Answer to Question #97941 in Calculus for Rachel

Question

Sketch each of the following parameterized curves.(a) g(t) =[cos(t2), sin(t2)],0≤t≤π ;(b) g(t) =[cosh(t), sinh(t)],−2≤t≤2 ;(c) g(t) =[2−3t, 5t+ 4],0≤t≤1

Answer 1

• $g(t) = (cos\,t^2, sin\,t^2), t \in [0, \pi]$

Notice that $|g| = 1$ for all $t$ and $\pi^2 > 2\pi$

Then the image of $[0, \pi]$ under $g$ is indeed the whole unit circle

• $g(t) = (cosh\,t,sinh\,t), t \in [-2, 2]$

We'll notice that $cosh^2t - sinh^2t = 1$ for all $t$

So the needed sketch is just a part of the hyperbola $x^2 - y^2 = 1$ with endpoints at $(cosh(-2), sinh(-2))\: and\: (cosh\,2,sinh\,2)$

• $g(t) = (2-3t,\,5t+4), t \in [0, 1]$

Both coordinates depend linearly on $t$ so the needed sketch will be a line segment with endpoints at $g(0)$ and $g(1)$