MATH 20C  Monday, October 18, 2010: first midterm
MATH 20C Lecture 11  Wednesday, October 20, 2010
Functions of several variables
Recall:
for a function of 1 variable, we can plot its graph, and the derivative is the slope of
the tangent line to the graph.
Plotting graphs of functions of 2 variables:
examples
z
=

y,
z
= 1

x
2

y
2
,
using slices by the coordinate planes.
(derived carefully).
Contour map: level
curves
f
(
x, y
) =
c.
Amounts to slicing the graph by horizontal planes
z
=
c.
Showed 2 examples from “real life”: a topographical map, and a temperature map, then did
the examples
z
=

y
and
z
= 1

x
2

y
2
.
Showed more examples of computer plots (
z
=
y
2

x
2
,
and another one).
Contour map gives some qualitative info about how
f
varies when we change
x, y.
(shown an
example where increasing
x
leads
f
to increase).
MATH 20C Lecture 12  Friday, October 22, 2010
Reviewed countour maps and level curves.
Limits
By substitution:
lim
(
x,y
)
→
(0
,
0)
3
x
2
+ 2
y

5 cos(4(
x
+
y
))
xy

e
x

y
=
0 + 0

5
0

1
= 5
Disclaimer: limits do not always exist!
For instance, take lim
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 Spring '08
 Helton
 Derivative, Slope, lim, Contour Map

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