Question 1

Which of the following is the correct derivative of the function $f(x) = 7x^4 - 9\sqrt{x} + 8\sqrt[3]{x} + 10$?

Accepted Answer

$f'(x) = 28x^3 - \dfrac{9}{2\sqrt{x}} + \dfrac{8}{3}x^{-2/3}$

Question 2

A company’s cost function is $C(x) = \sqrt{x}$ hundred dollars. What is the marginal cost when producing 100 items?

Accepted Answer

$MC(100) = \dfrac{1}{2\sqrt{100}} = 0.05$ hundred dollars

Question 3

If $f(x) = 5x^3 + 4x^2 + 0.5x^2 - 10x + 157$, what is $f'(x)$?

Accepted Answer

$f'(x) = 15x^2 + 8x + x - 10$

Question 4

Given the revenue function $R(x) = 6x^2$, what is the marginal revenue when $x = 5$?

Accepted Answer

$MR(5) = 12x = 60$

Question 5

Which rule allows you to find the derivative of $f(x) = ax^n$ as $f'(x) = a n x^{n-1}$?

Accepted Answer

Power Rule

Question 6

A book publisher has a cost function $C(x) = 1.5x^2 - 4x + 7$. What is the instantaneous rate of change of cost when 2 books have been produced?

Accepted Answer

$C'(2) = 2 \cdot 1.5 \cdot 2 - 4 = 6 - 4 = 2$

Question 7

If $f(x) = 10 - x^2$, what is the equation of the tangent line at $x = 2$?

Accepted Answer

$y = -4x + 14$

Question 8

Which of the following best describes the marginal profit $MP(x)$ in business calculus?

Accepted Answer

$MP(x)$ is the derivative of the profit function $P(x)$ with respect to $x$

Question 9

Given $f(x) = x^3$, what is $f''(x)$, the second derivative?

Accepted Answer

$f''(x) = 6x$

Question 10

If $C(x) = x^2 + 2x + 1$, what is the average rate of change of cost from $x = 1$ to $x = 3$?

Accepted Answer

$\dfrac{C(3) - C(1)}{3 - 1} = \dfrac{16 - 4}{2} = 6$

Question 11

Which of the following is the correct derivative of $f(x) = \dfrac{2x^2 + 11x}{2x}$?

Accepted Answer

$f'(x) = 1 + \dfrac{11}{2x}$

Question 12

A company’s profit function is $P(x) = R(x) - C(x)$. If $R(x) = 6x^2$ and $C(x) = x^2 + 2x + 1$, what is $P'(x)$?

Accepted Answer

$P'(x) = 12x - 2x - 2$

Question 13

If $f(x) = 3x^2 + 2x + 1$, what is the value of $f'(4)$?

Accepted Answer

$f'(4) = 3 \cdot 2 \cdot 4 + 2 = 24 + 2 = 26$

Question 14

Which of the following statements is true about the derivative of a constant function?

Accepted Answer

The derivative of a constant function is zero.

Question 15

If $f(x) = 4x^3$, what is the slope of the tangent line at $x = 2$?

Accepted Answer

$f'(2) = 12 \cdot 2^2 = 48$

Question 16

A company’s revenue function is $R(x) = 10x - x^2$. What is the marginal revenue when $x = 3$?

Accepted Answer

$R'(3) = 10 - 2 \cdot 3 = 4$

Question 17

If $f(x) = x^{-1}$, what is $f'(x)$?

Accepted Answer

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

Question 18

Which of the following is the correct application of the sum rule for derivatives?

Accepted Answer

$\dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

Question 19

A company’s cost function is $C(x) = 0.5x^2 + 2x + 100$. What is the marginal cost when $x = 10$?

Accepted Answer

$C'(10) = 0.5 \cdot 2 \cdot 10 + 2 = 10 + 2 = 12$

Formulas for Derivatives: Business Calculus Section 1.3 Study Notes

Study Guide - Practice Questions