$$\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$$

$$R'(10) = 100 - 2 \times 10 = 80$$

$$C'(2) = 12 \times 2^2$ + 4 \times 2 = 48 + 8 = 56$$

Question 1

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

Accepted Answer

$$\lim_{x 	o a} \frac{f(x)-f(a)}{x-a}$$

Question 2

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

Accepted Answer

$$6x + 2$$

Question 3

Suppose $$R(x) = 50\ln(x+2)$$ and $$C(x) = 3x + 2. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 50\ln(x+2) - 3x - 2$$

Question 4

If $$f(x) = x^3$ - 2x^2$ + x$$, what is the equation of the tangent line to $$f(x) at$$ $$x=1$$?

Accepted Answer

Question 5

Given $$R(x) = 50\ln(x+2)$$ and $$C(x) = 3x + 2$$, what is the marginal profit function $$P'(x)$$?

Accepted Answer

$$P'(x) = \frac{50}{x+2} - 3$$

Question 6

What is the marginal profit when $$x = 3$$ for $$R(x) = 50\ln(x+2)$$ and $$C(x) = 3x + 2$$?

Accepted Answer

$$P'(3) = \frac{50}{5} - 3 = 7$$

Question 7

If $$C(x) = 3x + 2$$, what is the average cost function $$AC(x)$$?

Accepted Answer

$$AC(x) = \frac{3x+2}{x}$$

Question 8

What is the marginal average cost function for $$C(x) = 3x + 2$$?

Accepted Answer

$$AC'(x) = \frac{-2}{x^2$}$$

Question 9

A pharmaceutical company models the cost to produce $$x$$ units of a drug as $$C(x) = 5x^2$ + 20x + 100. $$What is the marginal cost function?

Accepted Answer

$$C'(x) = 10x + 20$$

Question 10

A company’s revenue function is $$R(x) = 100x - x^2$. What is the marginal revenue when x = 10$?

Accepted Answer

$$R'(10) = 100 - 2 	imes 10 = 80$$

Question 11

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

Accepted Answer

$$2e^{2x}$$

Question 12

A business models its cost as $$C(x) = 4x^3$ + 2x^2$ + 7. $$What is the marginal cost when $$x = 2$$?

Accepted Answer

$$C'(2) = 12 	imes 2^2$ + 4 	imes 2 = 48 + 8 = 56$$

Question 13

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

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 14

A company’s profit function is $$P(x) = 20x - 5x^2$. What is the marginal profit when x = 4$?

Accepted Answer

Question 15

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 16

A company’s cost function is $$C(x) = 2x^2$ + 5x + 10. $$What is the average cost function?

Accepted Answer

$$AC(x) = \frac{2x^2$ + 5x + 10}{x}$$

Question 17

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

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 18

A company’s revenue function is $$R(x) = 80x - 4x^2$. What is the equation of the tangent line to R(x$$)$$ at x = 5$?

Accepted Answer

Question 19

If $$f(x) = \sin(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\cos(x)$$

Business Calculus Review: Derivatives, Tangent Lines, and Marginal Analysis

Study Guide - Practice Questions