Multivariable Functions

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

For the partial derivatives given in Exercises 51–54, find the

most general form for a function of two variables, , with

the given partial derivative

$\frac{{\partial }^{2}f}{\partial x^2}=0$

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

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.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function of two variables whose graph is a plane.

(b) A function of two variables whose graph is the surface of revolution formed when the graph of a function of a single variable is revolved around the y-axis.

(c) A function of two variables for which each level curve is a parabola.

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

Let f : R → R be a function of a single variable. Explain why the graph of f is a subset of $R^2$.

Let $f:R^2\to R$ be a function of two variables. Explain why the graph of f is a subset of $R^3$.

Let $f:R^3\to R$ be a function of three variables. Explain why the graph of f is a subset of $R^4$.

(a) Graph $f \left(x\right) = x$

(b) Graph the function $f (x, y) = x + y$. Explain why the graph of $f (x, y) = x + y$ contains the origin, $(0, 0, 0)$ 

(c) The graph of the function  $f (x, y, z) = x + y + z$ is a hyperplane of dimension $3$ in R4. Explain why “hyperplane” is a good name for the graph. (“Hyper” is from a Greek word meaning over or beyond. The Latin equivalent is super, meaning essentially the same thing.) Explain why the graph of $f$ contains the origin, $(0, 0, 0, 0)$

(d) Fill in the blanks: The graph of the function  $f (x_1, x_2, . . . , x_n) ={ x}_1 + x_2 +· · ·+x_n$ is a hyperplane of dimension in .

(a) Graph the function $f \left(x\right) = \sqrt{1 - x^2}$

(b) Graph the function $f (x, y) = \sqrt{1 - x^2 - y^2}$ (Hint: Let z =$\sqrt{1 - x^2 - y^2}$ and then square both sides of the equation.)

(c) The graph of the function f (x, y, z) =$\sqrt{1 - {x }^2- y^2 - z^2}$ is half of a hypersphere of dimension 3. Explain why “hypersphere” is a good name for the graph and why the graph is only half of the hypersphere. What equation

defines the entire hypersphere?

(d) Fill in the blanks: The graph of the function f (x 1, x 2, . . . , x n) =$\sqrt{1 - {x^2 }_1- {x^2 }_2-· · ·-{x^2}_n}$ is half of a .............of dimension......... in.........?

Find the range of the function $f\left(x,y\right)=-\frac{8x}{x^2+y^2+1}$from Example 4 by finding the largest value, M, and smallest value, m, of c for which the equation $-\frac{8x}{x^2+y^2+1}=c$ 

has a solution. (Hint: Find the two values for c such that the

level “curves” are a single point. Then show that the given

equation has a solution for every value of c between those two

values.)

Let z = f(x, y) be a function of two variables. Explain why the two sets {(x, y) | (x, y,z) ∈ Graph( f )} and Domain( f ) are identical.

Let z = f(x, y) be a function of two variables. Explain why the two sets 

{z | (x, y, z) ∈ Graph( f )} and Range( f ) are identical.

Let w = f(x, y, z) be a function of three variables. Explain why the two sets {(x, y, z) | (x, y, z, w) ∈ Graph( f )} and Domain( f ) are identical.

Let w = f(x, y, z) be a function of three variables. Explain why the two sets {w | (x, y, z, w) ∈ Graph( f )} and Range( f ) are identical.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

One level curve consists of a single point.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

One level curve consists of exactly two points.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

One level curve is a circle together with the point that is the center of the circle.

Provide a rough sketch of the graph of a function of two variables with the specified level "curve(s)." (There are many possible correct answers to each question.)
One level curve consists of all the points $$(m, n)$$

where \begin{equation} m \end{equation} and \begin{equation} n \end{equation} are both integers.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

All of the level curves are circles except one, which is a point.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

All of the level curves are circles.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

Some level curves consist of two concentric circles.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

Some level curves consist of infinitely many concentric circles.

In Exercise, provide a rough sketch of the graph of a function of two variables with the specified level “curve(s).”

All of the level curves are squares.

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$f\left(x,y\right)=x^2{y}^3,\left(\frac{1}{2},\frac{4}{3}\right),\left(0,\mathrm{\pi }\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$f\left(x,y\right)=x^2-y^2,\left(1,5\right),\left(-3,-2\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$g\left(x,y\right)=\frac{x^2+y^2}{x+y},\left(\mathrm{\pi },1\right),\left(4,5\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$g\left(x,y\right)=\frac{\ln \left(xy\right)}{\sqrt{1-x}},\left(-e,-1\right),\left(\frac{1}{2},2\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$f\left(x,y\right)=\frac{\sin \left(x-y\right)}{x-y},\left(0,\mathrm{\pi }\right),\left(\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{3}\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$f\left(x,y,z\right)=x^2+y^2+z^2,\left(1,0,-5\right),\left(\frac{1}{2},-1,\frac{1}{3}\right)$

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.

$f\left(x,y,z\right)=\frac{x+y+z}{xyz},\left(1,-2,6\right),\left(-3,-2,4\right)$

In exercise, 

Either simplify the specified composition or explain why the

composition cannot be formed.

$f_1\left(g_1\left(t\right),g_2\left(t\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

$g_1\left(f_1\left(x,y\right),f_2\left(x,y\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

$g_1\left(g_2\left(t\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

$f_1\left({\mathit{r}}_1\left(t\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

$f_3\left(g_2\left(t\right),g_1\left(t\right),g_3\left(t\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

$f_1\left(f_2\left(x,y\right),f_3\left(x,y,z\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

${\mathit{r}}_2\left(g_3\left(t\right)\right)$

In exercise, let

Either simplify the specified composition or explain why the composition cannot be formed.

${\mathit{r}}_2\left(f_3\left(x,y,z\right)\right)$

In Exercise, sketch the surface of revolution formed when the given function on the specified interval is revolved around the z-axis and find a function of two variables with the surface as its graph.

$f\left(x\right)=x,\left[0,3\right]$

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=\sqrt{x},[0,4]$

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=x^2 ,\left[0,2\right]$

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=x^{3/2} ,\left[0,1\right]$

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=\sin x ,\left[0,\frac{\pi }{2}\right]$

In Exercises 37–42, sketch the surface of revolution formed

when the given function on the specified interval is revolved

around the z-axis and find a function of two variables with the

surface as its graph.

$f\left(x\right)=\cos x ,\left[0,\frac{\pi }{2}\right]$

In Exercises 43–52, sketch the level curves$c=-3,-2,-1,0,1,2,3$

 if they exist for the specified function

$f(x, y)=x+y$

Sketch the level curves $c = -3, -2, -1, 0, 1, 2, 3$ if they exist for the specified function. $f (x, y) = \frac{3x}{y}$

Sketch the level curves $c = -3, -2, -1, 0, 1, 2, 3$ if they exist for the specified function. $f (x, y) = \frac{3x}{y^2}$

Sketch the level curves $c = -3, -2, -1, 0, 1, 2, 3$ if they exist for the specified function. $f (x, y) = \sqrt{xy}$