My solutions to the November 9 In-Class work

Math 104 Calculus II, November 10, 2017
T. Ratliff, Wheaton College
ratliff_thomas@wheatoncollege.edu

1. f(x,y) = 25 (x^4) - 30 (x^2) + 4 (y^2) - 12 y + 18

f=25x^4 - 30x^2 + 4y^2 - 12 y + 18
Solve[{D[f,x]==0,D[f,y]==0}]

ContourPlot[f,{x,-1,1},{y,1,2},Contours->30,ColorFunction->"Rainbow"]
Plot3D[f,{x,-1,1},{y,-1,3},MeshFunctions->{#3&},Mesh->30,ColorFunction->"Rainbow"]

2. g(x,y) = -((((x^2)-1))^2) - ((((x^2)y-x-1))^2)

g=-(x^2-1)^2 - (x^2 y - x - 1)^2
Solve[{D[g,x]==0,D[g,y]==0}]

ContourPlot[g,{x,-2,2},{y,-1,3},Contours->50,ColorFunction->"Rainbow"]
Plot3D[g,{x,-1.5,1.5},{y,-1,3},MeshFunctions->{#3&},Mesh->30,ColorFunction->"Rainbow"]

3. v(x,y) = ((x y (60 - x y))/(2x +2 y))

v = (x y (60-x y))/(2x + 2y)
Solve[{D[v,x]==0,D[v,y]==0}]

N[%]

ContourPlot[v,{x,3,6},{y,3,6},Contours->30,ColorFunction->"Rainbow"]
Plot3D[v,{x,3,6},{y,3,6},MeshFunctions->{#3&},Mesh->30,ColorFunction->"Rainbow"]

4. d(x,y) = (((x-3))^2) + (((y-2))^2)+((((x^2)+(y^2)-6))^2)

d=(x-3)^2 + (y-2)^2 +(x^2 + y^2 -6)^2
NSolve[{D[d,x]==0,D[d,y]==0},Reals]

ContourPlot[d,{x,-3,3},{y,-2,2},Contours->50,ColorFunction->"Rainbow"]
Plot3D[d,{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->50,ColorFunction->"Rainbow"]

d/. {{x->2.0755762465015084`,y->1.3837174976676723`},{x->-1.7971824486254846`,y->-1.1981216324169897`},{x->-0.27839379787602375`,y->-0.1855958652506825`}}

5. h(x,y) = 5 - 10 x y - 4 (x^2) + 3 y - (y^4)

h = 5 - 10 x y - 4x^2 + 3y - y^4
NSolve[{D[h,x]==0,D[h,y]==0},Reals]

ContourPlot[h,{x,-3,3},{y,-2,2},Contours->50,ColorFunction->"Rainbow"]
Plot3D[h,{x,1,3},{y,-3,0},MeshFunctions->{#3&},Mesh->50,ColorFunction->"Rainbow"]