Glossary

Critical Points

Criticality: 3

Points in the domain of a function where its derivative is either zero or undefined, and these are the only possible locations for relative extrema.

Example:

For $f(x) = |x|$ , the point $x=0$ is a critical point because its derivative is undefined there, even though it's a local minimum.

Decreasing Function

Criticality: 2

A function is decreasing over an interval if its output values (y-values) consistently fall as its input values (x-values) increase over that interval, meaning its derivative is negative.

Example:

The remaining battery life of your phone while you're watching videos is a decreasing function of time.

Derivative

Criticality: 3

The instantaneous rate of change of a function with respect to its independent variable, representing the slope of the tangent line to the function's graph at any given point.

Example:

If $s(t)$ is the position of a car, then $s'(t)$ is its velocity, which is the derivative of the position function.

First Derivative Test

Criticality: 3

A method used to determine the relative (local) extrema of a function by analyzing the sign changes of its derivative around critical points.

Example:

To find where $f(x) = x^3 - 3x$ has local extrema, you'd use the First Derivative Test by checking the sign of $f'(x)$ around its critical points.

Increasing Function

Criticality: 2

A function is increasing over an interval if its output values (y-values) consistently rise as its input values (x-values) increase over that interval, meaning its derivative is positive.

Example:

The amount of water in a bathtub while it's filling up is an increasing function of time.

Relative (Local) Extrema

Criticality: 3

The highest or lowest points of a function within a specific interval or neighborhood, representing local peaks or valleys.

Example:

On a roller coaster track, the highest point of a specific hill and the lowest point of a specific dip are examples of relative extrema.

Relative Maximum

Criticality: 3

A point where a function changes from increasing to decreasing, indicating a local peak in the function's graph.

Example:

If a company's profit function $P(t)$ shows $P'(t)$ changing from positive to negative at $t=5$ months, then $t=5$ corresponds to a relative maximum profit.

Relative Minimum

Criticality: 3

A point where a function changes from decreasing to increasing, indicating a local valley in the function's graph.

Example:

When a ball is thrown, its lowest point before it starts rising again is a relative minimum of its height function.