How is the secant line constructed to approximate the directional derivative in the video animation?
Conditions
- Point is fixed on the surface .
- A direction vector l is defined by angle alpha.
- A second point P is chosen at distance rho from along l.
- The coordinates of P are ( + rho*cos(alpha), + rho*sin(alpha)).
Reasoning, step by step
- Start with point on the surface.
- Define the direction vector l using angle alpha.
- Move a distance rho along l to find the projection of point P in the xy-plane.
- Find the corresponding height on the surface to locate point P.
- Draw a vertical segment representing the height difference between P and .
- Connect and P with a straight line to form the secant line.
- Calculate the slope of this secant line as the change in height divided by rho.
Example
The video shows a green ray extending from . A purple point P appears at distance rho. Yellow vertical segments mark the height difference, and a white line connects and P, forming the secant line.
Common misconceptions
- Thinking the secant line is drawn on the xy-plane instead of the cutting plane.
- Confusing the horizontal distance rho with the arc length along the surface.
- Believing the secant line represents the final directional derivative rather than an approximation.
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