Glossary
Derivative
The derivative of a function represents the instantaneous rate of change of the function at any given point, and it is formally defined as a limit of the difference quotient.
Example:
If a function describes the position of an object over time, its derivative would describe the object's velocity at any given instant.
Instantaneous rate of change
The instantaneous rate of change measures how quickly a function's output changes with respect to its input at a single, specific point. It is the core idea behind derivatives.
Example:
When you look at your car's speedometer, it's showing your instantaneous rate of change of position (speed) at that exact moment, not your average speed over your entire trip.
Limits
Limits are a foundational concept in calculus that describe the behavior of a function as its input approaches a particular value, without necessarily reaching it. They are crucial for defining continuity, derivatives, and integrals.
Example:
To find the value a function approaches as x gets closer and closer to 3, even if the function isn't defined at x=3, you would evaluate the limit of the function as x approaches 3.
Secant line
A secant line is a straight line that connects two distinct points on a curve, representing the average rate of change of the function between those two points.
Example:
If you plot your distance traveled over time, drawing a line between your position at 1 hour and your position at 3 hours would give you a secant line whose slope represents your average speed during that interval.
Tangent line
A tangent line is 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:
Imagine a car driving along a curved road; the direction the car is heading at any given moment can be represented by a tangent line to the road at that point.