Glossary

Chain Rule (for Related Rates)

Criticality: 3

A differentiation rule used to find the derivative of a composite function, often applied in related rates problems to connect rates of change of different variables with respect to time.

Example:

In a related rates problem, if you know $\frac{dx}{dt}$ and want $\frac{dy}{dt}$ , you can use the Chain Rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ .

Concave Down

Criticality: 2

A property of a function's graph where its slope is decreasing, meaning the graph 'spills water' or resembles an inverted cup opening downwards.

Example:

The graph of $y = -x^2$ is concave down everywhere because its second derivative, $y''=-2$ , is always negative.

Concave Up

Criticality: 2

A property of a function's graph where its slope is increasing, meaning the graph 'holds water' or resembles a cup opening upwards.

Example:

The graph of $y = x^2$ is concave up everywhere because its second derivative, $y''=2$ , is always positive.

Critical Points (on Implicit Functions)

Criticality: 3

Points on an implicit function where the first derivative (dy/dx) is either zero or undefined, indicating potential relative extrema or points where the tangent line is vertical.

Example:

For $x^2 + y^2 = 25$ , the critical points where $\frac{dy}{dx}=0$ are $(0, 5)$ and $(0, -5)$ .

Explicit Function

Criticality: 1

A function where the dependent variable (y) is isolated and expressed directly in terms of the independent variable (x), typically passing the vertical line test.

Example:

$y = x^3 - 2x + 1$ is an explicit function where y is clearly defined by x.

First Derivative Test

Criticality: 3

A method used to determine if a critical point corresponds to a relative maximum or minimum by analyzing the sign change of the first derivative around that point.

Example:

If $\frac{dy}{dx}$ changes from positive to negative at a critical point, the First Derivative Test indicates a relative maximum.

Implicit Differentiation

Criticality: 3

A technique used to find the derivative of an implicit function by differentiating both sides of the equation with respect to a variable (usually x) and applying the chain rule to terms involving the dependent variable.

Example:

To find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$ , you use implicit differentiation to get $2x + 2y \frac{dy}{dx} = 0$ .

Implicit Function

Criticality: 3

A function where the dependent variable (y) is not explicitly isolated on one side of the equation, often having multiple variables on the same side.

Example:

The equation $x^2 + y^2 = 25$ defines y as an implicit function of x, representing a circle.

Point of Inflection

Criticality: 2

A point on a function's graph where the concavity changes (from concave up to concave down or vice versa), and the second derivative is zero or undefined.

Example:

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

Related Rates

Criticality: 3

Problems that involve finding the rate at which a quantity changes by relating it to other known rates of change, typically using implicit differentiation with respect to time.

Example:

Determining how fast the top of a ladder is sliding down a wall given the rate at which its base is moving away is a classic related rates problem.

Relative Maximum

Criticality: 3

A point on a function's graph where the function's value is greater than or equal to the values at all nearby points, often occurring where the first derivative changes from positive to negative.

Example:

For $x^2 + y^2 = 25$ , the point $(0, 5)$ is a relative maximum because the derivative changes from positive to negative as x increases through 0 (for the upper semicircle).

Relative Minimum

Criticality: 3

A point on a function's graph where the function's value is less than or equal to the values at all nearby points, often occurring where the first derivative changes from negative to positive.

Example:

For $x^2 + y^2 = 25$ , the point $(0, -5)$ is a relative minimum because the derivative changes from negative to positive as x increases through 0 (for the lower semicircle).

Second Derivative

Criticality: 2

The derivative of the first derivative, denoted as d^2y/dx^2 or f''(x), which provides information about the concavity of a function.

Example:

If $\frac{dy}{dx} = -\frac{x}{y}$ , finding the second derivative $\frac{d^2y}{dx^2}$ requires implicit differentiation again, often involving both x, y, and $\frac{dy}{dx}$ .

Second Derivative Test

Criticality: 2

A method that uses the sign of the second derivative at a critical point to determine if it is a relative maximum or minimum, or to determine the concavity of a function.

Example:

If $f''(c) > 0$ at a critical point c, the Second Derivative Test indicates a relative minimum.