2.
Partial Differentiation
2A. Functions and Partial Derivatives
2A1
In the pictures below, not d l of the level curves are labeled. In (c) and (d), the
picture is the same, but the labelings are different. In more detail:
b) the origin is the level curve 0; the other two unlabeled level curves are .5 and 1.5;
c) on the left, two level curves are labeled; the unlabeled ones are 2 and 3; the origin is
the level curve 0;
d) on the right, two level curves are labeled; the unlabeled ones are
1
and 2; the origin
is the level curve
1;
The crude sketches of the graph in the first octant are at the right.
2A3
a) both sides are mnxm' yn'
xY
.
x
(Y
b)
f,
=
fx,
+
=
(fx),
=
'

f,

x)
+
=

+
f y x
=
(x
yI2
(x
yI3
'
(x
yI2
'
(x
+
Y ) ~
c)
f,
=
2xsin(x2+y),
f,,
=
(f,),
=
2xcos(x2 +y);
f,
=
sin(x2+y),
f,,
=
cos(x2+y).2x.
d) both sides are fl(x)g'(y).
2A4
(f,),
=
ax
+
6y,
(f,),
=
22
+
6y; therefore f,,
=
f,,
H
a
=
2.
By
inspection,
one sees that if a
=
2,
f
(x, y)
=
x2y
+
3xy2 is a function with the given
f,
and f,.
2A5
a) w,
=
aeax sin ay,
w,,
=
a2eax
sin ay;
W,
=
eaxa cos ay,
w,,
=
eaxa2
(
sin ay);
therefore wyy
=
w,,.
22
2(y2

x2)
b) We have w,
=

+
WXX
=
+
If we interchange x and y, the function
x2
y2
'
(x2
y2)2
'
w
=
ln(x2
+
y2) remains the same, while w,,
gets turned into w,,;
 
since the interchange
just changes the sign of the right hand side, it follows that w,,
=
w,,.
2B. Tangent Plane; Linear Approximation
2B1
a) z,
=
y2, z,
+
=
2xy; therefore at (1,1,1), we get
z,
=
1,
z,
=
2, so that the
tangent plane is z
=
1
(x

1)
+
2(y

I), or z
=
x
+
2y

2.