Week 10 Small Class
Topics: Sketching functions of two variables
Learning Objectives
•
Plot points on the
x
-
y
-
z
axes.
•
Given a function of two variables sketch level curves by setting
z
to be a constant.
•
Given a relation between three variables, sketch traces by setting one variable to be a constant.
A. Problems and takeaways
(Plotting in 3D):
1. Fold a lose piece of paper to represent the
x
-
y
-
z
axes. (Note: Pictorial instructions are at the end of
this document). Draw lines in the creases to represent the
x
axis,
y
axis, and
z
axis.
2. Plot the points (1
,
0
,
0), (0
,
1
,
0), and (0
,
0
,
1), marking them with dots.
Once teams are done with
those points have them plot the additional points (2
,
2
,
0), (2
,
0
,
2), and (0
,
2
,
2).
3. Using a pen or other small object represent the point (3
,
2
,
1) in space, for example, by holding the
base of the pen at the origin and the writing end at the point.
B. Problems and takeaways
(Sketching level curves and traces):
1. Consider
z
=
x
2
+
y
2
. Set
z
= 4 and plot the resulting equation in 2D. Do the same with
z
= 1,
z
= 0,
and
z
=
−
1.
2.
Definition
: A
level curve
is 2D horizontal slice of a larger 3D surface.
We find these by setting
z
equal to a constant.
3.
Takeaway
: We can get a good sense of what a function or surface in 3D looks like by plotting level
curves.
4. So far, we know our surface doesn't have points for
z <
0, is a single point at
z
= 0 and is a tower of
increasingly larger circles for
z >
0. How do the circles change? Is our surface (a) a bowl (b) a cone
(c) a wormhole?
5. To get a sense of which it is, let's plot 2D curves while fixing values of
x
and
y
this time. Sketch when
y
= 0,
x
= 0,
x
= 1, and
y
= 2.
6.
Definition
: A
trace
is just like a level curve only we can set any variable equal to a constant.
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