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Derivative and Unit Tangent Vector

Derivative and Unit Tangent Vector
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2020-08-19 21:40
VectorCalculusDerivativeParametric Equation
This example shows how to compute the derivative of a parametric curve and obtain its unit tangent vector. It involves concepts of derivative, tangent, vector, parametric equations, and limits. The construction uses GeoGebra's Textfield for input, CurveCartesian to define the curve, Vector to draw vectors, and sliders (inputBox1, inputBox2, inputBox3) to control parameters. It is suitable for high school or early college calculus students.

GeoGebra Commands & Parameters

  1. Textfield(g) → inputBox2
  2. Textfield(f) → inputBox1
  3. Textfield(a) → ta
  4. Textfield(b) → tb
  5. CurveCartesian(f(t), g(t), h(t), t, a, b) → c
  6. Textfield(h) → inputBox3
  7. Vector(((2 * cos(t / 4)) + cos(t), (2 * sin(t)) - 1, t^(3) - (3 * t)), ((2 * cos(t / 4)) + cos(t) - (1 / 2 * sin(t / 4)) - sin(t), (2 * sin(t)) - 1 + (2 * cos(t)), t^(3) - (3 * t) + (3 * t^(2)) - 3)) → u
  8. Vector(((2 * cos(t / 4)) + cos(t), (2 * sin(t)) - 1, t^(3) - (3 * t)), ((2 * cos(t / 4)) + cos(t) + x(unit), (2 * sin(t)) - 1 + y(unit), t^(3) - (3 * t) + z(unit))) → v

AI Drawing Prompt Reference

Draw a 3D parametric curve where the user can enter the three component functions x(t), y(t), z(t) and the parameter interval [a,b] via input boxes. At a chosen parameter value t, show the position vector, the tangent vector (the derivative), and the unit tangent vector, all originating from the corresponding point on the curve; the tangent vector points along the curve’s tangent direction and the unit tangent vector has length one.
The prompt above was compiled from the original drawing commands and is for reference only.