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I have a question from one my problems that ask me to graph this space curve using the "appropriate parameter range" to find the true nature of the graph. $r(t) = (t, \exp(t), \cos(t)).$

On Maple I did

with(plots): spacecurve([t, exp(t), cos(t)], t = 0 .. 2*Pi, axes = boxed) 

I chose $2\pi$ because that's the domain of $\cos(t)$, but when I extend this from $-2\pi$ to $2\pi$, I get a better view. SO what does it mean to have a "true nature of the curve"?

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    With t=-100..100, I got a cosinudal and a crooked (due to the nature of x = t I guess) exponential curve.2012-01-20

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About the true nature of the graph: There are two totally different regimes for $t\ll-1$ and $t\gg1$, and there is a zone of transition in between, where nothing spectacular happens. For $t\ll-1$ one has $y(t)=e^t\doteq0$; therefore $r(t)\doteq(t,0,\cos t)\qquad (t\ll-1)\ ,$ which is an ordinary cosine curve in the left half of the $(x,z)$-plane, extending to $-\infty$ in the $x$-direction.

For $t\gg1$ it is obvious that $y\gg1$ is the prominent variable; therefore we choose $y$ as new parameter. In this way we obtain the new parametrization $\tilde r(y)=\bigl(\log y, y, \cos(\log y)\bigr)\qquad(y\gg1)\ .$ Looking from the point $(\infty,0,0)$ we see in the $(y,z)$-plane the curve $z=\cos(\log y)$ which is an oscillation in $z$-direction becoming ever slower as $y\to\infty$. At the same time the moving point on the space curve $y\mapsto\tilde r(y)$ is also increasing its $x$-coordinate towards $\infty$, but at ever decreasing speed.

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If you think about each coordinate separately, the first is linear, the second is exponential with scale 1, and the third is periodic with period $2\pi$. You would like the range to show these. Without the specification of space curve, I would just graph each one on a 2D plot because they aren't related. I think you could defend $[0,2\pi]$ to $[-2\pi,4\pi]$ or anywhere in between. You could try a few and see what you like.