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Glossary

C

Concave Down Interval

Criticality: 3

An interval where the graph of a function resembles a cup opening downwards, indicated by a negative second derivative.

Example:

The function f(x)=−x2f(x) = -x^2 is concave down on its entire domain because f′′(x)=−2<0f''(x) = -2 < 0.

Concave Up Interval

Criticality: 3

An interval where the graph of a function resembles a cup opening upwards, indicated by a positive second derivative.

Example:

The parabola f(x)=x2f(x) = x^2 is concave up on its entire domain because f′′(x)=2>0f''(x) = 2 > 0.

Critical Point

Criticality: 3

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

Example:

For f(x)=x3−3xf(x) = x^3 - 3x, the critical points are found by setting f′(x)=3x2−3=0f'(x) = 3x^2 - 3 = 0, which gives x=±1x = \pm 1.

D

Decreasing Interval

Criticality: 3

An interval where the function's values are falling as x increases, indicated by a negative first derivative.

Example:

For f(x)=−x3f(x) = -x^3, the function is decreasing on (−∞,∞)(-\infty, \infty) because f′(x)=−3x2≤0f'(x) = -3x^2 \le 0.

Discontinuity

Criticality: 2

A point where a function is not continuous, meaning there is a break, hole, or jump in its graph.

Example:

The function f(x)=1/xf(x) = 1/x has a discontinuity at x=0x=0 because it is undefined there.

Domain

Criticality: 2

The complete set of all possible input values (x-values) for which a function is defined.

Example:

The domain of f(x)=xf(x) = \sqrt{x} is [0,∞)[0, \infty) because the square root of a negative number is not a real number.

E

Even Symmetry

Criticality: 1

A property of a function where its graph is symmetric with respect to the y-axis, meaning $f(-x) = f(x)$ for all x in its domain.

Example:

The function f(x)=cos⁡(x)f(x) = \cos(x) exhibits even symmetry because cos⁡(−x)=cos⁡(x)\cos(-x) = \cos(x).

F

First Derivative Test

Criticality: 3

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

Example:

If f′(x)f'(x) changes from positive to negative at a critical point, the First Derivative Test indicates a relative maximum.

I

Increasing Interval

Criticality: 3

An interval where the function's values are rising as x increases, indicated by a positive first derivative.

Example:

If f′(x)=x2+1f'(x) = x^2 + 1, then f(x)f(x) is always increasing because f′(x)f'(x) is always positive.

O

Odd Symmetry

Criticality: 1

A property of a function where its graph is symmetric with respect to the origin, meaning $f(-x) = -f(x)$ for all x in its domain.

Example:

The function f(x)=sin⁡(x)f(x) = \sin(x) exhibits odd symmetry because sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x).

P

Point of Inflection

Criticality: 3

A point on the graph of a function where the concavity changes (from concave up to concave down or vice versa).

Example:

For f(x)=x3f(x) = x^3, the origin (0,0)(0,0) is a point of inflection because f′′(x)=6xf''(x) = 6x changes sign at x=0x=0.

Polynomial Function

Criticality: 1

A function that can be expressed as a sum of terms, each consisting of a constant multiplied by a non-negative integer power of the variable.

Example:

f(x)=3x4−2x2+5f(x) = 3x^4 - 2x^2 + 5 is a polynomial function with a domain of all real numbers.

Product Rule

Criticality: 3

A differentiation rule used to find the derivative of a function that is the product of two or more differentiable functions.

Example:

To differentiate f(x)=xsin⁡(x)f(x) = x \sin(x), you would apply the Product Rule: f′(x)=1⋅sin⁡(x)+x⋅cos⁡(x)f'(x) = 1 \cdot \sin(x) + x \cdot \cos(x).

R

Relative Extrema (Local Extrema)

Criticality: 3

Points where a function reaches a maximum or minimum value within a specific neighborhood of its domain.

Example:

The function f(x)=x3−3xf(x) = x^3 - 3x has a relative maximum at x=−1x=-1 and a relative minimum at x=1x=1.

S

Second Derivative Test

Criticality: 2

A method used to classify relative extrema by evaluating the sign of the second derivative at a critical point.

Example:

If f′′(c)>0f''(c) > 0 at a critical point cc, the Second Derivative Test confirms a relative minimum at x=cx=c.

x

x-intercept

Criticality: 2

A point where the graph of a function crosses or touches the x-axis, meaning the function's value is zero at that point.

Example:

The function f(x)=x2−4f(x) = x^2 - 4 has x-intercepts at (−2,0)(-2,0) and (2,0)(2,0).

y

y-intercept

Criticality: 2

A point where the graph of a function crosses the y-axis, which occurs when the input value (x) is zero.

Example:

For f(x)=x2−4f(x) = x^2 - 4, the y-intercept is (0,−4)(0,-4) because f(0)=−4f(0) = -4.