key
Let p 2 1,0 Then Pfp f l 2 5
X 2 2 gti SZ
Q
Php
h o means that a smallcharge in thepositive x
direction
won't change the altitude
h_yE o means that a smallchange in the positive y direction
will decrease the attitude
Let p no 2 E ThenXCpt 02Y fp ytzlp tt23gep xtz pTyCp12
2 P T 2p xty p
2 3 1,0 00504 0
ps.nop 12Fr p
o p EE
1 i i I
to Approach along the line y mx wherem E IR
E.im i eimoDN The limit does notexist
because the value depends on M theSlope of the line e gIf 10,0 is approached on the y o then the answer is 1 but if
approachedalong 5 x then the value is Lz Thus differentpathsproducedifferent values i.e the limit is path dependent and therefore doesnot
The expression 4 t x yexist
is defined at3 t X 3gSo direct substitution in
fin It I 4xg in
Define F Z 3 5 y 2 3 Then 3 F
zwere F 3 22 35 and Fz X 3g Z Thus
3 2z 3y222
f 2x 2g fy y 2x 3 f y 2 fyf 2 fyg 2g D 4g 4 4cg l
f and fy are continuous on 1122 so
ZX 2g omg 2 3 0
y x x 2 2 3 0
X 3 Xtc O x 3 or x I
3,3 or f l l
D 3,3 8 O and f o localminimum at 3,3
D 1 1 8 0 saddle at CI
fly g Xy e objective function gcx.us fx2ttzy2 i oconstraintfunction
f 77gTy x 7T x y
y Fx x ay y f ayy Ey o
y l E o y o or 7 1 2
If y o then X o so o o But g 0,0 0 So this
is extraneous
If 7 1 2 this y It x
If y then tfX t IX 1 and so x z and y i
and therefore t2 Il
If y Ix then IX t x and so 2 andy I 1
and therefore t2 Il
f tz ti It 2 ti 2 Absolute maximum of 2 at Iz l
f t2 Il I 2 Ii 2 Absolute minimum of 2 at t2 Il
t
f 4 3 32 9 23g
Pf f4X 2gOf 4 3 f 16 6
Let it f IsDef Tf I
file 6 of Is E6 4ft
Pf 4,3 f 16 6
f 16 6L
The plane 2 Stax toy has normalvector T f 6,10 1
Define F x Y Z Z 2 y Then the surface 2 1 x2tyzis equivalent to the implicit surface 1 19,21 1 This
surface has normal Vector TF If the target place to
Flayz L is to beparallel to 2 St Cox toy then DF must
be parallel to rt that is they must be scalarmultiples of one
another Hence
FF IFZ It 3 C sf 2x 2g 1 7 6,10 1
Thus 2 1 and so3 and y 5 andtherefore the point is 3 5,35
2 It 7 x l tog
O9 It 25in o 2 I l t 7 09 1 26.2I 0.7 to 4
0.7g