Glossary
Average Rate of Change
Measures how much a function's output changes over a specific interval, calculated as the ratio of the change in output to the change in input.
Example:
If a plant grows from 10 cm to 25 cm in 5 days, its average rate of change in height is 3 cm per day.
Negative Rate of Change
Occurs when one quantity increases while the other decreases, indicating an inverse relationship between the two quantities.
Example:
When a hot cup of coffee cools down, its temperature exhibits a negative rate of change with respect to time, as time increases, temperature decreases.
Positive Rate of Change
Occurs when two quantities increase or decrease together, meaning as one quantity increases, the other also increases.
Example:
As the number of hours you practice a musical instrument increases, your skill level typically shows a positive rate of change.
Rate of Change
Describes how quickly a function's output is changing at an exact input value, representing the steepness of the curve at that specific point.
Example:
The rate of change of a rocket's altitude at the moment it runs out of fuel would indicate its instantaneous vertical speed at that precise time.
Slope of the Secant Line
The slope of the line connecting two distinct points on a function's graph, which visually represents the average rate of change over the interval between those points.
Example:
To find the slope of the secant line for a runner's distance-time graph between the 5-minute and 10-minute marks, you would connect the points corresponding to those times.
Slope of the Tangent Line
The slope of a line that touches a curve at a single point, representing the instantaneous rate of change at that specific point.
Example:
The instantaneous velocity of a car at a specific moment is represented by the slope of the tangent line to its position-time graph at that exact time.