Graphics Reference
In-Depth Information
∂
∂
∂
∂
a
+
b
x
y
which, when applied to a real-valued function f(x,y), gives
∂
∂
f
x
∂
∂
f
y
(
)
+
(
)
a
xy
,
b
xy
,
.
For example,
2
2
2
2
∂
∂
∂
∂
∂
∂
f
∂
∂∂
f
xy
xy
∂
∂
f
Ê
Ë
ˆ
¯
()(
)
=
2
(
)
+
(
)
+
2
(
)
a
+
b
fxy
,
a
xy
,
2
ab
,
b
xy
,
.
x
y
2
2
x
y
Let f(x,y) be of class C
n
in a neighborhood of a point (x
0
,y
0
). Then
Definition.
k
n
1
∂
∂
∂
∂
È
Í
˘
˙
Â
(
)
=
(
)
+
(
)
(
)
()(
)
gxy
,
fx y
,
xx
-
+-
yy
fxy
,
00
0
0
n
!
x
y
k
=
1
is called the
Taylor polynomial of f of degree n at x
0
.
4.3.23. Example.
To find the Taylor polynomial g(x,y) of degree 2 at (1,2) for the
function
(
)
=25
3
2
fxy
,
x
xy
.
Solution.
Now
2
2
2
∂
∂
f
x
∂
∂
f
y
∂
∂
f
∂
∂∂
f
xy
∂
∂
f
2
=
6
x
,
=
10
xy
,
=
12
x
,
=
0
,
and
=
10
x
,
2
2
x
y
so that
1
2
12
[
]
2
2
(
)
=+ -
(
)
+
(
)
+
(
)
(
)
(
)
+
(
)
gxy
,
18
6
x
1
20
y
-
2
x
-
1
+◊◊ -
2 0
x
1
y
-
2
10
y
-
2
.
(The Taylor Polynomial Theorem) Let f(x,y) be of class C
n+1
4.3.24. Theorem.
in a
neighborhood of a point (x
0
,y
0
). Then
k
n
1
∂
∂
∂
∂
È
Í
˘
˙
Â
(
)
=
(
)
+
(
)
(
)
()(
)
+
fxy
,
fx y
,
xx
-
+-
yy
fxy R
,
,
00
0
0
n
+
1
n
!
x
y
k
=
1
where
n
+
1
1
∂
∂
∂
∂
È
Í
˘
˙
(
)
(
)
()(
)
R
=
xx
-
+-
yy
fx y
*,
*
n
+
1
0
0
(
)
n
+
1!
x
y
