scumath

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These are answers submitted by scumath

Thanks for the examples. 

I hope I can further improve my skills of Maple plotting.

I want to know how to set the color and lightmodel  to achieve the quality of this image:

http://xahlee.org/surface/boys_bryant-kusner/boys_bryant-kusner.html

Thanks.

It seems that the surface integral can be converted to the double integral:

sqrt(3)*int(int(x^a1*y^a2*(1-x-y)^a3/((1-x)^b1*(1-y)^b2*(x+y)^b3), y=0..1-x),x=0..1);

If the ai's and bi's are not assigned with specific values, then Maple cannot produce a result with a closed form .
 



But if specific values are assigned to the  ai's and bi's,  then Maple can give a closed form result .

a1:=2:a2:=3:a3:=5:
b1:=2:b2:=3:b3:=5:
sqrt(3)*int(int(x^a1*y^a2*(1-x-y)^a3/((1-x)^b1*(1-y)^b2*(x+y)^b3),y=0..1-x),x=0..1);

Maybe this term is seldom used today. 
To consider the odevity of a function is to consider if it is an odd function or an even function.
You might find some clues here.

I think odevity=parity.
 

What did you mean by "in half "? 

Another way is to plot the surface z=x^2+y^2 parametrically.

with(plots):
surface:=plot3d([u*sin(theta), u*cos(theta), u^2], u=0..1.5, theta=0..2*Pi):
display(surface, axes=normal, scaling=constrained);

Thanks for your helpful explanation.

In fact, the same problem occurs in plotting explicit power functions with  fractional powers. The following can only produce half of the whole graph. What can't Maple plot the whole graph in the domain of the function y=x^(1/3)?

with(plots):
plot(x^(1/3),x=-2..2,thickness=3);

But I strongly suggest that Maplesoft introduce a special command for regular polygons.

"how many horizontal asymptotes can a graph have?"

At most two.
For example, y=arctan(x) has  two horizontal asymptotes: y=Pi/2 and y=-Pi/2.

"how many horizontal asymptotes can a graph have?"

Usually one, at most two.

For example, y=arctan(x) has  two horizontal asymptotes: y=Pi/2 and y=-Pi/2.

"Can the graph of of a function cross the graph of its horizontal or vertical asymptotes,?"

Yes for horizontal asymptotes.

For example, f(x)=sin(x)/x has a horizontal asymptote y=0 and the curve cross it many times.



No for vertical asymptotes.

Example 1

Example 2



Example 3

Example 4

 

"Yes, these work fine."

Thanks.

"does every undefined value of f(x) lead to a vertical asymptote?"

No! A vertical asymtote occurs at x=a only if limf(x)=infinity (as x->a).

The function f(x)=(6x^2)/(x^3+2x^2-5x) is not difined at x=0. However x=0 is not a vertical asymptote of the curve since limit f(x)=0 (as x->0).
 

with(plots):
u:=0.21e-1*x^3-0.947e-1*x^2-1.77*x+1.79:
v:=-0.451e-2*x^3-0.473e-1*x^2+.243*x+4.74:
U:=spacecurve([x,u(x),0], x=-8..11, color=blue, thickness=3):
V:=spacecurve([x,v(x),0], x=-8..11, color=red, thickness=3):
Region:=plot3d([x,y,0], x=-7..10.5, y=v(x)..u(x), color=grey, style=patchnogrid):
display(Region, U, V, axes=normal, orientation=[270,0], scaling=constrained);

See the following message for details:
http://www.mapleprimes.com/forum/easywayplotregionsbetweentwocurves

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