Functions
This unit covers the following ideas. In preparation
for the quiz and exam, make sure you have a lesson plan
containing examples that explain and illustrate the following
concepts.
 Describe uses for, and construct graphs of, space curves
and parametric surfaces. Find derivatives of space curves, and
use this to find velocity, acceleration, and find equations of
tangent lines.
 Describe uses for, and construct graphs of, functions of
several variables. For functions of the form $z=f(x,y)$, this
includes both 3D surface plots and 2D level curve plots. For
functions of the form $w=f(x,y,z)$, construct plots of level
surfaces.
 Describe uses for, and construct graphs of vector fields.
 If you are given a description of a vector field, curve, or
surface (instead of a function or parametrization), explain how
to obtain a function for the vector field, or a parametrization
for the curve or surface.
You'll have a chance to teach your examples to your peers
prior to the exam.
Function Terminology
A function is a set of instructions
involving two sets (called the domain and codomain). A function
assigns to each element of the domain $D$ exactly one element in
the codomain $R$. We'll often refer to the codomain $R$ as the
target space or range. We'll write $$f\colon D\to R$$ when we want to
be explicit about the domain and range of $f$.
In this class,
we will study what happens when the domain and target space are
subsets of ${\RR}^n$ (Euclidean $n$space). In particular,
we will study functions of the form $$f\colon {\RR}^n\to
{\RR}^m,$$ when $m$ and $n$ are 3 or less. The number
$n$ is the dimension of the input vector (or number of inputs).
The number $m$ is the dimension of the output vector (or number
of outputs). Our goal is to understand uses for each type of
function and be able to construct graphs to represent the
function. We will focus most of our time this semester on two
and threedimensional problems. However, many problems in the
real world require a higher number of dimensions.
When you hear
the word “dimension”, it does not always
represent a physical dimension, such as length, width, or height.
If a quantity depends on 30 different measurements, then the
problem involves 30 dimensions. As a quick illustration, the
formula for the distance between two points depends on 6 numbers,
so distance is really a 6dimensional problem. As another
example, if a piece of equipment has a color, temperature, age,
and cost, we can think of that piece of equipment being
represented by a point in fourdimensional space (where the
coordinate axes represent color, temperature, age, and cost).
A pebble falls from a 64 ft tall building. Its height
(in ft) above the ground $t$ seconds after it drops is given by
the function $y=f(t)=6416t^2$. What are $n$ and $m$ when we
write this function in the form $f\colon {\RR}^n\to
{\RR}^m$? Construct a graph of this function. How many
dimensions do you need to graph this function?
Parametric Curves: $\vec f\colon \RR \to \RR^m$
See also the Vectors chapter. There's a lot more practice of this idea in 11.1.
You'll also find more practice in 13.1: 18.
See also Larson 10.2. You can also
find more practice in Larson 12.1 and 12.3.
A horse runs around an elliptical track. Its position
at time $t$ is given by the function $\vec r(t)=(2\cos t, 3\sin
t).$ We could alternatively write this as $x=2\cos t, y=3\sin
t$.
 What are $n$ and $m$ when we write this function in the
form $\vec r\colon {\RR}^n\to {\RR}^m$?
 Construct a graph of this function.
 Label a few points on your graph with the corresponding time $t$. Include an
arrow for the horse's direction.
 How many dimensions do you need to graph this
function?
Notice in the problem above that we placed a vector symbol
above the function name, as in $\vec r\colon {\RR}^n\to
{\RR}^m$. When the target space (codomain) is
2dimensional or larger, we place a vector above the function
name to remind us that the output is a vector, not just a single number.
See
Sage or
Wolfram
Alpha.
The text has more practice in 13.1:
18.
See also Larson
12.3.
Consider the pebble from
the problem above. The pebble's height was given by
$y=6416t^2$. We later find out that the pebble also has some horizontal velocity: it's moving at 3 ft/s to the right. If we let the origin be
the base of the 64 ft building, then the position of the pebble
at time $t$ is given by $\vec r(t) = (3t, 6416t^2)$.
 What are $n$ and $m$ when we write this function in the
form $\vec r\colon {\RR}^n\to {\RR}^m$?
 At what time does the pebble hit the ground (the height
reaches zero)? Construct a graph of the pebble's path from
when it leaves the top of the building until it hits the
ground.

Find the pebble's velocity and acceleration vectors
at $t=1$. Draw these vectors on your graph with their base
at the pebble's position at $t=1$.
 At what speed is the pebble moving when it hits the
ground?
In the next problem, we keep the input as just a single
number $t$, but the output is now a vector in $\RR^3$.
See
Sage or
Wolfram Alpha.
The text has more
practice in 13.1: 914.More
practice is in Larson 12.1:9–12, 21–24, 27–32.
A jet begins spiraling upwards to gain height. The
position of the jet after $t$ seconds is modeled by the
equation $\vec r(t)=(2\cos t, 2\sin t, t).$ We could
alternatively write this as $x=2\cos t,\, y=2\sin t,\, z=t$.
 What are $n$ and $m$ when we write this function in the
form $\vec r\colon {\RR}^n\to {\RR}^m$?
 Construct a graph of this function by picking several
values of $t$ and plotting the resulting points $(2\cos t,
2\sin t, t)$.
 Label a few points on your graph with the corresponding time $t$. Include an
arrow for the jet's direction.
 How many dimensions do you need to graph this
function?
A key general principle in the problems above (and from our discussion in class) is that to fully represent a function visually, you need to use as many dimensions as the sum of the dimension of the domain and the dimension of the codomain. We can also use an arrow or an animation (i.e., time) to represent one dimension of information.
The text has more practice in 13.1:
1922.More practice in Larson
12.2:23–30.
Again, a jet spirals upwards. Its position is at time $t$ is
$$\vec r(t)=(2\cos t, 2\sin t, t).$$
Graph this function.
 Find the first and second derivative of $\vec r(t)$.
 Compute the velocity and acceleration vectors at
$t=\pi/2$. Place these vectors on your graph with their tails
at the point corresponding to $t=\pi/2$.
 Give an equation of the tangent line to this curve at
$t=\pi/2$.
Here's an interesting parametric curve drawing a cartoon character. You can see that they can get pretty complicated.
Parametric Surfaces: $\vec f\colon \RR^2 \to \RR^3$
We now increase the number of inputs from 1 to 2. This will allow us
to graph many space curves at the same time.
The jet from
the problem above is actually
accompanied by several jets flying side by side. As all the
jets fly, they leave a smoke trail behind them. The smoke from one jet spreads outwards to mix with the
neighboring jet, so that it looks like the jets are leaving a
rather wide sheet of smoke behind them as they fly. The
position of two of the many other jets is given by $\vec
r_3(t)=(3\cos t, 3\sin t, t)$ and $\vec r_4(t)=(4\cos t,4\sin
t,t)$. A function which represents the smoke stream is $\vec
r(a,t)=(a\cos t, a\sin t, t)$ for $0\leq t\leq 4\pi$ and $2\leq
a\leq 4$.
 What are $n$ and $m$ when we write the function $\vec
r(a,t)=(a\cos t, a\sin t, t)$ in the form $\vec r\colon
{\RR}^n\to {\RR}^m$?
 Start by graphing the position of the three jets $\vec
r(2,t)=(2\cos t, 2\sin t, t)$, $\vec r(3,t)=(3\cos t, 3\sin
t, t)$ and $\vec r(4,t)=(4\cos t,4\sin t,t)$.
 Let $t=0$ and graph the curve $r(a,0)=(a,0,0)$ for
$a\in[2,4]$. Then repeat this for $t=\pi/2,\pi,3\pi/2$.
 Describe the resulting surface.
The function above is called a parametric surface.
Parametric surfaces are formed by joining together many
parametric space curves. Most of 3D computer animation is done
using parametric surfaces. Woody's entire body in Toy Story is a collection of parametric surfaces. Car companies create
computer models of vehicles using parametric surfaces, and then
use those parametric surfaces to study collisions. Often the
mathematics behind these models is hidden in the software
program, but parametric surfaces are at the heart of just about
every 3D computer model.
Consider the parametric surface $\vec r(u,v)=(u\cos v,
u\sin v, u^2)$ for $0\leq u\leq 3$ and $0\leq v\leq 2 \pi$.
Construct a graph of this function. To do so, let $u$ equal a
constant (such as 1, 2, 3) and then graph the resulting space
curve. Then let $v$ equal a constant (such as 0, $\pi/2$, etc.)
and graph the resulting space curve until you can visualize the
surface. [Hint: Think satellite dish.]
Functions of Several Variables: $f\colon \RR^n \to
\RR$
In this section we'll focus on functions of the form
$f\colon \RR^2\to\RR^1$ and $f\colon
\RR^3\to\RR^1$; we'll keep the output as a real
number. In the next problem, you should notice that the input is
a vector $(x,y)$ and the output is a number $z$. There are two
ways to graph functions of this type. The next two problems show
you how.
See Larson 13.1:33–40.
A computer chip has been disconnected from electricity
and sitting in cold storage for quite some time. The chip is
connected to power, and a few moments later the temperature (in
Celsius) at various points $(x,y)$ on the chip is measured.
From these measurements, statistics is used to create a
temperature function $z=f(x,y)$ to model the temperature at any
point on the chip. Suppose that this chip's temperature
function is given by the equation $z=f(x,y)=9x^2y^2$. We'll
be creating a 3D model of this function in this problem, so
you'll want to place all your graphs on the same $x,y,z$ axes.
 What is the temperature at $(0,0)$, $(1,2)$, and
$(4,3)$?
See 14.1: 14.
 If you let $y=0$, construct a graph of the temperature
$z=f(x,0) = 9x^20^2$, or just $z=9x^2$. In the $xz$ plane
(where $y=0$) draw this upside down parabola.
 Now let $x=0$. Draw the resulting parabola in the $yz$
plane.
 Now let $z=0$. Draw the resulting curve in the $xy$
plane.
 Once you've drawn a curve in each of the three coordinate
planes, it's useful to pick an input variable (either $x$ or
$y$) and let it equal various constants. So now let $x=1$ and
draw the resulting parabola in the plane $x=1$. Then repeat
this for $x=2$.
 Describe the shape. Add any extra features to your graph
to convey the 3D image you are constructing.
See 14.1: 3748.
See Larson 13.1:45–56.
We'll be using the same function $z=f(x,y)=9x^2y^2$
as the previous problem. However, this time we'll construct a
graph of the function by only studying places where the
temperature is constant. We'll create a graph in 2D of the
surface (similar to a
topographical map).
See 14.1: 1316 and 3136.
Which points in the plane have zero temperature?
Just let $z=0$ in $z=9x^2y^2$. Plot the corresponding
points in the $xy$plane, and write $z=0$ next to this
curve. This curve is called a level curve. As long as you
stay on this curve, your temperature will remain level, it
will not increase nor decrease.
 Which points in the plane have temperature $z=5$? Add
this level curve to your 2D plot and write $z=5$ next to
it.
 Repeat the above for $z=8$, $z=9$, and $z=1$. What's
wrong with letting $z=10$?
See 14.1: 3748.
 Using your 2D plot, construct a 3D image of the function
by lifting each level curve to its corresponding height.
A level curve of a function $z=f(x,y)$ is a curve in the
$xy$plane found by setting the output $z$ equal to a constant.
Symbolically, a level curve of $f(x,y)$ is the curve $c=f(x,y)$
for some constant $c$. A 2D plot consisting of several level
curves is called a contour plot of $z=f(x,y)$.
Consider the function $f(x,y)=xy^2$.
 Construct a 3D surface plot of $f$. [So just graph in 3D
the curves given by $x=0$ and $y=0$ and then try setting $x$
or $y$ equal to some other constants, like $x=1$, $x=2$,
$y=1$, $y=2$, etc.]
 Construct a contour plot of $f$. [So just graph in 2D the
curves given by setting $z$ equal to a few constants, like
$z=0$, $z=1$, $z=4$, etc.]
See 14.1: 4952.
Which level curve passes through the point $(2,2)$?
Draw this level curve on your contour plot.
Notice that when we graphed the previous two functions (of
the form $z=f(x,y)$) we could either construct a 3D surface plot,
or we could reduce the dimension by 1 and construct a 2D contour
plot by letting the output $z$ equal various constants. The next
function is of the form $w=f(x,y,z)$, so it has 3 inputs and 1
output. We could write $f\colon \RR^3\to\RR^1$. We
would need 4 dimensions to graph this function, but graphing in
4D is not an easy task. Instead, we'll reduce the dimension and
create plots in 3D to describe the level surfaces of the
function.
See
Sage. Wolfram Alpha currently does not support drawing
level surfaces. You could also use Mathematica or
Wolfram Demonstrations.
You can access
more problems on drawing level surfaces in 12.6:144 or
14.1:5360.
See Larson 11.6 and 13.1:69–74, as
well as 13.1, Example 6.
Suppose that an explosion occurs at the origin
$(0,0,0)$. Heat from the explosion starts to radiate outwards.
Suppose that a few moments after the explosion, the temperature
at any point in space is given by $w=T(x,y,z)=100x^2y^2z^2.$
 Which points in space have a temperature of 99? To answer
this, replace $T(x,y,z)$ by $99$ to get $99=100x^2y^2z^2$.
Use algebra to simplify this to $x^2+y^2+z^2=1$. Draw this
object.
 Which points in space have a temperature of 96? of 84?
Draw the surfaces.
 What is your temperature at $(3,0,4)$? Draw the level
surface that passes through $(3,0,4)$.
 The 4 surfaces you drew above are called level surfaces.
If you walk along a level surface, what happens to your
temperature?
 As you move outwards, away from the origin, what happens
to your temperature?
Talk about graphing functions with 4 or more variables. Show
the class
OsiriX as
an example of graphing a 4d function (where opacity is the
density of material. Also, practice sliding a plane through a
3d object to get an idea of what the contour plots are telling
us.
See
Sage.
See Larson 11.6:7–16.
Consider the function $w=f(x,y,z)=x^2+z^2$. This
function has an input $y$, but notice that changing the input
$y$ does not change the output of the function.
 Draw a graph of the level surface $w=4$. [When $y=0$ you
can draw one curve. When $y=1$, you should draw the same
curve. When $y=2$, again you draw the same curve. This kind
of graph is called a cylinder, and is important in
manufacturing where you extrude an object through a
hole.]
 Graph the surface $9=x^2+z^2$ (so the level surface
$w=9$).
 Graph the surface $16=x^2+z^2$.
Most of our examples of function of the form $w=f(x,y,z)$
can be drawn by using our knowledge about conic sections. We can
graph ellipses and hyperbolas if there are only two variables. So
the key idea is to set one of the variables equal to a constant
and then graph the resulting curve. Repeat this with a few
variables and a few constants, and you'll know what the surface
is. Sometimes when you set a specific variable equal to a
constant, you'll get an ellipse. If this occurs, try setting that
variable equal to other constants, as ellipses are generally the
easiest curves to draw.
See
Sage.
Remember you can find more
practice in 12.6:144 or 14.1: 5364.
See Larson 11.6 and 13.1:69–74, as
well as 13.1, Example 6.
We'll have a few people present this problem.
Consider the function
$w=f(x,y,z)=x^2y^2+z^2$.
 Draw a graph of the level surface $w=1$. [You need to
graph $1=x^2y^2+z^2$. Let $x=0$ and draw the resulting
curve. Then let $y=0$ and draw the resulting curve. Let
either $x$ or $y$ equal some more constants (whichever gave
you an ellipse), and then draw the resulting ellipses.]
 Graph the level surface $w=4$. [Divide both sides by $4$
(to get a 1 on the left) and the repeat the previous
part.]
 Graph the level surface $w=1$. [Try dividing both sides
by a number to get a 1 on the left. If $y=0$ doesn't help,
try $y=1$ or $y=2$.]
 Graph the level surface that passes through the point
$(3,5,4)$. [Hint: what is $f(3,5,4)$?]
Vector Fields: $\vec f\colon
\RR^n\to\RR^n$
Now let's consider functions from $\RR^2$
to $\RR^2$ and functions from $\RR^3$ to
$\RR^3$. Depending on the application, we may view these
functions as either vector fields or transformations of 2d or 3d space. We'll study vector fields in this section and transformations in the next chapter.
A vector field $\vec F(x,y)=(M,N)$ is
visualized by putting the vector $(M,N)$ at the point $(x,y)$, so
you end up with a picture with a lot of vectors. These are often
used to visualize flows of air or liquids. For example, see
VisLab,
the NOAA visualizations,
Tim
Urness's research page, or the International
Visualization Challenge. There are also lots of other
visualizations challenges. Figuring out how to convey information
in an asthetic and effective way is a tough challenge.
See
Sage or
Wolfram
Alpha.
The computer shrinks the vectors by the same amount so they don't overlap.
See
16.2: 3944 for more practice.
See Larson 15.1:1–19.
Consider the vector field $\vec F(x,y)=(2x+y,x+2y)$. In
this problem, you will construct a graph of this vector field
by hand.
 Compute $\vec F(1,0)$. Then draw the vector $F(1,0)$ with
its base at $(1,0)$.
 Compute $\vec F(1,1)$. Then draw the vector $F(1,1)$ with
its base at $(1,1)$.
 Repeat the above process for the points $(0,1)$,
$(1,1)$, $(1,0)$, $(1,1)$, $(0,1),$ and $(1,1)$.
Remember, at each point draw a vector.
Change the Sage code in the link in the last problem to check your work. See 16.2:
3944 for more practice.
See Larson 15.1:1–19.
Consider the vector field $\vec F(x,y)=(y,x)$.
Construct a graph of this vector field. Remember, the key to
plotting a vector field is: at the point $(x,y)$, draw
the vector $\vec F(x,y)$ with its base at $(x,y)$.
Plot at least 8 vectors (a few in each quadrant), so we can see
what this field is doing.
Ask me in class about other ways to visualize 2d vector fields, like streamline plots.
We can also visualize 3d vector fields like $\vec
F(x,y,z)=(y,z,x)$ by plotting a grid of 3d vectors in $\RR^3$. Here's an example.
How do we get the equation of a vector field from a description of what the field does? The following problem will help you
develop the gravitational vector field.
Use
Sage to plot your vector fields.
See
16.2: 3944 for more practice. See Larson 15.1:1–19.
Do the following:
 Let $P=(x,y,z)$ be a point in space. At the point $P$,
let $\vec F(x,y,z)$ be the vector which points from $P$ to
the origin. Give a formula for $\vec F(x,y,z)$.
 Give an equation of the vector field where at each point
$P$ in the plane, the vector $\vec F_2(P)$ is a unit vector
that points towards the origin.
 Give an equation of the vector field where at each point
$P$ in the plane, the vector $\vec F_3(P)$ is a vector of
length 7 that points towards the origin.
 Give an equation of the vector field where at each point
$P$ in the plane, the vector $\vec G(P)$ points towards the
origin, and has a magnitude equal to $1/d^2$ where $d$ is the
distance to the origin.
Summary
We've covered the following types of functions
in the problems above.
Domain/Range  Example  Description 
$f\colon \RR\to\RR$  $y=f(x)$  singlevariable function, like in calculus 1 
$f\colon \RR\to\RR^2$  $\vec r(t)=(x,y)$  2D parametric curves 
$f\colon \RR\to\RR^3$  $\vec r(t)=(x,y,z)$  3D parametric curves 
$f\colon
\RR^2\to\RR^3$  $\vec r(u,v)=(x,y,z)$  parametric surfaces 
$f\colon \RR^2\to\RR$  $z=f(x,y)$  functions of two variables 
$f\colon \RR^3\to\RR$  $w=f(x,y,z)$  functions of three variables 
$\vec F\colon \RR^2\to\RR^2$  $\vec F(x,y)=(u,v)$  2D vector fields 
$\vec F\colon \RR^3\to\RR^3$  $\vec F(x,y,z)=(u,v,w)$  3D vector fields 