Derivatives of vector-valued functions

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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.

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.

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.

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?

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.