chapter 5 partial derivatives introduction small increments & rates of change implicit functions...
TRANSCRIPT
![Page 1: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/1.jpg)
CHAPTER 5PARTIAL DERIVATIVES
INTRODUCTIONSMALL INCREMENTS & RATES OF CHANGE
IMPLICIT FUNCTIONSCHAIN RULE
JACOBIAN FUNCTIONHESSIAN FUNCTIONSTATIONARY POINT
![Page 2: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/2.jpg)
INTRODUCTION
• Consider the following functions where
are independent variables.
• If we differentiate f with respect variable , then we assume thati. as a single variableii. as constants
nxxxf ,...,, 21
nxxx ,...,, 21
ix
ix
nii xxxxx ,...,,,..., 1121
![Page 3: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/3.jpg)
NotationIf • First order partial derivatives:
• Second order partial derivatives:
,, yxff
y
f
x
f
and
x
f
yxy
fy
f
xyx
fy
f
yy
fx
f
xx
f
2
2
2
2
2
2
![Page 4: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/4.jpg)
Example 1Write down all partial derivatives of the following function
xyyxyxyxf ln2cos2, 233
![Page 5: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/5.jpg)
Example 1Write down all partial derivatives of the following function
SolutionFirst order PD
xyyxyxyxf ln2cos2, 233
xyyxyxy
fx
yyyx
x
f
ln22sin43
2cos23
23
232
![Page 6: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/6.jpg)
Second order PD
xyxyx
xyyxyxy
y
f
yy
fx
yxy
x
yyyx
x
x
f
xx
f
ln22cos86
ln22sin43
6
2cos23
3
23
2
2
2
23
232
2
2
![Page 7: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/7.jpg)
Second order PD (mixed partial)
x
yyyx
x
yyyx
y
x
f
yxy
fx
yyyx
xyyxyxx
y
f
xyx
f
22sin49
2cos23
22sin49
ln22sin43
22
232
2
22
23
2
![Page 8: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/8.jpg)
In example 1, we observed that
This properties hold for all functions provided that certain smoothness properties are satisfies.
The mixed partial derivative must be equal whenever f is continuous.
xy
f
yx
f
22
![Page 9: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/9.jpg)
Example 2Write down all partial derivatives of the following functions:
yxxyxxeyxfii
xxyexyxfixy
y
2
342
sin,235,
![Page 10: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/10.jpg)
Solution 2
340340
801210
3206310
235,
42
42
422
24
2
2
4224
342
yy
yy
yy
y
xexy
fxe
yx
f
exy
fxe
x
f
xexy
fxyxe
x
f
xxyexyxfi
![Page 11: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/11.jpg)
Solution 2
xxyxxyyxxeexyxy
f
xxyxxyyxxeexyyx
f
xyxexy
f
yxyyxyxyyeexyx
f
xxyxexy
f
xyxyxyexyex
f
yxxyxxeyxfii
xyxy
xyxy
yx
xyxy
xy
xyxy
xy
2cos2sin2
2cos2sin2
sin
2cos2sin2
cos
2cos
sin,
222
222
332
2
222
2
222
2
![Page 12: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/12.jpg)
SMALL INCREMENTS & RATES OF CHANGE
Notation for small increment is Let then
i. A small increment in z, is given by
Where are small increments of the stated variablesii. Rate of change z wrt time, t is given by
,,...,, 21 nxxxfz
.
nxxx ,...,, 21
z
nn
xx
zx
x
zx
x
zz
...2
21
1
t
x
x
z
t
x
x
z
t
x
x
z
t
z n
n
...2
2
1
1
![Page 13: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/13.jpg)
Example 3
The measurements of closed rectangular box are length, x = 5m, width y = 3m, and height, z = 3.5m, with a possible error of in each measurement.
What is the maximum possible error in the calculated value of the volume, V and the surface, S area of the box?
cm10
1
![Page 14: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/14.jpg)
Solution 3
Volume of rectangular box:Possible error of the volume:
xyzV
x
y
z
V
zz
Vy
y
Vx
x
VV
![Page 15: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/15.jpg)
Solution 3
Possible error of the volume:
xyz
Vxz
y
Vyz
x
VxyzV
43000
1.0351.05.351.05.33
zxyyxzxyz
zz
Vy
y
Vx
x
VV
![Page 16: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/16.jpg)
Solution 3
Surface Area of rectangular box:Possible error of the volume:
yzxzxyA 222
x
y
z
A
zz
Ay
y
Ax
x
AA
![Page 17: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/17.jpg)
Solution 3
Possible error of the surface area:
yxz
Azx
y
Azy
x
AyzxzxyA
222222
222
4601.032521.05.32521.05.3232
222222
zyxyzxxzy
zz
Ay
y
Ax
x
AA
![Page 18: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/18.jpg)
Example 4
The radius r of a cylinder is increasing at the rate of 0.2cms-1 while the height, h is increasing at 0.5cms-1. Determine the rate of change for its volume when r=8cm and h =12cm.
![Page 19: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/19.jpg)
Solution 4Volume of cylinder:
Rate of change:
hrV 2
3
1
t
r
r
V
t
h
h
V
t
V
h
r
125.0
82.0
?,
hdt
dh
rdt
drt
V
(1)
(2)
![Page 20: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/20.jpg)
Solution 4From (1)Differentiate partially wrt t:
Substitute in (2):
2
3
1
3
2r
h
Vrh
r
V
4.70
2.01283
25.08
3
1
2.03
25.0
3
1
2
2
rhrt
V
![Page 21: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/21.jpg)
IMPLICIT FUNCTIONS
DefinitionLet f be a function of two independent variables x and y, given by
constant. To determine the derivative of this implicit function:LetHence,
ccyxf ,,
yz
xz
dx
dy
dy
dy
y
z
dx
xd
x
z
x
z0
dx
dy
., cyxfz
![Page 22: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/22.jpg)
Example 5Assume that y is a differentiable of x that satisfies the given function. Find using implicit differentiation.
2sin
323
2
342
yxxyxxeii
xxyexi
xy
y
dx
dy
![Page 23: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/23.jpg)
Solution 5(i) Let
Then,
Therefore ,
323 342 xxyexz y
xexy
z
xyxex
z
y
y
34
632
42
24
xex
xyxe
dx
dyy
y
34
63242
24
![Page 24: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/24.jpg)
THE CHAIN RULE
DefinitionLet z be a function of two independent variables x and y, while x and y are functions of two independent variables u and v.
The derivatives of z with respect to u and v as follows: Hence,
dv
dy
y
z
dv
xd
x
z
v
z
du
dy
y
z
du
xd
x
z
u
z
![Page 25: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/25.jpg)
Example 6Let , where and
Find and
yexyz x22 vvux 32 2 uuvy 32
u
z
v
z
![Page 26: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/26.jpg)
Solution 6
yexyz x22 x
x
exy
z
yeyx
z
2
2
2
22
vvux 32 2
32
4
2
uv
x
uvu
x
uuvy 322
3
6
12
uvv
y
vu
y
![Page 27: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/27.jpg)
Solution 6Therefore
3222232223
2222
322233223
322
22
22
322613222623222
32212128122422
uuv
xx
uuv
xx
euuveuuuvuvexuyey
v
y
y
z
v
x
x
z
v
z
euveuuvuvvexuvyey
u
y
y
z
u
x
x
z
u
z
![Page 28: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/28.jpg)
JACOBIAN FUNCTION
DefinitionLet be n number of functions of n variables
nfff ...,,, 21 nxxx ...,,, 21
nnn
n
n
xxxff
xxxffxxxff
,...,,
,...,,,...,,
21
2122
2111
![Page 29: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/29.jpg)
JACOBIAN FUNCTION
Jacobian for this system of equations is given by:
OR
n
n
nn
n
n
x
f
x
f
x
f
x
f
x
f
x
fx
f
x
f
x
f
J
21
22
2
2
1
11
2
1
1
n
nnn
n
n
x
f
x
f
x
f
x
f
x
f
x
fx
f
x
f
x
f
J
11
2
2
2
1
2
1
2
1
1
1
![Page 30: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/30.jpg)
Example 7Given and , determine the Jacobian for the system of equation.
yeu x 2sin2 yev x 2cos2
![Page 31: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/31.jpg)
Solution 7Given and , determine the Jacobian for the system of equation.
yeu x 2sin2 yev x 2cos2
y
yyyy
yeye
yeyey
v
y
ux
v
x
u
J
xx
xx
4cos42cos2sin42cos42sin4
2sin22cos2
2cos22sin2
22
22
22
22
![Page 32: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/32.jpg)
INVERSE FUNCTIONS FOR PARTIAL DERIVATIVES
DefinitionLet u and v be two functions of two independent variables x and y. . . Partial derivatives and are given by:
yxfvyxfu ,,, ,,,v
x
u
y
u
x
v
y
Jxu
v
y
J
yu
v
x
Jxv
u
y
J
yv
u
x
![Page 33: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/33.jpg)
Example 8Given and , Find and
yeu x 2sin2 yev x 2cos2
,,,v
x
u
y
u
x
v
y
![Page 34: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/34.jpg)
Solution 8Given and , Find and
yeu x 2sin2 yev x 2cos2
,,,v
x
u
y
u
x
v
y
y
ye
v
y
y
ye
v
x
y
ye
u
y
y
ye
u
x
xx
xx
4sin2
2sin
4sin2
2cos
4sin2
2cos
4sin2
2sin
22
22
![Page 35: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/35.jpg)
Example 9Let and Find and
2223 43,3 vuxyxyxz
u
z
v
z
.52 vuy
![Page 36: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/36.jpg)
HESSIAN FUNCTION
DefinitionLet f be a function of n number of variables . Hessian of f is given by the following determinant:
2
2
2
22
1
12
2
2
22
22
12
12
1
2
21
22
21
12
n
n
nn
n
n
n
n
x
f
xx
f
xx
f
xx
f
x
f
xx
fxx
f
xx
f
x
f
H
nxxx ...,,, 21
![Page 37: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/37.jpg)
HESSIAN FUNCTION
Hessian of a function of 2 variables:Let f be a function of 2 independent variables x and y. Then the Hessian of f is given by:
22
2
2
2
2
2
22
2
2
2
yx
f
y
f
x
f
y
f
xy
fyx
f
x
f
H
![Page 38: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/38.jpg)
HESSIAN FUNCTION
Stationary PointDefinition Given a function . The stationary point of occurs when and
Properties of Stationary Point
22
2
2
2
2
2
22
2
2
2
yx
f
y
f
x
f
y
f
xy
fyx
f
x
f
H
yxff , yxff ,0
x
f0
y
f
![Page 39: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/39.jpg)
HESSIAN FUNCTION
Properties of Stationary Pointa) If H<0, then stationary point is a SADDLE
POINT
b) If H>0a) MAXIMUM POINT if
b) MINIMUM POINT if
c) If H=0, then TEST FAILS or NO CONCLUSION
0,02
2
2
2
y
f
x
f
0,02
2
2
2
y
f
x
f
![Page 40: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/40.jpg)
Example 10Find and classify the stationary points of
223 23, yxyxxyxf
![Page 41: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/41.jpg)
Solution 10Find stationary point(s):
1022
22
10263
2632
2
yxyx
yxy
f
yxx
yxxx
f
![Page 42: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/42.jpg)
Substitute (2) in (1)
Stationary points:
3
4,0
3
4,0
0430430263
2
2
xx
yy
yyyyyyy
3
4,
3
4,0,0
![Page 43: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/43.jpg)
Find the Hessian function:
4662
2
2
66
2
2
2
2
2
xH
yx
fy
f
xx
f
![Page 44: CHAPTER 5 PARTIAL DERIVATIVES INTRODUCTION SMALL INCREMENTS & RATES OF CHANGE IMPLICIT FUNCTIONS CHAIN RULE JACOBIAN FUNCTION HESSIAN FUNCTION STATIONARY](https://reader036.vdocuments.us/reader036/viewer/2022081420/5697c0151a28abf838ccdd86/html5/thumbnails/44.jpg)
Determine the properties of SP:
Point Hessian: Conclusion
SP is a maximum point
SP is a saddle point
0,0
3
4,
3
4
4662 xH
02
06
08
2
2
2
2
y
fx
fH
08 H