Lab05 - Partial derivatives - part 1

[Your name(s) here...]

A partial derivative of a multivariable function is a derivative with respect to one of the variables, while holding the other variables constant.

If z is a function of two variables-- z=z(x,y) -- then holding y constant, for example, y=1, leaves us with a function that depends on only one variable:  z(x,1.0).  This is a “vertical” trace of the function.  It’s the intersection of the surface y=1with the surface given by z(x,y).  

Below, a surface is plotted together with the planes x=π/2 and y=1.

z[x_,y_]:=y Cos[x (y^3)]

plot1=Plot3D[    z[x,y],
{x,0,π},{y,0,2},ViewPoint->{10,10,20},
AxesLabel->{x,y,"z"},LabelStyle->Directive[Bold, Medium]];
plot2=Graphics3D[{Opacity[.5],Blue,Polygon[{{0,1,-2},{0,1,2},{π,1,2},{π,1,-2}}],Gray,Polygon[{{(π/2),0,-2},{(π/2),0,2},{(π/2),2,2},{(π/2),2,-2}}]}];
Show[plot1,plot2]

Problem 1 4pts

Rotate the plot above as needed, and make a hand sketch of the function z(x,1).  Think carefully about identifying the positive direction for the independent variable x!

Rotate the plot above as needed, and make a hand sketch of the function z(π/2,y).
(Insert a snapshot of your sketches)

Problem 2 2pts

Now, use Mathematica’s Plot[...] function to plot each of the 2 functions above.  (And then see how well your hand sketches worked out.)  Think carefully about identifying the positive direction for the independent variable y!

Problem 3 4 pts

Calculate the functional form of (Subscript[f, x])and (Subscript[f, y])by hand, and then evaluate (Subscript[f, x])(π/2,1) and (Subscript[f, y])(π/2,1). (Insert a snapshot of your calculations).

Problem 4 2pts

Using Mathematica, Calculate (Subscript[f, x])and (Subscript[f, y]), using D[ ... , x] and D[ ... , y] functions and evaluate (Subscript[f, x])(π/2,1) and (Subscript[f, y])(π/2,1).