91Ó°ÊÓ

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.

A function $f$of two or three variables has a stationary point at a point $P$.

Letf be a function of two variables that is continuous everywhere.

(a) Explain why the function $\frac{f(x,y)}{x-y}$is continuous if and only if $x≠y$.

(b) Use Definition $12.15$ to explain why $\underset{(x,y)→(a,a)}{\lim }\frac{f(x,y)}{x-y}$ does not exist for any real numbera.

Chain rule: If $$f$$ is a function of $$x$$ and $$x$$ is a function of $$t$$, how is the chain rule used to find the rate of change of $$f$$ with respect to $$t$$?

First-Derivative Test: Review the first-derivative test for functions of a single variable. Explain how the test works, what conditions a function must satisfy to make the test useful, and when, if ever, the test might fail.

Optimizing a function of two variables subject to a constraint: If you wish to find the maximum and minimum of the function $$f(x, y) = xy$$ subject to the constraint $$x^{2}+4y^{2}=16$$, you may eliminate one variable of $$f$$ by solving the constraint equation for either $$x$$ or $$y$$ and rewriting $$f$$ in terms of a single variable. Do this and then use the techniques of Chapter 3 to find the maximum and minimum of the resulting function.

A Double Summation: Let $\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}f(i,j)=\underset{i=1}{\overset{m}{∑}}\left(\underset{j=1}{\overset{n}{∑}}f(i,j)\right)$

Evaluate:$\underset{i=1}{\overset{15}{∑}}\underset{j=1}{\overset{10}{∑}}i{j}^2.$

The gradient at a maximum: If a function of two variables, $$f(x, y)$$, is differentiable at a point $$f(x_{0}, y_{0})$$ where the function has a maximum, what is $$\bigtriangledown f(x_{0}, y_{0})$$ ?

A kind of derivative for a function of two variables: Explain why the derivative of the function $$\dfrac{3x}{y^{4}}$$ is $$\dfrac{3}{y^{4}}$$ if $$x$$ is the variable and $$y$$ is a constant and is $$\dfrac{-12x}{y^{5}}$$ if $$y$$ is the variable and $$x$$ is a constant. What is the derivative if both $$x$$ and $$y$$ are constants?

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The domain of a function of two variables is a subset of R2.

(b) True or False: The range of a function of two variables is a subset of R2.

(c) True or False: The graph of a function of two variables is a subset of R3.

(d) True or False: The domain of a function of three variables is a subset of R.

(e) True or False: The range of a function of three variables is a subset of R.

(f) True or False: The graph of a function of three variables is a subset of R4.

(g) True or False: The graph of a linear function of two variables is a plane.

(h) True or False: If a function f : R → R is continuous on an interval [0, p] then the surface formed when the graph of f is rotated about the y-axis may be

expressed as a function of two variables.

Explain why Definition 0.2 is not general enough to define the domain or range of a function of two or three variables.