3D plot & limits
Pablo Antolin
-How to 'get' a limit of two varibles?? like that;
Lim = 2x^2-y^2
(x,y)-->(0,0) ------------
x^2+2y^2
I would know the general way to do that, not the result of this
case...thanks!
Another one;
-How to plot the 3d 'curve' (??);
0<x<1 (x*cos(2(PI)x, x*sin(2(PI)x, x) ???
where-- i j k
or x y z
Thanks at all!
Pablo
Nick Karagiaouroglou
Hi Pablo!
Answering your second question first. There is no special plot type
for 3D-Curve but you can use the plot type Pr-Surface (Parametric
surface) for this curve.
Press shift-blue and F4 at the same time. You see the PLOT SETUP
screen.
Enter { 'X*cos(2*PI*X)' 'X*sin(2*PI*X)' 'X') for EQ
Now press shift-blue and F2 at the same time. You see the PLOT WINDOW
- PR-SURFACE screen. Type the values:
X-Left: -1 X-Right: 1
Y-Near: -1 Y-Far: 1
Z-Low: 0 Z-High: 1.5
XE: .5 YE:-2 ZE: .5
Step Indep: 15 Depnd: 2
Now press the menu button XXYY (F4) go to input field XXLeft and type
the values:
XXLeft: 0 XXRight: 1
You can leave the rest unchanged.
Press the menu button ERASE (F5) and then DRAW (F6)
After some seconds the HP49G draws the curve. Actually it should draw
a surface, but because the EQ depends only on X it simply draws the
same curve 2 times. (Number of steps for Y (Depnd) is 2)
Trying now to answer your first question. For such limits it is
generally important to know from what direction (x,y) approaches
(0,0). If some dependence y(x) is known, then you can substitude this
for y. For example, if it is known that y=x-1 then
2x^2-y^2 2x^2-(x-1)^2 x^2+2*x-1
-------- = -------------- = ------------
x^2+2y^2 x^2+2*(x-1)^2 3*x^2-4*x+2
and you can take the limit of this for x->0, which results in -1/2.
If no dependence of the two variables is known, then you can
parametrize x and y so that they are dependend on the same parameter.
Parametrization :
x=r*cos(t)
y=r*sin(t)
Substitude this for x and y in
2x^2-y^2
--------
x^2+2y^2
and use EXPAND to obtain:
SIN(t)^2-2*COS(t)^2
- --------------------
2*SIN(t)^2+COS(t)^2
which already shows that the limit doesn't depend on r (magnitude of
the compex (x,y)) but it depends on t, that is the angle or argument
of (x,y) = ATAN(y/x). You see that the limit depends on the ratio of y
and x, which represents the direction on the x,y-plane from which
(0,0) is approached. Now, you can play with different values for t.
For example for t=0, the limit is 2. Substitude t=2 and use EXPAND to
get this result. Remember that t=2 means that ATAN(y/x)=2 which means
that y=TAN(2)*x. Approaching (0,0) from angle t=2 already defines a
dependency of y from x. Try t=PI/3 which implies ATAN(y/x)=PI/3 =>
y=sqrt(3)*x.
The limit is then -1/7. You can also plot the expression
SIN(t)^2-2*COS(t)^2
- --------------------
2*SIN(t)^2+COS(t)^2
with independend variable t from 0 to 2*PI and investigate how the
limit behaves for different approaches to (0,0).
Greetings and I hope I could help you a little,
Nick