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

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

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