Glossary

Derivative

Criticality: 3

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

Criticality: 3

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 $C(t)$ represents the number of customers in a store at time $t$ hours, then the function meaning of $C(t)$ is the count of customers at a specific hour.

Instantaneous Rate of Change

Criticality: 3

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 $A(t)$ , then $A'(5)$ gives the instantaneous rate of change of its altitude exactly 5 seconds after launch.

Rates of Change in Applied Contexts (Non-Motion)

Criticality: 3

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

Criticality: 2

*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

Criticality: 3

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).

Glossary

Derivative

Criticality: 3

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

Criticality: 3

Example:

If $C(t)$ represents the number of customers in a store at time $t$ hours, then the function meaning of $C(t)$ is the count of customers at a specific hour.

Instantaneous Rate of Change

Criticality: 3

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 $A(t)$ , then $A'(5)$ gives the instantaneous rate of change of its altitude exactly 5 seconds after launch.

Rates of Change in Applied Contexts (Non-Motion)

Criticality: 3

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

Criticality: 2

*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

Criticality: 3

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).