Business Calculus: Functions, Limits, Continuity, and Applications Study Guide

Functions and Their Properties

Definition and Representation of Functions

A function is a relation that assigns exactly one output value to each input value from its domain. Functions can be represented by equations, graphs, or tables.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Example: If $f(-10) = -3$ and $f(2) = -3$, this is still a function because each input has only one output, even if outputs repeat.

• Non-Function Example: If a single input is assigned two different outputs, the relation is not a function.

Equations and Functions

• Not all equations represent functions. For example, $4x^2 - \frac{y}{2} = 4$ may not pass the vertical line test, depending on how $y$ is expressed in terms of $x$.

• To determine if an equation represents a function, solve for $y$ in terms of $x$ and check if each $x$ yields only one $y$.

Limits and Continuity

Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$.

• Notation: $\lim_{x \to a} f(x)$

• If the left-hand and right-hand limits are equal, the limit exists.

Continuity

A function $f(x)$ is continuous at $x = a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

If any of these conditions fail, the function is not continuous at $x = a$.

One-Sided Limits

• Left-hand limit: $\lim_{x \to a^-} f(x)$

• Right-hand limit: $\lim_{x \to a^+} f(x)$

• If both one-sided limits exist and are equal, the two-sided limit exists.

Examples and Applications

• Example: $\lim_{x \to -4} H(x)$ asks for the value $H(x)$ approaches as $x$ nears $-4$.

• Continuity Check: To determine if $H$ is continuous at $x = -4$, check the three conditions above.

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• Example: For $f(x) = \frac{x-2}{6x-12}$, set $6x-12 \neq 0 \implies x \neq 2$.

Quadratic Functions

Standard Form and Properties

A quadratic function has the form $f(x) = ax^2 + bx + c$.

• Vertex: The vertex is at $x = -\frac{b}{2a}$.

• Line of Symmetry: $x = -\frac{b}{2a}$

• Opens Up/Down: If $a > 0$, opens up; if $a < 0$, opens down.

• y-intercept: Set $x = 0$ to find $f(0)$.

• x-intercepts: Solve $f(x) = 0$.

Example: $f(x) = x^2 - 8x + 7$

• Vertex: $x = -\frac{-8}{2 \cdot 1} = 4$; $f(4) = 16 - 32 + 7 = -9$; vertex at $(4, -9)$

• Line of Symmetry: $x = 4$

• Opens Up (since $a = 1 > 0$)

• y-intercept: $f(0) = 7$

• x-intercepts: Solve $x^2 - 8x + 7 = 0$

Limits at Infinity

Evaluating Limits at Infinity

To find $\lim_{x \to \infty} \frac{5x^2 - 9x + 8}{4x^2 - 8x + 1}$, compare the degrees of the numerator and denominator:

• If degrees are equal, the limit is the ratio of leading coefficients: $\frac{5}{4}$.

• If numerator degree is less, limit is 0; if greater, limit is $\infty$ or $-\infty$.

Exponential Growth and Compound Interest

Continuous Compounding Formula

For an initial amount $P$ compounded continuously at rate $r$ for $t$ years:

• Formula: $A = Pe^{rt}$

• Example: $P = 9900$, $r = 0.09$, $t = 4$

Derivatives and Tangent Lines

Finding the Derivative

The derivative $f'(x)$ gives the instantaneous rate of change of $f(x)$ with respect to $x$.

• For $f(x) = 2x^2 - 9x + 79$, $f'(x) = 4x - 9$

Equation of the Tangent Line

• At point $(a, f(a))$, the tangent line has slope $f'(a)$ and passes through $(a, f(a))$.

• Equation: $y - f(a) = f'(a)(x - a)$

Business Applications: Cost, Revenue, and Profit

Cost, Revenue, and Profit Functions

• Cost Function: $C(x)$ gives the total cost to produce $x$ units.

• Revenue Function: $R(x)$ gives the total revenue from selling $x$ units.

• Profit Function: $P(x) = R(x) - C(x)$

• Average Unit Profit: $\frac{P(x)}{x}$

Example Table: Cost, Revenue, and Profit

Function	Formula	Description
Cost	$C(x) = 77x + 1900$	Total cost for $x$ keyboards
Revenue	$R(x) = 146x$	Total revenue from $x$ keyboards
Profit	$P(x) = R(x) - C(x)$	Total profit from $x$ keyboards
Average Unit Profit	$\frac{P(x)}{x}$	Profit per keyboard

Average Rate of Change

Definition

The average rate of change of a function $f$ from $x = a$ to $x = b$ is:

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Represents the slope of the secant line between $(a, f(a))$ and $(b, f(b))$

Example:

• Given $p(t) = 0.6t^2 + 0.4t + 75$, find the average rate of change from $t = 5$ (2015) to $t = 14$ (2024):

• Compute $p(14)$ and $p(5)$, then use the formula above.