# Chapter 12: Multivariable Functions

1 TB.

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?

1 THINKING BACK

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 $$x = 2$$

1 THINKING FORWARD

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.

2 TB

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

2 TB

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.

Q 0.

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

Q. 0

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

Q. 0

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

Q. 00

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

Q 1.

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