1
. (a) The dimensions
x,y,z
of a rectangular box are measured as 70 cm,
50 cm and 40 cm respectively with a possible error of 0.1 in each dimension.
Use diﬀerentials to estimate the maximum error in the calculated volume
V
=
xyz
of the box.
Answer
. The diﬀerential
dV
=
V
x
dx
+
V
y
dy
+
V
z
dz
=
yz
(
dx
) +
xz
(
dy
) +
xy
(
dz
) and the calculated error
Δ
V
≈
yz
(Δ
x
) +
xz
(Δ
y
) +
xy
(Δ
z
) = 50
·
40(0
.
1) + 70
·
40(0
.
1) + 70
·
50(0
.
1)
= 830 (cm
3
)
.
(b) Find the linearization of
f
(
x,y
) = ln(
x
2

3
y
) at (1
,
0) and use it to
approximate
f
(1
.
1
,

0
.
1). Write the equation of the tangent plane to
z
=
ln(
x
2

3
y
) at (1
,
0
,
0).
Answer
. Linearization
L
(
x,y
) =
f
(1
,
0) +
f
x
(1
,
0)(
x

1) +
f
y
(1
,
0)
y.
We ﬁnd
f
x
=
2
x
x
2

3
y
,f
y
=

3
x
2

3
y
,
f
(1
,
0) = 0
,f
x
(1
,
0) = 2
,f
y
(1
,
0) =

3
,
and
L
(
x,y
) = 2(
x

1)

3
y.
The approximation
f
(1
.
1
,

0
.
1)
≈
L
(1
.
1
,

0
.
1) =
2(0
.
1)

3(

0
.
1) = 0
.
5
.
The equation of the tangent plane is