# 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) + v_{h}* 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: