Business Calculus: Functions, Limits, Continuity, and Applications

Functions and Their Properties

Definition and Identification of Functions

A function is a relation in which each input (from the domain) is assigned to exactly one output (in the range). Functions are fundamental in calculus and business applications, as they model relationships between variables such as cost, revenue, and profit.

• Vertical Line Test: A graph represents a function 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 different inputs share the same output.

• Non-Function Example: If $f(-10) = -3$ and $f(-10) = 2$, then $f$ is not a function because the same input has two different outputs.

Equations Representing Functions

• Some equations, such as $4x^2 - \frac{y}{2} = 4$, can be solved for $y$ in terms of $x$ to determine if they define $y$ as a function of $x$.

• Equations like $2x^4 + 5y^4 = -4$ may not represent a function if, for some $x$, there are multiple $y$ values or no real $y$ values.

Limits and Continuity

Definition of a Limit

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

• $\lim_{x \to a} f(x)$

Properties of Limits and Continuity

• Existence of Limit: If $\lim_{x \to a} f(x)$ exists, it does not necessarily mean $f(a)$ exists or that $f$ is continuous at $a$.

• Continuity: A function $f$ is continuous at $x = a$ if:

  • $f(a)$ is defined

  • $\lim_{x \to a} f(x)$ exists

  • $\lim_{x \to a} f(x) = f(a)$

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

Examples and Applications

• Example: If $\lim_{x \to 8} f(x) = f(8)$, then $f$ is continuous at $x = 8$.

• Example: If $f$ is continuous at $x = -10$, then $\lim_{x \to -10} f(x)$ exists and equals $f(-10)$.

Domain of Functions

Finding the Domain

The domain of a function is the set of all real numbers 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 and Their Graphs

Standard Form and Key Features

A quadratic function has the form $f(x) = ax^2 + bx + c$. Its graph is a parabola.

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

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

• Direction: Opens upward if $a > 0$, downward if $a < 0$.

• y-intercept: $f(0) = c$.

• x-intercepts: Solve $ax^2 + bx + c = 0$.

Example

• For $f(x) = x^2 - 8x + 7$:

  • Vertex: $x = 4$, $y = f(4) = 4^2 - 8 \times 4 + 7 = 16 - 32 + 7 = -9$

  • 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 and Continuity: Practice Problems

• Evaluate $\lim_{x \to -4} H(x)$, $\lim_{x \to -4^-} H(x)$, $\lim_{x \to -4^+} H(x)$, and $H(-4)$.

• Determine if $H$ is continuous at $x = -4$ by checking if the left and right limits and the function value agree.

Limits at Infinity

Rational Functions

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 $-\infty$.

Applications in Business Calculus

Continuous Compounding

For an account with principal $P$, annual rate $r$, and time $t$ (in years), the amount after $t$ years with continuous compounding is:

• $A = Pe^{rt}$

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

Derivatives and Tangent Lines

• The derivative $f'(x)$ gives the slope of the tangent line to $f(x)$ at $x$.

• The equation of the tangent line at $(a, f(a))$ is $y = f'(a)(x - a) + f(a)$.

• Example: For $f(x) = 3x^2 - 9x - 66$ at $(6, -12)$, find $f'(6)$ and use the point-slope form.

Profit, Cost, and Revenue 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: If $C(x) = 77x + 1900$ and $R(x) = 146x$, then $P(x) = 146x - (77x + 1900)$

Average Rate of Change

• The average rate of change of $f$ from $x = a$ to $x = b$ is $\frac{f(b) - f(a)}{b - a}$.

• Example: For $p(t) = 0.6t^2 + 0.4t + 75$, the average rate of change from $t_1$ to $t_2$ is $\frac{p(t_2) - p(t_1)}{t_2 - t_1}$.

Table: Summary of Function Properties

Property	Definition	Example
Function	Each input has one output	$f(x) = x^2$
Domain	Set of all valid inputs	For $f(x) = \frac{1}{x}$, domain is $x \neq 0$
Continuity	No breaks, jumps, or holes	$f(x) = x^2$ is continuous everywhere
Limit	Value as $x$ approaches a point	$\lim_{x \to 0} x^2 = 0$
Derivative	Instantaneous rate of change	$f'(x) = 2x$ for $f(x) = x^2$