Question

The following curve lies on a surface S in 3 space. Find the rectangular equations of S and describe its behavior.

λ(t) = (cos(4t)e^(3t), sin(4t)e^(3t), e^(6t) - 10) tϵℝ and C = trace(λ)

Answer 1

this is a circle in x-y plane (BIG clue is cos and sin terms). so let x = cos4te^3t and y = sin4te^3t and see what happens.

what is x^2 + y^2? yep e^6t!!

thus:
z = x^2 + y^2

looks like this:
https://www.google.co.uk/search?q=z+%3D+x%5E2+%2B+y%5E2&rlz=1C5CHFA_enGB524GB524&oq=z+%3D+x%5E2+%2B+y%5E2&aqs=chrome..69i57.346j0j4&sourceid=chrome&ie=UTF-8

this means you have surface ø = x^2 + y^2 - z = 0

so grad ø = <2x, 2y, -1>, or -2x, -2y, 1>, the normal vectors to the surface. you now own it!!

Answer 2

Wow!!

Answer 3

thank you! @IrishBoy123