다변수 미분 전체 문제

True or False: If we evaluate $\lim_{(x,y)\to (0,0)}f(x)$ along several paths and each time the limit is $1,$ we can conclude that $\lim_{(x,y)\to (0,0)}f(x)=1.$

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

$\left(\frac{3}{4},\frac{9}{16}\right)$ is a critical point of $g(x,y)=4{x}^{3}-2{x}^{2}y+{y}^{2}-2.$

Give two examples (other than those given in the text) of “real world” functions that require more than one input.

The graph of a function of two variables is a _____.

Most people are familiar with the concept of level curves in the context of _____ maps.

T/F: Along a level curve, the output of a function does not change.

The analogue of a level curve for functions of three variables is a level _____.

What does it mean when level curves are close together? Far apart?

Describe in your own words the difference between boundary and interior points of a set.

Use your own words to describe (informally) what $\displaystyle \lim_{(x,y)\to (1,2)} f(x,y) = 17$ means.

Give an example of a closed, bounded set.

Give an example of a closed, unbounded set.

Give an example of a open, bounded set.

Give an example of a open, unbounded set.

What is the difference between a constant and a coefficient?

Given a function $z=f(x,y)$, explain in your own words how to compute $f_x$.

In the mixed partial fraction $f_{xy}$, which is computed first, $f_x$ or $f_y$?

In the mixed partial fraction $\displaystyle \frac{\partial^2f}{\partial x\partial y}$, which is computed first, $f_x$ or $f_y$?

T/F: If $f(x,y)$ is differentiable on $S$, the $f$ is continuous on $S$.

T/F: If $f_x$ and $f_y$ are continuous on $S$, then $f$ is differentiable on $S$.