Glossary
Average Rate of Change
The overall change in a function's value over an interval, calculated as the slope of the secant line connecting two points on the function.
Example:
If a car travels 180 miles in 3 hours, its average rate of change (speed) is 60 mph.
Derivative
A fundamental concept in calculus that represents the instantaneous rate of change of a function with respect to one of its variables. It gives the slope of the tangent line at any point.
Example:
For f(x) = x^2, the derivative f'(x) = 2x, which tells you the slope of the tangent line at any x.
Instantaneous Rate of Change
The rate at which a function's value is changing at a specific single point, found using the derivative.
Example:
The reading on your car's speedometer at any given moment is its instantaneous rate of change (speed).
Limit Definition of the Derivative
The formal definition of the derivative, expressing it as the limit of the difference quotient as the change in x approaches zero.
Example:
Using to find the slope of the tangent line for f(x) = x^2 at x=2 yields 4.
Secant Line
A line that connects two distinct points on a curve. Its slope represents the average rate of change of the function between those two points.
Example:
Drawing a line from (1, f(1)) to (3, f(3)) on the graph of f(x) = x^2 creates a secant line.
Slope
A measure of the steepness and direction of a line, calculated as the 'rise over run' or the change in y divided by the change in x.
Example:
A ramp that rises 3 feet for every 10 feet of horizontal distance has a slope of 3/10.
Tangent Line
A line that touches a curve at a single point and has the same slope as the curve at that point. Its slope represents the instantaneous rate of change.
Example:
The line that just grazes the top of a parabola at its vertex is a tangent line.