The material here takes us to the end of the course. I have included an extra section on multivariable calculus - simply an idea of what a multivariable function is and how to take partial derivatives.
- Section 12.7: As discussed earlier, movement through space has obvious interpretations with vector-valued functions. The example video allows us to see that we have been doing math like this previously, but that our new setting makes things easier.
- Section 12.8: In this section, we learn how to measure the length of the graph of a vector-valued function, which is a far more useful concept than the arclength of the functions we saw at the end of the first semester. It also gives us much better arithmetic. A fascinating use of this measure is the ability to change the parameter to describe precisely the amount traveled from any one point.
- Section 12.9: This is an odd section. It will either be necessary for a lot of what you do in life, or completely useless, which I think is a great way to end the course. The concepts here are an incredibly balanced combination of calculus and geometry.