Glossary

Concave Down

Criticality: 3

A portion of a function's graph where the second derivative is negative, indicating that the slope of the tangent line is decreasing.

Example:

The function $f(x) = -x^2$ is always concave down because its second derivative, $f''(x) = -2$ , is always negative.

Concave Up

Criticality: 3

A portion of a function's graph where the second derivative is positive, indicating that the slope of the tangent line is increasing.

Example:

The function $f(x) = x^2$ is always concave up because its second derivative, $f''(x) = 2$ , is always positive.

Concavity (Upward/Downward)

Criticality: 3

Describes the direction in which the graph of a function bends; upward concavity means it opens like a cup, downward concavity means it opens like a frown.

Example:

A graph that looks like a smile has upward concavity, while one that looks like a frown has downward concavity.

Critical Point

Criticality: 3

A point in the domain of a function where the first derivative is either zero or undefined.

Example:

For $f(x) = x^3 - 12x$ , setting $f'(x) = 3x^2 - 12 = 0$ gives $x = \pm 2$ , which are the critical points.

First Derivative

Criticality: 3

The derivative of a function, representing its instantaneous rate of change and indicating where the function is increasing or decreasing.

Example:

If $f(x)$ models the temperature over time, then $f'(x)$ tells you how fast the temperature is changing, acting as the first derivative.

Global Extremum

Criticality: 2

The absolute maximum or minimum value of a function over its entire domain or a specified closed interval.

Example:

On the interval $[0, 5]$ , the function $f(x) = x^2$ has a global extremum at $x=5$ (global maximum) and $x=0$ (global minimum).

Inconclusive (Second Derivative Test)

Criticality: 2

A situation where the Second Derivative Test cannot determine if a critical point is a local maximum or minimum because the second derivative at that point is zero or undefined.

Example:

For $f(x) = x^4$ , $f'(0)=0$ and $f''(0)=0$ , making the Second Derivative Test inconclusive at $x=0$ , requiring the First Derivative Test to classify it.

Inflection Point

Criticality: 3

A point on the graph of a function where the concavity changes, provided the function is continuous at that point.

Example:

For $f(x) = x^3$ , the point $(0,0)$ is an inflection point because the graph changes from concave down to concave up there.

Local Maximum

Criticality: 3

A point where the function's value is greater than or equal to the values at all nearby points, often found where the function changes from increasing to decreasing.

Example:

The highest point on a hill in a landscape represents a local maximum of its elevation function.

Local Minimum

Criticality: 3

A point where the function's value is less than or equal to the values at all nearby points, often found where the function changes from decreasing to increasing.

Example:

The lowest point in a valley on a roller coaster track represents a local minimum of its height function.

Relative Extrema (Local Maximum/Minimum)

Criticality: 3

Points on a function's graph where the function reaches a peak (local maximum) or a valley (local minimum) within a specific interval.

Example:

The function $f(x) = \sin(x)$ has relative extrema at $x = \frac{\pi}{2}$ (local maximum) and $x = \frac{3\pi}{2}$ (local minimum) within the interval $[0, 2\pi]$ .

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, which provides information about the concavity of the original function.

Example:

If a car's velocity is given by $v(t)$ , then its acceleration, $a(t) = v'(t)$ , is the second derivative of its position function.

Second Derivative Test

Criticality: 3

A method that uses the sign of the second derivative at a critical point to classify it as a local maximum or local minimum.

Example:

If you find a critical point 'c' and $f''(c) > 0$ , the Second Derivative Test tells you that 'c' corresponds to a local minimum.