Glossary

Average Rate of Change

Criticality: 2

The overall rate at which a function's value changes over an interval, calculated as the slope of the secant line connecting the endpoints of the interval.

Example:

If a population grew from 100 to 150 individuals over 5 years, the average rate of change was $(150-100)/5 = 10$ individuals per year.

Continuous (over an interval)

Criticality: 3

A function is continuous over an interval if its graph can be drawn without lifting the pen, meaning there are no holes, asymptotes, or jumps within that interval.

Example:

The function $f(x) = x^2$ is continuous over the interval $[-5, 5]$ because its graph is a smooth, unbroken curve.

Derivative

Criticality: 3

A fundamental concept in calculus that measures the instantaneous rate at which a function's output changes with respect to its input, often interpreted as the slope of the tangent line.

Example:

The derivative of $f(x) = x^3$ is $f'(x) = 3x^2$ , which tells you the slope of the tangent line at any point $x$ .

Differentiable (over an interval)

Criticality: 3

A function is differentiable over an interval if its derivative exists at every point in that interval, implying the function is smooth and has no sharp corners or vertical tangent lines.

Example:

The function $f(x) = \sin(x)$ is differentiable over $(-\infty, \infty)$ because its graph is smooth everywhere, allowing a unique tangent line at any point.

Instantaneous Rate of Change

Criticality: 3

The rate at which a function's value is changing at a specific point, represented by the derivative of the function at that point.

Example:

If a ball's height is given by $h(t) = -16t^2 + 64t$ , its instantaneous rate of change at $t=1$ second (its velocity) is $h'(1) = 32$ ft/s.

Mean Value Theorem

Criticality: 3

A theorem stating that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point 'c' in (a, b) where the instantaneous rate of change equals the average rate of change over the interval.

Example:

If your car traveled 100 miles in 2 hours, the Mean Value Theorem guarantees that at some point during your trip, your instantaneous speed was exactly 50 mph.

Secant Line

Criticality: 2

A straight line that connects two distinct points on a curve, whose slope represents the average rate of change between those two points.

Example:

Drawing a secant line between the points $(0,0)$ and $(2,4)$ on the graph of $y=x^2$ gives a slope of 2, which is the average rate of change over that interval.

Tangent Line

Criticality: 2

A straight line that touches a curve at a single point and has the same slope as the curve at that point, representing the instantaneous rate of change.

Example:

The tangent line to the parabola $y=x^2$ at $x=1$ has a slope of 2, indicating how steeply the parabola is rising at that exact point.

Glossary

Average Rate of Change

Criticality: 2

The overall rate at which a function's value changes over an interval, calculated as the slope of the secant line connecting the endpoints of the interval.

Example:

If a population grew from 100 to 150 individuals over 5 years, the average rate of change was $(150-100)/5 = 10$ individuals per year.

Continuous (over an interval)

Criticality: 3

A function is continuous over an interval if its graph can be drawn without lifting the pen, meaning there are no holes, asymptotes, or jumps within that interval.

Example:

The function $f(x) = x^2$ is continuous over the interval $[-5, 5]$ because its graph is a smooth, unbroken curve.

Derivative

Criticality: 3

A fundamental concept in calculus that measures the instantaneous rate at which a function's output changes with respect to its input, often interpreted as the slope of the tangent line.

Example:

The derivative of $f(x) = x^3$ is $f'(x) = 3x^2$ , which tells you the slope of the tangent line at any point $x$ .

Differentiable (over an interval)

Criticality: 3

A function is differentiable over an interval if its derivative exists at every point in that interval, implying the function is smooth and has no sharp corners or vertical tangent lines.

Example:

The function $f(x) = \sin(x)$ is differentiable over $(-\infty, \infty)$ because its graph is smooth everywhere, allowing a unique tangent line at any point.

Instantaneous Rate of Change

Criticality: 3

The rate at which a function's value is changing at a specific point, represented by the derivative of the function at that point.

Example:

If a ball's height is given by $h(t) = -16t^2 + 64t$ , its instantaneous rate of change at $t=1$ second (its velocity) is $h'(1) = 32$ ft/s.

Mean Value Theorem

Criticality: 3

Example:

If your car traveled 100 miles in 2 hours, the Mean Value Theorem guarantees that at some point during your trip, your instantaneous speed was exactly 50 mph.

Secant Line

Criticality: 2

A straight line that connects two distinct points on a curve, whose slope represents the average rate of change between those two points.

Example:

Drawing a secant line between the points $(0,0)$ and $(2,4)$ on the graph of $y=x^2$ gives a slope of 2, which is the average rate of change over that interval.

Tangent Line

Criticality: 2

A straight line that touches a curve at a single point and has the same slope as the curve at that point, representing the instantaneous rate of change.

Example:

The tangent line to the parabola $y=x^2$ at $x=1$ has a slope of 2, indicating how steeply the parabola is rising at that exact point.