Increasing and Decreasing Functions: Business Calculus Study Notes

Increasing and Decreasing Functions

Definitions and Basic Properties

Understanding whether a function is increasing or decreasing on an interval is fundamental in calculus, especially for analyzing business models and optimization problems.

• Increasing Function: A function $f$ is increasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) < f(x_2)$.

• Decreasing Function: A function $f$ is decreasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) > f(x_2)$.

Examples of Increasing and Decreasing Functions

Different types of functions exhibit increasing or decreasing behavior depending on their form and parameters.

Type	Increasing	Decreasing
Exponential	$y = a^x$, $a > 1$	$y = a^x$, $0 < a < 1$
Linear	$y = mx + b$, $m > 0$	$y = mx + b$, $m < 0$
Logarithmic	$y = ext{log}_a x$, $a > 1$	$y = ext{log}_a x$, $0 < a < 1$

Test for Intervals Where $f(x)$ is Increasing or Decreasing

Using the Derivative

The derivative of a function provides a powerful tool for determining where a function is increasing or decreasing.

• If $f'(x) > 0$ for each $x$ in an interval, then $f$ is increasing on that interval.

• If $f'(x) < 0$ for each $x$ in an interval, then $f$ is decreasing on that interval.

• If $f'(x) = 0$ for each $x$ in an interval, then $f$ is constant on that interval.

The sign of the derivative at a point indicates the slope of the tangent line to the graph at that point:

• Positive slope ($m > 0$): Function is increasing.

• Negative slope ($m < 0$): Function is decreasing.

• Zero slope ($m = 0$): Function is constant or at a critical point.

Critical Numbers

Definition and Importance

Critical numbers are essential for identifying where a function changes from increasing to decreasing or vice versa.

• Critical Number: A value $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist.

• Critical Point: The point $(c, f(c))$ corresponding to a critical number.

Applying the Test for Increasing and Decreasing

To determine intervals of increase or decrease:

1. Find all critical numbers by solving $f'(x) = 0$ and identifying where $f'(x)$ is undefined.

2. Divide the domain into intervals using these critical numbers.

3. Choose test points in each interval and evaluate the sign of $f'(x)$ at those points.

4. If $f'(x) > 0$ at a test point, $f$ is increasing on that interval; if $f'(x) < 0$, $f$ is decreasing.

Worked Examples and Applications

Example 1: Graphical Analysis

Given a graph, identify intervals where the function is increasing or decreasing by observing the slope.

• Increasing: Where the graph rises as $x$ increases.

• Decreasing: Where the graph falls as $x$ increases.

Example 2: Polynomial Function

Given $y = 1 - 0.3x - 0.3x^2$:

• Find $f'(x) = -0.3 - 0.6x$.

• Set $f'(x) = 0$ to find critical numbers: $-0.3 - 0.6x = 0 \Rightarrow x = -0.5$.

• Test intervals: For $x < -0.5$, $f'(x) > 0$ (increasing); for $x > -0.5$, $f'(x) < 0$ (decreasing).

Intervals:

• Increasing: $(-\infty, -0.5)$

• Decreasing: $(-0.5, \infty)$

Example 3: Rational Function

Given $y = \frac{6x}{x^2 + 1}$:

• Find $f'(x)$ using the quotient rule: $f'(x) = \frac{6(x^2 + 1) - 12x^2}{(x^2 + 1)^2}$

• Simplify and solve $f'(x) = 0$ for critical numbers.

• Test intervals to determine increasing/decreasing behavior.

Example 4: Application in Business Context

Suppose the total cost $C(x)$ to manufacture a quantity $x$ of weed killer is given by $C(x) = \frac{x^2}{9 - x^2} + 8x + 50$.

• Find $C'(x)$ to determine where cost is increasing or decreasing.

• Set $C'(x) = 0$ and solve for $x$ to find critical numbers.

• Test intervals around critical numbers to classify increasing/decreasing intervals.

Summary Table: Steps for Finding Intervals of Increase/Decrease

Step	Description
1	Find the derivative $f'(x)$.
2	Solve $f'(x) = 0$ and $f'(x)$ undefined for critical numbers.
3	Divide the domain into intervals using critical numbers.
4	Choose test points in each interval and evaluate $f'(x)$.
5	Classify intervals as increasing ($f'(x) > 0$) or decreasing ($f'(x) < 0$).

Key Terms

• Increasing function

• Decreasing function

• Critical number

• Derivative

• Test point

Additional info:

• These notes are directly relevant to Business Calculus, focusing on the analysis of functions for optimization and cost analysis.

• Examples and problems are typical of those found in business calculus courses, including applications to cost functions and graphical analysis.