Why does the secant line become a tangent line as rho approaches zero?
Conditions
- The two points lie on a smooth curve (the section curve).
- One point is fixed at .
- The other point moves along the curve towards .
- rho represents the distance between the two points.
Reasoning, step by step
- Start with two distinct points and P on the curve.
- Draw the secant line connecting them.
- Reduce the distance rho between P and .
- Observe the change in the slope of the secant line.
- Take the limit as rho -> 0.
- Identify the resulting line as the tangent line.
- Connect this geometric limit to the analytical definition of the derivative.
Example
The video animates the purple point P moving closer to the red point . The white secant line adjusts its slope until it becomes the white tangent line when rho is infinitesimally small.
Common misconceptions
- Thinking the tangent line is just a secant line with a very small rho, rather than a limit.
- Believing the tangent line touches the curve at only one point globally (it is a local property).
- Confusing the tangent line to the section curve with the tangent plane to the surface.
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