Applet: The derivative of a parametrized curve
The function $\dllp(t)=(3\cos t)\vc{i} + (2 \sin t) \vc{j}$ parametrizes an ellipse. For given values of $t$ and $h$ (changeable by the sliders), the blue vector is $\dllp(t)$ and the green vector is $\dllp(t+h)$. The red vector is an estimate for the derivative: $\dllp'(t) \approx (\dllp(t+h) - \dllp(t))/h$. As $h$ decreases to zero, this estimate approaches the value of the derivative $\dllp'(t)$, which is tangent to the ellipse.
Applet file: derivative_parametrized_curve.ggb
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