Glossary

Derivative

Criticality: 3

The derivative, denoted as f'(x), represents the instantaneous rate of change of a function at any given point. It is equivalent to the slope of the line tangent to the curve at that specific location.

Example:

If a car's position is given by $s(t) = t^2$ , its derivative $s'(t) = 2t$ gives its instantaneous velocity at time $t$ .

Derivative Notation

Criticality: 2

Various symbols used to represent the derivative of a function, including $y'$, $f'(x)$, and $\frac{dy}{dx}$, all indicating the rate of change of the dependent variable with respect to the independent variable.

Example:

If $y = x^3$ , then $y'$ , $f'(x)$ , and $\frac{dy}{dx}$ all represent $3x^2$ , demonstrating different derivative notations for the same concept.

Horizontal tangent

Criticality: 2

A tangent line to a curve that has a slope of zero, indicating a local maximum, local minimum, or a saddle point on the function's graph.

Example:

For $f(x) = x^3 - 3x$ , the horizontal tangents occur where $f'(x) = 3x^2 - 3 = 0$ , which is at $x = \pm 1$ .

Instantaneous rate of change

Criticality: 3

The rate at which a function's value is changing at a specific moment or point, as opposed to an average rate over an interval.

Example:

The speedometer in your car shows the instantaneous rate of change of your position with respect to time, which is your current speed.

Limit Definition of a Derivative

Criticality: 3

The formal definition of the derivative, expressed as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$, which calculates the slope of a secant line as the distance between two points approaches zero.

Example:

Using the limit definition of a derivative, you can prove that the derivative of $f(x) = x^2$ is $f'(x) = 2x$ .

Point-slope form

Criticality: 2

An algebraic form for the equation of a straight line, given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is its slope.

Example:

To write the equation of a line with slope 3 passing through $(2,5)$ , you would use point-slope form: $y - 5 = 3(x - 2)$ .

Slope of the curve

Criticality: 3

At any given point on a curve, this refers to the slope of the line tangent to the curve at that specific point, indicating the steepness and direction of the curve.

Example:

For the parabola $y=x^2$ , the slope of the curve at $x=2$ is $2(2)=4$ , meaning the tangent line at $(2,4)$ has a slope of 4.

Tangent line

Criticality: 3

A straight line that touches a curve at a single point and has the same slope as the curve at that point.

Example:

The line $y = -x + 2$ is the tangent line to the curve $f(x) = 1/x$ at the point $(1,1)$ .