1
. a) Differentiating implicitly
yz
= ln(
x
+
z
) (assuming this equation defines
z
as a function
x
and
y
), find the rate of change of
z
in the direction of
x
axis,
when
x
= 0
, y
= 0
, z
= 1.
Answer
.
Differentiating both sides of
yz
= ln(
x
+
z
) (thinking of
z
as a
function of
x, y
) we get
y
∂z
∂x
=
1
x
+
z
(1 +
∂z
∂x
)
.
At
x
= 0
, y
= 0
, z
= 1
,
1
0 + 1
(1 +
∂z
∂x
) = 0 or
∂z
∂x
=

1
.
b) Show that if
u
(
t, x
) =
f
(
x
+
ct
) +
g
(
x

ct
)
,
then
u
tt
=
c
2
u
xx
.
Answer
. By chain rule,
u
t
(
t, x
) =
cf
0
(
x
+
ct
)

cg
0
(
x

ct
)
, u
x
(
t, x
) =
f
0
(
x
+
ct
) +
g
0
(
x

ct
)
,
u
tt
(
t, x
) =
c
2
f
00
(
x
+
ct
) +
c
2
g
00
(
x
+
ct
)
, u
xx
(
t, x
) =
f
00
(
x
+
ct
) +
g
00
(
x
+
ct
)
.
2
. The temperature at a point (
x, y
) is
T
(
x, y
) and
T
x
(5
,
1) = 2
, T
y
(5
,
1) = 3.
A bug crawls so that its position after
t
seconds is given by
x
= 1 +
t
2
, y
=
cos (
πt
). How fast is the temperature rising on the bug’s path in 2 seconds.
Answer
.
d
dt
T

t
=2
=
T
x
(5
,
1)
dx
dt

t
=2
+
T
y
(5
,
1)
dy
dt

t
=2
= 2(2
t
)

t
=2
+3(

sin
πt
)
π

t
=2
=
8
.
3
. A right circular cylinder is measured to have a radius of 10
±
0
.
02 inches
and a height of 6
±
0
.
01 inches. Estimate the error in the volume calculation
(the formula
V
=
πr
2
h
was used)
.
Answer
.
Differential of
f
(
r, h
) =
πr
2
h
is
dV
=
f
r
(
r, h
)
dr
+
f
h
(
r, h
)
dh
=
2
πrhdr
+
πr
2
dh.
The error
Δ
V
≈
f
r
(10
,
6)0
.
02 +
f
h
(10
,
6)0
.
01 = 2
π
·
10
·
6
·
0
.
02 +
π
10
2
0
.
01 = 3
.
4
π
4
. Write the equation of the tangent plane to the level surface
S
:
f
(
x, y, z
) =
ln
(
xy
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 '07
 KAMIENNY
 Math, Calculus, Derivative, Rate Of Change, Trigraph, answer.

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