MATH 20C — MIDTERM #2 REVIEW SHEET SOLUTIONS
You are strongly encouraged to solve the problems yourself before reading the solutions here. You will
learn much more in that way and prepare better for the test.
This solutions sheet is not guaranteed to be free of errors; please let me know if you find one.
1.
Sketch the contour map of
f
(
x, y
) = 3
x
+ 2
y
with contour interval
m
= 2
.
For any
c
, the level curve at level
c
has equation 3
x
+ 2
y
=
c
, or
y
=

3
2
x
+
1
2
c
.
This is a line with
slope

3
2
and intercept
1
2
c
. Thus the contour curves with interval
m
= 2 will be lines with slope

3
2
and
intercepts 0, 1, 2,

1, etc.
2.
What do the vertical traces of the graph of the function
f
(
x, y
) = sin(
x/y
)
look like?
The trace in the plane
y
=
b
is
z
= sin(
x/b
), which is a sine curve scaled by a factor of
b
. The trace in
the place
x
=
a
is
z
= sin(
a/y
), which is a sine curve that stretches out towards infinity and becomes
infinitely oscillatory towards zero, as shown below.
3.
Sketch the contour maps of
f
(
x, y
) = sin(
x
2
+
y
2
)
with contour intervals
m
= 1
and
1
2
.
With contour interval 1 the only contour curves will be at levels

1, 0, and 1, since
f
(
x, y
) is always
between

1 and 1. Similarly, for contour interval
1
2
we have curves at levels

1,

1
2
, 0,
1
2
, and 1. Every
level curve consists of a sequence of concentric circles, since sin is periodic. The countour line at level
0, for instance, consists of the circles of radius
√
nπ
, for all integers
n
, since on those circles we have
x
2
+
y
2
=
nπ
which are the places where sin vanishes.
4.
True or false?
(a)
If the partial derivatives
f
x
and
f
y
both exist at a point
(
x, y
) = (
a, b
)
, then the function
f
is differ
entiable at
(
a, b
)
.
False.
(b)
If
lim
(
x,y
)
→
(0
,
0)
f
(
x, y
)
exists, and the onevariable limit
lim
x
→
0
f
(
x,
0) = 4
, then
lim
(
x,y
)
→
(0
,
0)
f
(
x, y
) =
4
.
True. If a twovariable limit exists, then it must also be the limit approaching that point along any
direction.
(c)
If for some
(
a, b
)
, the onevariable limits
lim
x
→
a
f
(
x, b
)
and
lim
y
→
b
f
(
a, y
)
both exist but are
not
equal, then
lim
(
x,y
)
→
(
a,b
)
f
(
x, y
)
does not exist.
True, for the same reason.
(d)
If
f
is differentiable at
(
a, b
)
and
f
x
(
a, b
) =
f
y
(
a, b
) = 0
, then the tangent plane to the graph of
f
at
(
a, b
)
is horizontal (parallel to the
xy
plane).
True, from the equation of the tangent plane to the graph of a differentiable function in terms of the
partial derivatives.
(e)
If
f
is differentiable and
f
x
(
a, b
) =
f
y
(
a, b
) = 0
, while
f
xx
(
a, b
) =
f
yy
(
a, b
) = 3
, then
f
has a local
minimum at
(
a, b
)
.
False, or at least not necessarily true. If
f
xy
(
a, b
) is sufficiently negative, then
f
might have a saddle
point.
(f)
A continuous function of two variables on a closed and bounded domain achieves a maximum value.
True.
(g)
A differentiable function of two variables is necessarily also continuous.
True.
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(h)
If
f
is continuous at
(
a, b
)
, then
lim
(
x,y
)
→
(
a,b
)
f
(
x, y
) =
f
(
a, b
)
.
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 Fall '07
 BoonYeap
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