MATH 261, Spring 2001 Due Date: Name(s):
Extra Project 15.3: Partial Derivatives
Objective
In this project we discuss the computation and geometry of partial derivatives.
Narrative
If you have not already done so, read Section 15.3 in the text. In this project we discuss the computation
and geometry of partial derivatives.
Tasks
a) Type the command lines below into Maple; they produce a plot of the graph of f(x, y) = -5x/(x2
+y2
+1).
> # Project 15.3: Partial Derivatives
> restart: with(plots): with(plottools):
> setoptions3d(axes=normal,orientation=[20,80],grid=[50,50],lightmodel=light1,
scaling=constrained,style=patchnogrid,shading=zgrayscale);
> f := (x,y) -> (24-x^2-2*y^2)/8;
> plot0 := plot3d(f(x,y),x=-3..3,y=-3..3,color=green): display(plot0);
b) Continue by typing the command lines below into Maple; they compute the value val of f at P(a, b).
> a := 2; b := 2.5;
> val := subs({x=a,y=b},f(x,y));
c) Continue by typing the command lines below into Maple; they plot the graph of f over a slightly smaller
domain, plot the x-curve of f through P(a, b) (in blue), compute fx and fx(a, b), and then draw the tangent
line to the x-curve of f through P(a, b) (in red). (The curve of intersection of the graph of z = f(x, y) and
a plane whose equation is of the form x = a is called an x-curve, and the curve of intersection of the graph
of z = f(x, y) and a plane whose equation is of the form y = b is called an y-curve.)
> plot1 := plot3d(f(x,y),x=-3..3,y=-3..b,color=green):
> curve1 := spacecurve([t,b,f(t,b),t=-3..3],color=blue,thickness=2):
> f1 := (x,y) -> diff(f(x,y),x);
> slope1 := subs({x=a,y=b},f1(x,y));
> tanline1 := spacecurve({[t+a,b,slope1*t+val,t=-3..3]},color=red,thickness=2):
> display({plot1,curve1,tanline1});
d) Continue by typing the command lines below into Maple; they again plot the graph of f over a slightly
smaller domain, plot the y-curve of f through P(a, b) (in blue), compute fy and fy(a, b), and then draw the
tangent line to the y-curve of f through P(a, b) (in red).
> plot2 := plot3d(f(x,y),x=-3..a,y=-3..3,color=green):
> curve2 := spacecurve({[a,t,f(a,t),t=-3..3]},color=blue,thickness=2):
> f2 := (x,y) -> diff(f(x,y),y);
> slope2 := subs({x=a,y=b},f2(x,y));
> tanline2 := spacecurve({[a,t+b,slope2*t+val,t=-3..3]},color=red,thickness=2):
> display({plot2,curve2,tanline2});
e) Finally, type the command line below into Maple; it plots both the x-curve and its tangent line at P(a, b),
and the y-curve and its tangent line at P(a, b). Adjust the graphic to get a good view.
> display({plot0,curve1,tanline1,curve2,tanline2});
At this point, make a hard copy of your typed input and Maple's responses. Then, ...
f) By hand, label each x-curve on the graphics you created as "x-curve", the tangent to each x-curve as
"tangent to x-curve", each y-curve on the graphics you created as "y-curve", and the tangent to each y-curve
as "tangent to y-curve".