Average: Over an interval, slope of secant line | Instantaneous: At a point, slope of tangent line.

Secant: Intersects curve at two points | Tangent: Touches curve at one point, represents derivative.

Average: Change in position over a time interval | Instantaneous: Velocity at a specific time.

Slope formula: for secant lines (average rate of change) | Derivative: for tangent lines (instantaneous rate of change).

Average: $\frac{f(b)-f(a)}{b-a}$ | Instantaneous: $\lim_{h \to 0} \frac{f(c+h)-f(c)}{h}$

Average acceleration: Change in velocity over a time interval | Instantaneous acceleration: Acceleration at a specific time.

Secant lines: Approximate function behavior over an interval | Tangent lines: Approximate function behavior near a point.

Limit definition: Fundamental definition, used for proof | Derivative rules: Shortcuts for finding derivatives.

Average rate of change: Slope of secant line | Mean Value Theorem: Guarantees a point where instantaneous rate of change equals average rate of change.

Average rate of change: Used for overall motion analysis | Instantaneous rate of change: Used for specific moment analysis.

The average rate of change shows the average amount that a function changes over a given interval.

The instantaneous rate of change shows the exact amount that a function is changing at a specific point.

The slope of the secant line connecting two points on the graph of the function.

The slope of the tangent line at a specific point on the graph of the function.

The instantaneous rate of change is the limit of the average rate of change as the interval approaches zero.

The derivative of a function at a point is equal to the instantaneous rate of change at that point.

It provides a rigorous way to define the derivative and understand its meaning.

The sign of the derivative indicates whether the function is increasing (positive) or decreasing (negative).

It can be used to calculate average speeds, growth rates, or changes in quantities over time.

It can be used to determine velocity at a specific time, reaction rates in chemistry, or the rate of change of a stock price.

Slope of the secant line between two points on a curve.

Slope of the tangent line at a single point; found using the derivative.

A line that intersects a curve at two or more points.

A line that touches a curve at a single point, having the same slope as the curve at that point.

A measure of the instantaneous rate of change of a function.

The value that a function approaches as the input approaches some value.

The instantaneous rate of change of $f(x)$ at $x = c$.

The derivative gives the slope of the tangent line at a specific point.

The average rate of change of position with respect to time over an interval.

The instantaneous rate of change of position with respect to time at a specific moment.