Business Calculus Exam 1: Core Concepts and Formulas Study Guide

Business Terms and Fundamental Formulas

Key Business Functions and Relationships

Understanding the basic business terms and their mathematical relationships is essential in business calculus. These concepts form the foundation for modeling and solving real-world business problems.

• x: Number of units produced or sold.

• p: Price per unit.

• Demand Function: The price required to sell x units, typically written as $p = f(x)$.

• Total Revenue (R): The total income from selling x units. $R = x p$

• Total Cost (C): The sum of variable and fixed costs. $C = \text{Variable Cost} + \text{Fixed Cost}$

• Average Cost per Unit: $\overline{C} = \frac{C}{x}$

• Total Profit (P): The difference between total revenue and total cost. $P = R - C$

• Equilibrium Point: The point where supply equals demand, denoted as $(x_0, p_0)$.

Example: Calculating Profit

• If a company sells 100 units at $5 each, and the total cost is $400, then:

• Total Revenue: $R = 100 \times 5 = 500$

• Total Profit: $P = 500 - 400 = 100$

Properties of Exponents

Exponent Rules

Exponent rules are essential for simplifying algebraic expressions, especially in calculus and business applications.

• Negative Exponent: $x^{-n} = \frac{1}{x^n}$

• Product of Powers: $x^m x^n = x^{m+n}$

• Power of a Power: $(x^n)^m = x^{nm}$

• Quotient of Powers: $\frac{x^n}{x^m} = x^{n-m}$

• Power of a Product: $(xy)^n = x^n y^n$

• Power of a Quotient: $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$

• Zero Exponent: $x^0 = 1$ (for $x \neq 0$)

Example: Simplifying Exponents

• Simplify $x^3 x^{-2}$:

• $x^3 x^{-2} = x^{3 + (-2)} = x^1 = x$

Factorization Techniques

Common Factoring Patterns

Factoring polynomials is a key algebraic skill for solving equations and simplifying expressions.

• Difference of Squares: $x^2 - a^2 = (x - a)(x + a)$

• Perfect Square Trinomial: $x^2 \pm 2ax + a^2 = (x \pm a)^2$

• Sum/Difference of Fourth Powers: $x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) = (x - a)(x + a)(x^2 + a^2)$

• Factoring by Grouping: $ax^2 + adx + bcx + bd = (ax^2 + adx) + (bcx + bd) = a x(x + d) + b c(x + d) = (x + d)(a x + b c)$

Example: Factoring a Trinomial

• Factor $x^2 + 6x + 9$:

• Recognize as a perfect square trinomial: $(x + 3)^2$

Quadratic Equations and Properties

Square Root Property

If $a^2 = b$, then $a = \pm \sqrt{b}$.

Quadratic Formula

The quadratic formula solves equations of the form $ax^2 + bx + c = 0$:

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solving a Quadratic Equation

• Solve $2x^2 - 4x - 6 = 0$:

• $x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm 8}{4}$

• So $x = 3$ or $x = -1$

Operations with Fractions

Adding and Subtracting Fractions

• To add or subtract fractions: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$

Multiplying Fractions

• Multiply numerators and denominators: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

Dividing Fractions

• Multiply by the reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$

Example: Fraction Operations

• $\frac{2}{3} + \frac{1}{4} = \frac{2 \times 4 + 1 \times 3}{3 \times 4} = \frac{8 + 3}{12} = \frac{11}{12}$

Rationalization Techniques

Rationalizing the Denominator or Numerator

• If the denominator is $\sqrt{a}$, multiply numerator and denominator by $\sqrt{a}$.

• If the denominator is $a + \sqrt{b}$ or $a - \sqrt{b}$, multiply by the conjugate $a - \sqrt{b}$ or $a + \sqrt{b}$, respectively.

Example: Rationalizing a Denominator

• $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Equations of Lines

Slope of a Line

• Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is:

• $m = \frac{y_2 - y_1}{x_2 - x_1}$

Point-Slope Form

• The equation of a line with slope $m$ passing through $(x_1, y_1)$:

• $y - y_1 = m(x - x_1)$

Example: Finding the Equation of a Line

• Given points (1,2) and (3,6): $m = \frac{6-2}{3-1} = 2$

• Equation: $y - 2 = 2(x - 1)$

Evaluating and Finding the Domain of Functions

Evaluating Functions

• To evaluate $f(x)$ at $x = a$, substitute $a$ into the function.

Finding the Domain

• The domain of a function is the set of all real numbers for which the function is defined.

• Common restrictions: division by zero, even roots of negative numbers.

Example: Domain of $f(x) = \frac{1}{x-2}$

• Domain: all real $x$ except $x = 2$

Business Applications and Equilibrium

Equilibrium Point

• The equilibrium point occurs where supply equals demand: $\text{Supply} = \text{Demand}$

• Graphically, this is the intersection point of the supply and demand curves.

Example: Finding Equilibrium

• If $\text{Demand}: p = 20 - 2x$ and $\text{Supply}: p = 4 + x$, set equal:

• $20 - 2x = 4 + x \implies 16 = 3x \implies x = \frac{16}{3}$

• Substitute back to find $p$.

Appendix and Practice Recommendations

Recommended Practice

• Factoring Polynomials: Appendix 3 (Examples 1-5, Problems 1-5, Exercises 1-56)

• Finding Real Zeros: Appendix 7 (Examples 1-3, Problems 1-3, Exercises 1-38)

• Simplifying Expressions: Appendix 4, 5, 6 (Examples, Problems, Exercises as listed)

• Rationalizing: Appendix 6 (Examples 7-8, Problems 7-8, Exercises 55-66)

• Evaluating Functions: Section 1.1 (Examples 4,6; Problems 4,6; Exercises 61-80)

• Finding Domain: Section 1.1 (Example 5, Problem 5; Section 1.3 Exercises 47-52)

• Equations of Lines: Section 1.3 (Example 1, Problem 1; Exercises 9-17, 33-40)

• Business Applications: Section 1.3, 1.4, Chapter 1 Review (as listed)

Summary Table: Key Algebraic Properties

Property	Formula	Example
Product of Powers	$x^m x^n = x^{m+n}$	$x^2 x^3 = x^5$
Quotient of Powers	$\frac{x^n}{x^m} = x^{n-m}$	$\frac{x^5}{x^2} = x^3$
Power of a Power	$(x^n)^m = x^{nm}$	$(x^2)^3 = x^6$
Difference of Squares	$x^2 - a^2 = (x - a)(x + a)$	$x^2 - 9 = (x - 3)(x + 3)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 4x + 3 = 0$

Additional info: The study guide references specific textbook appendices and sections for further practice, which are standard in business calculus courses. Students are encouraged to review these for comprehensive preparation.