3431.sagews

var('x' 'y')
f(x,y)=x^2*y+y^2
fx=diff(f(x,y),x)
fy=diff(f(x,y),y)
fxx=diff(fx,x)
fxy=diff(fx,y)
fyy=diff(fy,y)
fyx=diff(fy,x)
print "fx=",fx 
print "fy=",fy
print "fxx",fxx
print "fxy",fxy
print "fyx",fyx
print "fyy",fyy

fx= 2*x*y fy= x^2 + 2*y fxx 2*y fxy 2*x fyx 2*x fyy 2

var('x' 'y')
f(x,y)=e^x*cos(y)
fx=diff(f(x,y),x)
fy=diff(f(x,y),y)
fxx=diff(fx,x)
fxy=diff(fx,y)
fyy=diff(fy,y)
fyx=diff(fy,x)
print "fx=",fx 
print "fy=",fy
print "fxx",fxx
print "fxy",fxy
print "fyx",fyx
print "fyy",fyy
h=0
k=0
var('x''y''z')
tanplane(x,y)=(x-h)*fx(h,k)+(y-k)*fy(h,k)+f(h,k)
print tanplane(.1,.1)
a=implicit_plot3d(tanplane==z,(x,-10,10),(y,-10,10),(z,-10,10), color='purple')
b=implicit_plot3d(f(x,y)==z,(x,-10,10),(y,-10,10),(z,-10,10))
a+b

fx= e^x*cos(y) fy= -e^x*sin(y) fxx e^x*cos(y) fxy -e^x*sin(y) fyx -e^x*sin(y) fyy -e^x*cos(y) 1.10000000000000

var('x' 'y')
f(x,y)=x*y^2-x^3+10
fx=diff(f(x,y),x)
fy=diff(f(x,y),y)
fxx=diff(fx,x)
fxy=diff(fx,y)
fyy=diff(fy,y)
fyx=diff(fy,x)
print "fx=",fx 
print "fy=",fy
print "fxx=",fxx
print "fxy=",fxy
print "fyx=",fyx
print "fyy=",fyy
h=1
k=1.5
var('x''y''z')
tanplane(x,y)=(x-h)*fx(h,k)+(y-k)*fy(h,k)+f(h,k)
print tanplane(1.1,1.2)
a=implicit_plot3d(tanplane==z,(x,-10,10),(y,-10,10),(z,-10,10), color='purple')
b=implicit_plot3d(f(x,y)==z,(x,-10,10),(y,-10,10),(z,-10,10))
a+b

fx= -3*x^2 + y^2 fy= 2*x*y fxx= -6*x fxy= 2*y fyx= 2*y fyy= 2*x 10.2750000000000