Question 1

Which of the following best describes the concept of a 'limit' in calculus, as introduced in Lesson 2?

Accepted Answer

The value a function approaches as the input approaches a certain point.

Question 2

A business wants to model its profit $$P(x) as a $$function of units sold $$x. If$$ $$P(x) = 5x^2$ - 20x + 100$$, what is the marginal profit when $$x = 10$$ units?

Accepted Answer

$$P'(x) = 10x - 20$$, so at $$x = 10$$, $$P'(10) = 80$$

Question 3

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

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

Question 4

A company’s revenue function is $$R(x) = 100x - x^2$. What is the elasticity of demand at x = 20$?

Accepted Answer

$$E(x) = \frac{R'(x) \cdot x}{R(x)}$$, so $$E(20) = \frac{(100 - 2 \cdot 20) \cdot 20}{100 \cdot 20 - 20^2$} = \frac{60 \cdot 20}{1600} = 0.75$$

Question 5

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

Accepted Answer

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

Question 6

A manufacturer’s cost function is $$C(x) = 50 + 10x + 0.5x^2$. What is the average cost per unit when producing 20 units?

Accepted Answer

$$\frac{C(20)}{20} = \frac{50 + 10 \cdot 20 + 0.5 \cdot 400}{20} = \frac{50 + 200 + 200}{20} = \frac{450}{20} = 22.5$$

Question 7

Which of the following is the correct integral for finding the area under the curve $$y = 3x^2$ from x = 1$ to x = 4$?

Accepted Answer

$$\int_{1}^{4} 3x^2$ \, dx$$

Question 8

If $$f(x) = x^3$ - 6x^2$ + 9x$$, find the critical points and determine if they are maxima or minima.

Accepted Answer

Critical points at $$x = 1 ($$max) and $$x = 3 ($$min)

Question 9

A company’s demand function is $$p(x) = 100 - 2x. $$What is the revenue-maximizing quantity?

Accepted Answer

Set $$R(x) = x \cdot p(x)$$, $$R(x) = 100x - 2x^2$, maximize: R$$'(x) = 100 - 4x = 0 \Rightarrow x = 25$$

Question 10

Which of the following is the correct application of the chain rule for $$f(x) = \sin(3x)$$?

Accepted Answer

$$f'(x) = 3 \cos(3x)$$

Question 11

If the marginal cost $$MC(x) of $$producing $$x$$ units is $$MC(x) = 4x + 10$$, what is the total cost function $$C(x)$$, assuming fixed costs are $$100$$?

Accepted Answer

$$C(x) = 2x^2$ + 10x + 100$$

Question 12

Which of the following best describes the concept of elasticity of demand in economics?

Accepted Answer

It measures the responsiveness of quantity demanded to changes in price.

Question 13

Evaluate the definite integral $$\int_{0}^{2} (4x + 1) \, dx.$$

Accepted Answer

$$10$$

Question 14

A business wants to minimize its cost function $$C(x) = x^2$ - 8x + 20. $$What is the minimum cost and at what value of $$x$$ does it occur?

Accepted Answer

Minimum at $$x = 4$$, $$C(4) = 4^2$ - 8 \cdot 4 + 20 = 4$$

Question 15

Which of the following is the correct formula for the derivative of $$f(x) = \ln(x)$$?

Accepted Answer

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

Question 16

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

Accepted Answer

$$f'(x)g(x) + f(x)g'(x) = 2x(2x + 1) + x^2$(2) = 4x^2$ + 2x + 2x^2$ = 6x^2$ + 2x$$

Question 17

A company’s profit function is $$P(x) = -x^2$ + 12x - 20. $$What is the maximum profit and at what value of $$x$$ does it occur?

Accepted Answer

Maximum at $$x = 6$$, $$P(6) = -(6^2$) + 12 \cdot 6 - 20 = 16$$

Question 18

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

Accepted Answer

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

MATH 209 - Fundamental Mathematics II (Business Calculus) Syllabus and Study Guide

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