Business Calculus Test 2 Review: Study Guide (Sections 2.1–2.3, 2.5–2.7, 3.1, 3.2, Factoring)

Test 2 Review: Business Calculus (Math 16)

Overview

This study guide summarizes the key topics and skills covered in Test 2 for Business Calculus, based on Sections 2.1–2.3, 2.5–2.7, 3.1, 3.2, and the factoring supplement. The focus is on foundational calculus concepts, algebraic manipulation, and problem-solving techniques essential for business applications.

Section 2.1–2.3: Functions and Their Properties

Functions: Definitions and Types

• Function: A rule that assigns to each element in a set (domain) exactly one element in another set (range).

• Domain and Range: The set of all possible input values (domain) and output values (range) for a function.

• Types of Functions: Linear, quadratic, polynomial, rational, exponential, and logarithmic functions are commonly used in business calculus.

• Function Notation: $f(x)$ denotes the value of the function $f$ at input $x$.

Example: If $f(x) = 2x + 3$, then $f(4) = 2 \times 4 + 3 = 11$.

Operations with Functions

• Addition/Subtraction: $(f \pm g)(x) = f(x) \pm g(x)$

• Multiplication: $(fg)(x) = f(x)g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = f(g(x)) = (x + 1)^2$.

Graphing and Analyzing Functions

• Intercepts: Points where the graph crosses the axes.

• Asymptotes: Lines that the graph approaches but never touches (common in rational functions).

• Increasing/Decreasing Intervals: Where the function values rise or fall as $x$ increases.

Section 2.5–2.7: Algebraic Techniques and Factoring

Factoring Polynomials

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Factoring Trinomials: Express as $(ax + b)(cx + d)$.

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

• Factoring with Rational and Negative Exponents: Apply exponent rules to factor expressions with fractional or negative exponents.

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

Rational Expressions

• Simplifying: Reduce by factoring numerator and denominator and canceling common factors.

• Operations: Addition, subtraction, multiplication, and division of rational expressions require common denominators and factoring.

Example: $\frac{x^2 - 4}{x^2 - x - 6} = \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} = \frac{x - 2}{x - 3}$ (for $x \neq -2, 3$)

Section 3.1–3.2: Introduction to Limits

Limits: Basic Concepts

• Limit: The value that a function approaches as the input approaches a certain value.

• Notation: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right).

• Infinite Limits: When $f(x)$ increases or decreases without bound as $x$ approaches $a$.

Example: $\lim_{x \to 2} (3x + 1) = 7$

Properties of Limits

• Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$

• Product: $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

Techniques for Evaluating Limits

• Direct Substitution: Substitute $x = a$ into $f(x)$ if $f$ is continuous at $a$.

• Factoring: Factor numerator and denominator to cancel common terms before substitution.

• Rationalization: Multiply by a conjugate to simplify expressions with square roots.

Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{x \to 1} (x + 1) = 2$

Factoring Supplement: Factoring with Rational and Negative Exponents

Exponent Rules

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

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

• Fractional Exponent: $a^{1/n} = \sqrt[n]{a}$

Example: $x^{3/2} - 4x^{1/2} = x^{1/2}(x - 4)$

Test Preparation Strategies

• Review all formulas, definitions, theorems, and methods from lecture notes and textbook.

• Practice factoring, simplifying rational expressions, and evaluating limits without a calculator.

• Complete assigned textbook and worksheet problems for hands-on practice.

• Prepare a one-sided notes sheet with essential formulas and definitions (no worked examples).

Sample Table: Properties of Exponents

Property	Algebraic Form	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^{5}$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^{3}$
Power Rule	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^{6}$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = 1/x^2$
Fractional Exponent	$a^{1/n} = \sqrt[n]{a}$	$x^{1/2} = \sqrt{x}$

Additional Info

• Sections 2.4 and 3.3 are not included in this test.

• Calculator use is restricted to arithmetic and approximation in Part II of the test.

• Factoring with rational and negative exponents is emphasized for algebraic manipulation.