Answer to Question #97941 in Calculus for Rachel

Question #97941
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
1
Expert's answer
2019-11-05T10:21:45-0500
  • "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)"

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