We think about the derivative of vector-valued functions.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Previously in calculus course you learned the following metaphor for the derivatives:

Given a function \(f:\R \to \R \), the derivative of \(f\) is the slope of the tangent line at any point on the graph \(y = f(x)\).

This is really a great metaphor for functions that map from \(\R \) to \(\R \). However, now we are studying vector-valued functions. A new metaphor is needed:

Given a vector-valued function \(\vec {f}:\R ^n\to \R \), the derivative of \(\vec {f}\) is a tangent vector at any point on the graph of \(\vec {f}\).

Let’s see if we can figure what this is saying.

1 Lines

Suppose you have a line given by the vector valued function \(\vecl \).

Pencil-in some tangent vectors for \(\vecl \) above.
Someone has plotted \(\vecl '\) below: Make sense of this plot. Explain what is going on to someone else.

2 Circles

The a circle of radius \(2\) centered at \((3,1\) is given by

\[ \vec {c}(t) = \vector {3 + 2 \cos (t), 1+2\sin (t)} \]

and here is a plot:

Pencil-in some tangent vectors for \(\vec {c}\) above.
Someone has plotted \(\vec {c}'\) below: Make sense of this plot. Explain what is going on to someone else.

3 Projectile motion

Vector-valued functions are excellent for modeling projectile motion. The function below models the path of a calculus book being thrown from an initial height of \(1\unit {m}\) at an initial velocity of \(5\unit {m}/{s}\) at a \(45^\circ \) angle:

\[ \vec {p}(t)=\vector {\frac {5t}{\sqrt {2}}, 1+ \frac {5t}{\sqrt {2}}-5t^2} \]

For your viewing pleasure here is a plot:

Pencil-in some tangent vectors for \(\vec {p}\) above.
Someone has plotted \(\vec {p}'\) below: Make sense of this plot (note, this is not a complete plot). Explain what is going on to someone else.
If we were to plot \(\vec {p}''\) what would it look like?
Rather than plotting \(\vec {p}''\), what should you do?

4 The moral of the story

The moral of the story is this: When studying, functions from \(\R \) to \(\R \), it makes a lot of sense to plot their derivatives. When dealing with vector-valued functions, plotting their derivatives might not be the best idea. Instead you should be plotting tangent vectors.