Q: A company has a demand function given by $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the total revenue $$R$$ when 10 units are sold?

$$R = (50 - 2 \times 10) \times 10 = 300$$

Q: Given the cost function $$C(x) = 5x + 200$$, what is the average cost per unit when 20 units are produced?

$$\frac{C(x)}{x} = \frac{5 \times 20 + 200}{20} = 15$$

Q: A business has a revenue function $$R(x) = 20x$$ and a cost function $$C(x) = 8x + 120. $$What is the profit when 15 units are sold?

$$P = R - C = 20 \times 15 - (8 \times 15 + 120) = 300 - 240 = 60$$

Q: Which of the following is the correct application of the quadratic formula to solve $$2x^2 - 4x - 6 = 0$?

$$x = \frac{4 \pm \sqrt{(-4)^2$ - 4 \times 2 \times (-6)}}{2 \times 2}$$

Question 1

A company has a demand function given by $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the total revenue $$R$$ when 10 units are sold?

Accepted Answer

$$R = (50 - 2 	imes 10) 	imes 10 = 300$$

Question 2

Given the cost function $$C(x) = 5x + 200$$, what is the average cost per unit when 20 units are produced?

Accepted Answer

$$\frac{C(x)}{x} = \frac{5 	imes 20 + 200}{20} = 15$$

Question 3

Which of the following correctly applies the difference of squares to factor $$x^2 - 9$?

Accepted Answer

$$x^2 - 9 = (x - 3)(x + 3$$)$$

Question 4

A business has a revenue function $$R(x) = 20x$$ and a cost function $$C(x) = 8x + 120. $$What is the profit when 15 units are sold?

Accepted Answer

$$P = R - C = 20 	imes 15 - (8 	imes 15 + 120) = 300 - 240 = 60$$

Question 5

Simplify the expression $$\frac{x^2$ - 4}{x - 2}$$ for $$x 
eq 2.$$

Accepted Answer

$$x + 2$$

Question 6

Find the real zeros of the polynomial $$x^2 - 5x + 6$.

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 7

Which of the following is the correct domain of the function $$f(x) = \frac{1}{x - 4}$$?

Accepted Answer

All real numbers except $$x = 4$$

Question 8

Rationalize the denominator of $$\frac{5}{\sqrt{3}}.$$

Accepted Answer

$$\frac{5\sqrt{3}}{3}$$

Question 9

Given two points $$(2, 5)$$ and $$(6, 13)$$, what is the slope of the line passing through them?

Accepted Answer

$$m = \frac{13 - 5}{6 - 2} = 2$$

Question 10

If the supply and demand functions for a product are $$S(x) = 2x + 10$$ and $$D(x) = 30 - x$$, find the equilibrium quantity $$x_0.$$

Accepted Answer

$$x_0 = 30$$ - $$x_0 = 2x_0 + 10$$ \Rightarrow $$x_0 = 10$

Question 11

Which of the following is the correct application of the quadratic formula to solve $$2x^2 - 4x - 6 = 0$?

Accepted Answer

$$x = \frac{4 \pm \sqrt{(-4)^2$ - 4 	imes 2 	imes (-6)}}{2 	imes 2}$$

Question 12

Simplify the expression $$\frac{2x^2$ - 8}{4x}$$ for $$x 
eq 0.$$

Accepted Answer

$$\frac{x^2$ - 4}{2x}$$

Question 13

A company’s cost function is $$C(x) = 3x^2$ + 12x + 100. $$What is the fixed cost?

Accepted Answer

$$100$$

Question 14

Which of the following is the correct factorization of $$x^2 + 6x + 9$?

Accepted Answer

$$(x + 3)^2$

Question 15

Evaluate the function $$f(x) = 2x^2$ - x + 5 at$$ $$x = -2.$$

Accepted Answer

$$f(-2) = 2(-2)^2$ - (-2) + 5 = 8 + 2 + 5 = 15$$

Question 16

Which of the following is the correct result of multiplying $$\frac{3}{4}$$ and $$\frac{2}{5}$$?

Accepted Answer

$$\frac{6}{20}$$

Question 17

If $$a^2 = 25$, what are the possible values of a$?

Accepted Answer

$$a = \pm 5$$

Question 18

A business sells a product at a price $$p$$ per unit. The demand function is $$p = 100 - 4x. $$What price should the company set to sell 15 units?

Accepted Answer

$$p = 100 - 4 	imes 15 = 40$$

Question 19

Which of the following is the correct point-slope form for a line with slope $$m = -3$$ passing through $$(2, 7)$$?

Accepted Answer

$$y - 7 = -3(x - 2)$$

Business Calculus Exam 1: Core Concepts and Formulas Study Guide

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