Fundamental Concepts in Business Calculus: Functions, Difference Quotients, and Applications

Functions and Their Properties

Definition and Evaluation of Functions

A function is a rule that assigns to each input exactly one output. In business calculus, functions are used to model relationships such as cost, revenue, and profit.

• Notation: If $f(x)$ is a function, then $f(a)$ represents the value of the function when $x = a$.

• Example: If $f(x) = x^2 - 6$, then $f(4) = 4^2 - 6 = 10$.

Difference Quotient

The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval. It is the basis for the derivative.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• This expression measures the change in the function's value as $x$ increases by $h$.

• Example: For $f(x) = x^2 - 6$:

  • $f(x+h) = (x+h)^2 - 6 = x^2 + 2xh + h^2 - 6$

  • $f(x+h) - f(x) = (x^2 + 2xh + h^2 - 6) - (x^2 - 6) = 2xh + h^2$

  • Difference quotient: $\frac{2xh + h^2}{h} = 2x + h$ (for $h \neq 0$)

Linear Functions and Their Graphs

Linear Function Properties

A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope ($m$): Measures the rate of change of the function. Calculated as $m = \frac{\Delta y}{\Delta x}$.

• Y-intercept ($b$): The value of $f(x)$ when $x = 0$.

• X-intercept: The value of $x$ when $f(x) = 0$.

• Example: For $f(x) = 5x - 5$:

  • Slope: $5$

  • Y-intercept: $-5$

  • X-intercept: Solve $5x - 5 = 0 \implies x = 1$

Evaluating and Simplifying Expressions

• Function evaluation: Substitute the given value into the function.

• Example: For $f(x) = 5x - 5$:

  • $f(x+h) = 5(x+h) - 5 = 5x + 5h - 5$

  • $f(x+h) - f(x) = (5x + 5h - 5) - (5x - 5) = 5h$

  • Difference quotient: $\frac{f(x+h) - f(x)}{h} = \frac{5h}{h} = 5$ (for $h \neq 0$)

Applications: Volume of a Box

Expressing Volume as a Function

In business calculus, functions can model physical quantities such as volume. For a box with variable dimensions, the volume can be expressed as a function of one variable.

• Volume formula: $V(x) = \text{length} \times \text{width} \times \text{height}$

• Example: If the box's dimensions depend on $x$, express $V(x)$ in terms of $x$.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined, often determined by physical or practical constraints.

• Example: If $x$ represents a length, then $x > 0$ and may be further restricted by the box's construction.

Tabular and Graphical Representation

Functions can be represented in tables and graphs to visualize their behavior and identify key features such as maximum or minimum values.

x	V(x)
1	V(1)
2	V(2)
3	V(3)

Additional info: The actual expressions for V(1), V(2), and V(3) depend on the specific formula for V(x) given in the problem, which is not fully visible in the image.

• Plotting these points on a graph helps to visualize how the volume changes as $x$ varies.

Summary Table: Key Concepts

Concept	Definition	Example
Function	Rule assigning each input one output	$f(x) = x^2 - 6$
Difference Quotient	Average rate of change	$\frac{f(x+h) - f(x)}{h}$
Linear Function	Function of the form $mx + b$	$f(x) = 5x - 5$
Domain	Set of allowable input values	$x > 0$
Volume Function	Product of dimensions	$V(x) = \text{length} \times \text{width} \times \text{height}$