crazynut52 said:r(t)= <t^2, cost, sint>
Does anyone have a graphing program to make a picture of this, thanks.
Tom Mattson said:The implied range is that x>0, and that y and z must both be between -1 and 1 (inclusive). I don't have a graphing utility handy, but what I would do is find the 2D curve in each coordinate plane by eliminating the parameter. So, in the xy plane, you have y(x)=arccos(x1/2), in the xz plane you have z(x)=arcsin(x1/2), and in the yz plane you have y2+z2=1.
Basically, the curve is constrained to the unit cylinder y2+z2=1, and as it goes around it moves forward on the x-axis, starting from x=0.
graphic7 said:If you need a larger range, just request it.
Tom Mattson said:There is no larger range. The implied range that I stated is the maximal range.
graphic7 said:Only in the y and z directions, though. I just replotted from -1000 to 1000 and you really get to see the unit cyclinder take form.
Tom Mattson said:Ah, I see what you're saying. You mean a larger range in your picture. What I was saying is that the range implied by the equations is the maximal range, and that if there is any modification to that range in the problem, it can only be smaller, not bigger.
The graph of this equation is a three-dimensional curve that resembles a twisted ribbon. The t2 component creates a parabola-like shape in the x-y plane, while the cos t and sin t components create sinusoidal curves in the x-z and y-z planes, respectively.
The value of t determines the position of the point on the curve. As t increases, the curve moves along the parabola in the x-y plane and also oscillates along the sinusoidal curves in the x-z and y-z planes.
The domain for this equation is all real numbers, as t can take on any value. The range depends on the amplitude of the cosine and sine functions, but it will also be all real numbers.
Yes, there are several important points on this graph. The point (0,1,0) is the highest point on the curve, and the points (1,1,0) and (-1,1,0) are the lowest points. The curve also intersects the x-y plane at (0,0,0) and the x-z and y-z planes at (1,0,0) and (0,0,1), respectively.
This equation describes a particle moving along a curved path in three-dimensional space. The particle's position is constantly changing as t increases, following the shape of the curve. The cosine and sine components also indicate that the particle is oscillating in the z-direction as it moves along the curve.