Unit 3: Derivatives, but worse

By: Harsh Singh

Level Curves and Domain

Levels curves are essentially when you take a function in $\mathbb{R}³$, and slice it with planes that intersect the $z$ axis.

Unlike the rest of the math in this course, level curves come up quite a lot in the real world, specifically in contour maps that tell the elevation at certain points. Therefore, levels curves may and will also be referred to as contour plots.   

To calculate the contours of a contour plot (see, I used it!), plug the level into$z$ in the function and solve for the rest of the function.

Finding the domain of the function is also important to understand. The idea of domain and range still remains essentially the same, but one of the major differences is that usually, $z$ is the range and $x$,$y$ is the domain. To solve a domain and range problem, identify when the range would become invalid, such as by dividing by $0$ or having a negative under a square root, and then solve the domain to avoid reaching those discontinuities.

Limits and Continuity (Again)

If you recall Unit 1 of Calculus BC, we learned about what a limit is and how to define continuity. In Calc 3, the principles remain the same, but it is now extended into 3 dimensions. To solve a limit problem, plug the numbers in and solve as all the properties of limits from Calc BC still apply.

To define whether or not a limit exists, there is a thing known as the 2 Path Test, where you approach a point via 2 different paths and if they result in different values, than the limit doesn’t exist.

If you remember continuity from Calc BC, then you will realize that the only difference that occurs when you jump into Calc 3 is that there is an extra dimension added. The steps are provided below:

Another thing we have to cover before going onto the next lesson is the idea of interior and boundary points. An interior point is a point that is inside a shape, while boundary points are on the edge of the shape. An Open Set of points contains only interior points, but a Closed Set contains both interior and boundary points 

Where $P_0$ is a boundary point and $P$ are interior points