Glossary

Critical Points

Criticality: 3

Points in the domain of a function where its derivative is either zero or undefined, or where the function itself is undefined. These are potential locations where a function can change from increasing to decreasing or vice versa.

Example:

For $f(x) = x^3 - 3x$ , the critical points are found by setting $f'(x) = 3x^2 - 3 = 0$ , which gives $x = \pm 1$ .

Decreasing Function

Criticality: 3

A function is decreasing on an interval if its derivative is negative throughout that interval, meaning the function's output values are getting smaller as the input values increase.

Example:

When a balloon is deflating, its volume $V(t)$ is a decreasing function of time, as indicated by $V'(t) < 0$ .

Derivative

Criticality: 3

The derivative of a function measures the instantaneous rate of change of the function at any given point. It indicates the slope of the tangent line to the function's graph.

Example:

If a car's position is given by $s(t)$ , then its velocity, $s'(t)$ , is the derivative of its position function, telling us how fast it's moving at any moment.

Function Behavior Analysis

Criticality: 3

The process of using the first derivative of a function to determine the intervals where the function is increasing or decreasing, by examining the sign of the derivative.

Example:

Performing a function behavior analysis on $g(x) = \frac{1}{x}$ involves finding its derivative, $g'(x) = -\frac{1}{x^2}$ , and noting that $g'(x) < 0$ for all $x eq 0$ , indicating $g(x)$ is always decreasing on its domain.

Increasing Function

Criticality: 3

A function is increasing on an interval if its derivative is positive throughout that interval, meaning the function's output values are getting larger as the input values increase.

Example:

If a company's profit function $P(x)$ has $P'(x) > 0$ for $x \in (100, 500)$ , it means the profit is an increasing function when between 100 and 500 units are sold.

Test Point

Criticality: 2

A specific x-value chosen within an interval to evaluate the sign of the derivative, thereby determining whether the original function is increasing or decreasing on that entire interval.

Example:

To check the behavior of $f(x)$ on the interval $(-3, 3)$ , we might choose $x=0$ as a test point to plug into $f'(x)$ .