Chapter 12: Multivariable Functions

Intervals: What is meant by an open interval in R? What is meant by a closed interval in R? Is every interval either open or closed?

Finding a direction vector for a tangent line: Find a direction vector for the line tangent to the curve \begin{equation}y=x^{3}\end{equation} when $$x = 2$$

The derivative along a cut edge: Let

\begin{equation}f(x,y)=x^2y^3\end{equation}

Find the rate of change of f in the (positive) y direction when the surface is cut by the plane with equation x = 2. Find the rate of change of f in the (positive) x direction when the surface is cut by the plane with equation y = 1.

Limits: If f is a function of a single variable, what is the intuitive interpretation of $\underset{x\to a}{\lim }f\left(x\right)$? What is the formal definition?

Finding the equation of the plane containing two intersecting lines: Show that the lines given by \begin{equation}\mathbf{r}1(t)=(3t-4,-4t+1,t) \end{equation} \begin{equation}\mathbf{r}2(t)=(-t+2,2t-9,-2t+7)\end{equation}

intersect, and find the equation of the plane containing the lines.

Read the section and make your own summary of the material.

Problem Zero: Read the section of Chain Rule and make your own summary of the material.

Problem Zero: Read the section and make your own summary of the material.

Problem Zero: Read the section and make your own summary of the material.

Explain why Definition 0.1 is general enough to include functions of two and three variables.