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18 160 "Helvetica-BoldOblique" 18}{SCP_R 1 0 92{COM_R 2 0{TEXT 1 
73 "STATIONARY VALUES\015Alastair Rough: ROUGH@MANADON-ENGINEERIN
G-COLLEGE.AC.UK"}}{COM_R 3 0{TEXT 1 0 ""}}{COM_R 4 0{TEXT 1 3 "AI
M"}}{COM_R 5 0{TEXT 1 0 ""}}{COM_R 6 0{TEXT 1 90 "The aim of this
 tutorial is to reinforce the lectures on maxima, minima and sadd
le points."}}{COM_R 7 0{TEXT 1 0 ""}}{COM_R 8 0{TEXT 1 0 ""}}
{COM_R 9 0{TEXT 1 10 "EXERCISE 1"}}{COM_R 10 0{TEXT 1 0 ""}}
{COM_R 11 0{TEXT 1 40 "Find the turning points of the function:"}
}{INP_R 12 0 "> "{TEXT 0 37 "z1:= x^2 + x*y + y^2 + 5*x - 5*y + 3
;"}}{COM_R 13 0{TEXT 1 0 ""}}{COM_R 14 0{TEXT 1 41 "First define 
and plot the function using:"}}{INP_R 15 0 "> "{TEXT 0 33 "plot3d
(z1, x=-10..10, y=-10..10);"}}{COM_R 16 0{TEXT 1 0 ""}}{COM_R 17 
0{TEXT 1 69 "Now identify stationary points by solving the simult
aneous equations:"}}{COM_R 18 0{TEXT 1 23 "To solve the equations
:"}}{INP_R 19 0 "> "{TEXT 0 53 "exp1:= solve(\{diff(z1,x) = 0, di
ff(z1,y) = 0\},\{x,y\});"}}{COM_R 20 0{TEXT 1 0 ""}}{COM_R 21 0
{TEXT 1 64 "The resulting solution is at the stationary point x =
 -5, y = 5."}}{COM_R 22 0{TEXT 1 0 ""}}{COM_R 23 0{TEXT 1 77 "Now
 identify the nature of the stationary point using the following 
criteria:"}}{COM_R 24 0{TEXT 1 67 "dz/dx = 0 , dy/dx = 0, rt > s^
\{2\} and  r < 0 ,  (x,y)  is a MAXIMUM"}}{COM_R 25 0{TEXT 1 80 "
                                    rt > s^\{2\} and  r > 0 ,  (x
,y)  is a MINIMUM"}}{COM_R 26 0{TEXT 1 74 "                      
              rt < s^\{2\} ,  (x,y) is a SADDLE POINT."}}{COM_R 
27 0{TEXT 1 62 "where  r = d2z/dx2,  s= d2z/dxdy , t= d2z/dy2 , d
= rt - s^\{2\}."}}{COM_R 28 0{TEXT 1 0 ""}}{INP_R 29 0 "> "{TEXT 
0 57 "d:= z -> diff(z,x$2)*diff(z,y$2) - (diff(diff(z,y),x))^2;"}
}{INP_R 30 0 "> "{TEXT 0 21 "r:= z -> diff(z,x$2);"}}{COM_R 31 0
{TEXT 1 21 "                     "}}{COM_R 32 0{TEXT 1 40 "The st
ationary value of the function is:"}}{INP_R 33 0 "> "{TEXT 0 24 "
subs(\{x= -5, y= 5\}, z1);"}}{COM_R 34 0{TEXT 1 0 ""}}{COM_R 35 0
{TEXT 1 48 "Now identify the nature of the stationary point:"}}
{INP_R 36 0 "> "{TEXT 0 27 "subs(\{x= -5, y= 5\}, d(z1));"}}
{INP_R 37 0 "> "{TEXT 0 27 "subs(\{x= -5, y= 5\}, r(z1));"}}
{COM_R 38 0{TEXT 1 0 ""}}{COM_R 39 0{TEXT 1 71 "The stationary po
int x = -5, y = 5, z_\{1\} = -22 is therefore a minimum."}}{COM_R
 40 0{TEXT 1 0 ""}}{COM_R 41 0{TEXT 1 10 "EXERCISE 2"}}{COM_R 42 
0{TEXT 1 0 ""}}{COM_R 43 0{TEXT 1 76 "Repeat the procedure above 
to investigate the saddle  point of the function:"}}{INP_R 44 0 "
> "{TEXT 0 40 "z2 := 5*x*y - 4*x^2 + y^2 - 2*x - y + 5;"}}{INP_R 
45 0 "> "{TEXT 0 45 "plot3d(z2, x=-10..10, y=-10..10, axes=FRAME)
;"}}{COM_R 46 0{TEXT 1 30 "To find the stationary points:"}}
{INP_R 47 0 "> "{TEXT 0 53 "exp2:= solve(\{diff(z2,x) = 0, diff(z
2,y) = 0\},\{x,y\});"}}{COM_R 48 0{TEXT 1 0 ""}}{COM_R 49 0{TEXT 
1 40 "The stationary value of the function is:"}}{INP_R 50 0 "> "
{TEXT 0 29 "subs(\{x= 1/41, y= 8/41\}, z2);"}}{COM_R 51 0{TEXT 1 
0 ""}}{COM_R 52 0{TEXT 1 47 "Now identify the nature of the stati
onary point"}}{INP_R 53 0 "> "{TEXT 0 32 "subs(\{x= 1/41, y= 8/41
\}, d(z2));"}}{INP_R 54 0 "> "{TEXT 0 32 "subs(\{x= 1/41, y= 8/41
\}, r(z2));"}}{COM_R 55 0{TEXT 1 0 ""}}{COM_R 56 0{TEXT 1 70 "The
 point x =  1/ 41 , y = 8 / 41, z_\{2\}= 8095/1681 is a saddle po
int."}}{COM_R 57 0{TEXT 1 0 ""}}{COM_R 58 0{TEXT 1 0 ""}}{COM_R 
59 0{TEXT 1 10 "EXERCISE 3"}}{COM_R 60 0{TEXT 1 0 ""}}{COM_R 61 0
{TEXT 1 80 "Repeat the procedure above to investigate the station
ary points of the function:"}}{INP_R 62 0 "> "{TEXT 0 24 "z3 := x
^3 - 6*x*y + y^3;"}}{COM_R 63 0{TEXT 1 0 ""}}{INP_R 64 0 "> "
{TEXT 0 33 "plot3d(z3, x=-10..10, y=-10..10);"}}{INP_R 65 0 "> "
{TEXT 0 53 "exp3:= solve(\{diff(z3,x) = 0, diff(z3,y) = 0\},\{x,y
\});"}}{INP_R 66 0 "> "{TEXT 0 44 "subs(\{x=0,y=0\},d(z3));subs(\
{x=0,y=0\},r(z3));"}}{INP_R 67 0 "> "{TEXT 0 44 "subs(\{x=2,y=2\}
,d(z3));subs(\{x=2,y=2\},r(z3));"}}{COM_R 68 0{TEXT 1 0 ""}}
{COM_R 69 0{TEXT 1 27 "Stationary points occur at:"}}{COM_R 70 0
{TEXT 1 43 "                x = 0, y = 0 a saddle point"}}{COM_R 
71 0{TEXT 1 45 "                x = 2, y = 2 a local minimum."}}
{COM_R 72 0{TEXT 1 32 "The remaining roots are complex."}}{COM_R 
73 0{TEXT 1 0 ""}}{COM_R 74 0{TEXT 1 0 ""}}{COM_R 75 0{TEXT 1 10 
"EXERCISE 4"}}{COM_R 76 0{TEXT 1 0 ""}}{COM_R 77 0{TEXT 1 80 "Rep
eat the procedure above to investigate the stationary points of t
he function:"}}{INP_R 78 0 "> "{TEXT 0 24 "z4 := x^3 - 3*x + x*y^
2;"}}{INP_R 79 0 "> "{TEXT 0 33 "plot3d(z4, x=-10..10, y=-10..10)
;"}}{INP_R 80 0 "> "{TEXT 0 55 "exp4:= [solve(\{diff(z4,x) = 0, d
iff(z4,y) = 0\},\{x,y\})];"}}{COM_R 81 0{TEXT 1 0 ""}}{COM_R 82 0
{TEXT 1 27 "Stationary points occur at:"}}{COM_R 83 0{TEXT 1 42 "
   x = 0, y = +-sqrt\{3\} are saddle points;"}}{COM_R 84 0{TEXT 1
 35 "   x = 1, y = 0 is a local minimum;"}}{COM_R 85 0{TEXT 1 36 
"   x = -1, y = 0 is a local maximum."}}{COM_R 86 0{TEXT 1 0 ""}}
{COM_R 87 0{TEXT 1 38 "To determine the nature of the points:"}}
{INP_R 88 0 "> "{TEXT 0 32 "exp4 := map(allvalues,exp4,'d');"}}
{INP_R 89 0 "> "{TEXT 0 22 "for i from 1 to 4 do; "}}{INP_R 90 0 
"> "{TEXT 0 12 "    exp4[i];"}}{INP_R 91 0 "> "{TEXT 0 45 "    su
bs(exp4[i],d(z4)); subs(exp4[i],r(z4));"}}{INP_R 92 0 "> "{TEXT 0
 3 "od;"}}{INP_R 93 0 "> "{TEXT 0 0 ""}}}{END}
