Practice Problems 9, Partial Differentiation (c)

Lagrange Multipliers, Example 1

Manipulate[
ContourPlot[
{5x-3y==k,(x^2)+(y^2)==136},{x,-15,15},{y,-15,15},
ImageSize->600
],
{{k,0,"parabola"},-100,100,1}
]

Lagrange Multipliers, Example 2

Manipulate[
ContourPlot[
{8(x^2)-2y==k,(x^2)+(y^2)==1},{x,-5,5},{y,-5,5},
ImageSize->600
],
{{k,-3,"parabola"},-5,12,.125}
]

Problem 1

(∂_l)((l^2)Sin[θ]Cos[θ]+(24l-2(l^2))Sin[θ])

(∂_θ)((l^2)Sin[θ]Cos[θ]+(24l-2(l^2))Sin[θ])

Solve[(24-4 l) +2 l Cos[θ] ==0&&(24 l-2(l^2)) Cos[θ]+(l^2) ((Cos[θ])^2)-(l^2) ((Sin[θ])^2)==0,{l,θ}]

Problem 2

Solve[x +x (y^2)+5y==0&&y+(x^2)y+5x==0,{x,y}]

Solve[x +x (y^2)+5y==y+(x^2)y+5x,{x,y}]

Problem 4

Plot3D[
2(x^2)-3(y^2)-2x+10,
{x,-1,1},{y,-1,1},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
ImageSize->800,
AxesStyle->Black,
Mesh->None]

ContourPlot[
2(x^2)-3(y^2)-2x+10,
{x,-1,1},{y,-1,1},
ColorFunction->"Rainbow",
Contours->100,
AspectRatio->1,
ImageSize->800,
AxesStyle->Black]

Another Temperature Example

Plot3D[
(Sqrt[(x^2)+(y^2)+(((x y+5))^2)]),
{x,-4,4},{y,-4,4},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
ImageSize->800,
AxesStyle->Black,
Mesh->None]

ContourPlot[
(Sqrt[(x^2)+(y^2)+(((x y+5))^2)]),
{x,-4,4},{y,-4,4},
ColorFunction->"Rainbow",
Contours->100,
AspectRatio->1,
ImagePadding->20,
ImageSize->800,
AxesStyle->Black]