Partial derivative plots

Greg Warrington - 08.05.05

Open up two copies of this worksheet (you may need to do this by double-clicking from Windows rather than by trying to open the second version from within Maple). Make the windows narrow and place them side by side. Go up to "Edit -> Execute -> Worksheet" so that Maple reads in everything in each worksheet.

The purpose of this worksheet is to help you visualize how the graphs of the

partial derivatives relate to the graph of the original function.

I suggest looking at f in the lefthand window and then one of

f_x, f_y, f_xx, f_xy, f_yy in the right.

You might also want to place f_x or f_y in the left window, however.

Instructions on how to change the function you're using are given below. Make sure you're using the same function in each window!

restart: with(plots): setoptions3d(axesfont=[TIMES,ROMAN,14], labelfont=[TIMES,ROMAN,14], font=[TIMES,BOLD,14], titlefont=[TIMES,BOLD,18]);

This function does the actual computing of the plots of the partial derivatives.

You can safely ignore it.

plofun := proc(f,cl) local a: a := array(1..3,1..3): a[1,1] := plot3d(f(x,y),x=-cl..cl,y=-cl..cl): a[2,1] := plot3d(diff(f(x,y),x),x=-cl..cl,y=-cl..cl): a[1,2] := plot3d(diff(f(x,y),y),x=-cl..cl,y=-cl..cl): a[3,1] := plot3d(diff(f(x,y),x,x),x=-cl..cl,y=-cl..cl): a[1,3] := plot3d(diff(f(x,y),y,y),x=-cl..cl,y=-cl..cl): a[2,2] := plot3d(diff(f(x,y),x,y),x=-cl..cl,y=-cl..cl): return a: end proc:

It should be clear how to add your own functions. Remember to hit "enter" so that Maple knows about your new function.

f := (x,y) -> sin(x*y^2): g := (x,y) -> (1 - x^2 - y^2)^(1/2): h := (x,y) -> x^2 - y^2 + x*y: j := (x,y) -> x^3 - y^3 + x*y: k := (x,y) -> exp(x-y^2): l := (x,y) -> sin(x)*cos(y): m := (x,y) -> sin(x*y): n := (x,y) -> 1/(1 + x^2 + 3*y^2): p := (x,y) -> x^3 + x^2*y^3 - 2*y^2:

This is where you change the function you want to use. Change plofun(l,vw) to plofun(m,vw) or plofun(p,vw) or

whatever you want (as long as it's defined above).

vw determines which x and y values we're interested in. Feel free to change it.

xyp is just the xy-plane displayed as a red mesh.

vw := 2: pls := plofun(l,vw): ############ This is the line to change xyp := plot3d([x,y,0],x=-vw..vw,y=-vw..vw,grid=[10,10],color=RED,style=WIREFRAME): display({pls[1,1],xyp},axes=BOXED,title="f"); display({pls[2,1],xyp},axes=BOXED,title="fx"); display({pls[1,2],xyp},axes=BOXED,title="fy"); display({pls[3,1],xyp},axes=BOXED,title="fxx"); display({pls[1,3],xyp},axes=BOXED,title="fyy"); display({pls[2,2],xyp},axes=BOXED,title="fxy");

This just displays the plots without the labels for quizzing purposes.

pls2 := plofun(p,vw): display({pls2[1,1],xyp},axes=BOXED); display({pls2[2,1],xyp},axes=BOXED); display({pls2[1,2],xyp},axes=BOXED); display({pls2[3,1],xyp},axes=BOXED); display({pls2[1,3],xyp},axes=BOXED); display({pls[2,2],xyp},axes=BOXED);

This is for zooming in to a surface.

cenx := 1.5: ceny := 1.5: myeps := 1/10: plot3d(sin(x*y^2),x=cenx-myeps..cenx+myeps, y=ceny-myeps..ceny+myeps, axes=BOXED);