Glossary
Derivative
The *derivative* of a function measures the sensitivity of the function's output with respect to its input, representing the instantaneous rate of change or the slope of the tangent line.
Example:
To find how fast the area of a circle is growing as its radius increases, you would take the derivative of the area formula with respect to the radius.
Function Meaning
In applied problems, understanding the *function meaning* involves identifying what the dependent variable represents (e.g., volume, population, temperature) and what the independent variable represents (e.g., time, radius).
Example:
If represents the number of customers in a store at time hours, then the function meaning of is the count of customers at a specific hour.
Instantaneous Rate of Change
The *instantaneous rate of change* is the rate at which a quantity is changing at a specific moment in time or at a particular value of the independent variable, represented by the derivative.
Example:
If a rocket's altitude is given by , then gives the instantaneous rate of change of its altitude exactly 5 seconds after launch.
Rates of Change in Applied Contexts (Non-Motion)
This concept extends the idea of derivatives to describe how quantities change over time or with respect to another variable in real-world scenarios beyond just movement.
Example:
If a function models the amount of water in a pool, then its rate of change in applied contexts would describe how quickly the water level is rising or falling.
Rectilinear Motion
*Rectilinear motion* refers to the movement of an object along a straight line, where calculus concepts like position, velocity, and acceleration are applied.
Example:
Analyzing a car's speed and direction as it drives straight down a highway is an example of studying its rectilinear motion.
Units of Derivatives
The *units of derivatives* are always the units of the dependent variable divided by the units of the independent variable, providing context to the rate of change.
Example:
If a function describes the temperature in degrees Celsius over time in hours, then the units of derivatives would be degrees Celsius per hour (°C/hr).