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Midterm 2 Review
Edward Carter
April 13, 2005
1
Partial Derivatives
1.1
Things to Know
•
Deﬁnition of the limit of a multivariable function.
•
Deﬁnitions of partial derivatives.
•
Clairaut’s Theorem: Suppose
f
is deﬁned on a disk
D
that contains the point (
a, b
). If the functions
f
xy
and
f
yx
are both continuous on
D
, then
f
xy
(
a, b
) =
f
yx
(
a, b
).
•
Formula for tangent plane.
•
Deﬁnition of diﬀerentiability of a multivariable function.
•
If the partial derivatives
f
x
and
f
y
exist near (
a, b
) and are continuous at (
a, b
), then
f
is diﬀerentiable
at (
a, b
).
•
If
z
=
f
(
x, y
),
x
=
g
(
t
), and
y
=
h
(
t
), then
dz
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
.
•
If
z
=
f
(
x, y
),
x
=
g
(
s, t
), and
y
=
h
(
s, t
), then
∂z
∂s
=
∂z
∂x
∂x
∂s
+
∂z
∂y
∂y
∂s
and
∂z
∂t
=
∂z
∂x
∂x
∂t
+
∂z
∂y
∂y
∂t
.
•
If
u
=
f
(
x
1
, . . . , x
n
) and each
x
i
is a diﬀerentiable function of
t
1
, . . . , t
m
, then
∂u
∂t
i
=
n
X
j
=1
∂u
∂x
j
∂x
j
∂t
i
.
•
Let
f
be a function of
x
and
y
. The gradient of
f
at (
a, b
),
∇
f
(
a, b
), is deﬁned to be the vector
h
f
x
(
a, b
)
, f
y
(
a, b
)
i
. The deﬁnition is analogous for functions with more than two variables. The gradient
vector points in the direction of steepest ascent for
f
. The vector
∇
f
(
a, b
) is always perpendicular to
the level curve passing through (
a, b
).
1
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The directional derivative of a function of two variables is the derivative of the one variable function
you get by taking a slice of the two variable function along a certain direction. It can be computed
by taking the dot product of a unit vector for the direction you’re looking at and the gradient of the
function at that point.
•
A function
f
(
x, y, z
) of three variables has level surfaces rather than level curves. At any point (
a, b, c
),
the normal vector for the plane tangent to the level surface of
f
passing through (
a, b, c
) is given by
∇
f
(
a, b, c
).
•
At any local minimum or maximum of any function, the gradient either does not exist or is equal to
the zero vector. Points where the gradient vector does not exist or is equal to the zero vector are called
critical points, like in 1A.
•
Let
f
(
x, y
) have continuous partial derivatives on a disk centered at (
a, b
), and suppose
f
x
(
a, b
) =
f
y
(
a, b
) = 0. Let
D
=
f
xx
(
a, b
)
f
yy
(
a, b
)

[
f
xy
(
a, b
)]
2
. Then if
D >
0 and
f
xx
(
a, b
)
>
0, then (
a, b
) is a
local maximum for
f
. If
D >
0 and
f
xx
(
a, b
)
<
0, then (
a, b
) is a local minimum. If
D <
0, then (
a, b
)
is neither a local maximum nor a local minimum.
•
Recall from 1A that if you have a continuous function on a closed and bounded set, it must achieve
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 Fall '08
 Lanzara
 Physics

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