 120x - x^2$. What is the maximum revenue and at what value of $x$ does it occur?

Accepted Answer

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

Business Calculus Study Guide: Functions, Limits, Derivatives, Integration, and Applications

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