Week 4 Discussion Notes

Table of Contents

• Differentiation
• Arc Length

Differentiation

If $\vec{r}\p{t}$ is a curve, then the tangent vector to $\vec{r}$ at $t = a$ is $\vec{r}'\p{a}$. On the other hand, the tangent line is the line that passes through $\vec{r}\p{a}$ and has direction vector $\vec{r}'\p{a}$, i.e.,

is a parametrization of the tangent line.

Arc Length

Here's a quick mnemonic to remember the arc-length formula:

From 1D, you should recall that $\Delta f \approx f'\p{x} \,\Delta{x}$. This is just a different way to say that the tangent line to $f$ is the best linear approximation to $f$. With vector-valued functions instead, we end up with

If we take lengths and write $s$ for the arc-length function of $\vec{r}$, then

So, in the limit, we end up with

From here, to actually use the formula, you'll also need to plug in the bounds for the integral.