Interpretation of the Derivative
Beyond the formula: Understanding the derivative as slope, rate of change, and velocity.
Deep Dive
The derivative provides three layers of meaning:
It is the slope of the curve.
If , the graph is rising.
If , it is falling.
It measures the instantaneous rate of change of with respect to .
"How fast is the function changing right now?"
If describes position, then describes velocity.
Acceleration is the derivative of velocity!
Worked Example: Particle Motion
The position of a particle is given by meters. Find the velocity at seconds.
We use the limit definition of the derivative:
Substitute :
Expand and :
Distribute and cancel terms:
Factor out from the numerator:
As , terms with vanish:
Interpretation: The particle has momentarily stopped moving at .
Interactive Visualization
Visualizing Motion
Explore the relationship between Position (Blue) and Velocity (Red). Notice that when Velocity is 0 (at x=0 and x=4), the Position graph has a horizontal tangent (a turning point).
Practice Quiz
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