Tangency: chords, tangents and velocity of a parametric curve

This applet explores the velocity vector of a parametric curve, and its relationship to the chord r(t+h)-r(t) and the difference quotient \(\frac{{\textbf{r}}(t+h) – {\textbf{r}}(t)}{h}\).

The velocity vector of a vector function r(t) is given by

\( \displaystyle {\textbf{r}}'(t) = \lim_{h\to 0}\frac{{\textbf{r}}(t+h) – {\textbf{r}}}{h} \).

This expression contains

  • \({\textbf{r}}(t+h) – {\textbf{r}}\), which is a chord from a point r(t) on the curve to another point r(t+h), and
  • \(\frac{{\textbf{r}}(t+h) – {\textbf{r}}}{h}\), sometimes called the difference quotient.

Enable the Show velocity checkbox to show the velocity vector \({\textbf{r}}'(t)\). Enable Show chord to show the vector r(t+h)-r(t), drawn starting at the point r(t). Enable Show r(t) + vh to show the difference quotient.

Change the value of h using the slider. What happens to the length of the chord (the blue vector) as you change h? What happens to the length of the difference quotient (the red vector)?

What happens as h approaches 0?

What happens if h becomes negative?

You can choose a different example of a vector function using the dropdown menu, or enter your own by entering its i and j components. You can also use the t slider to change the value of t.

Other resources:

Geogebratube page for this applet