(10) Let r(t) = (e'cos 2t, e sin 2t, e?t )a) The curve lies on a quadric surface. Give the equation of the surface and sketch its graph.Describe the graph of the curve.b) Find parametric equations for the tangent line to the curve at t = 0.

Name: (10) Let r(t) = (e'cos 2t, e sin 2t, e?t )a) The curve lies on a quad
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Description: (10) Let r(t) = (e'cos 2t, e sin 2t, e?t )a) The curve lies on a quadric surface. Give the equation of the surface and sketch its graph.Describe the graph of the curve.b) Find parametric equations for the tangent line to the curve at t = 0.

Answered: (10) Let r(t) = (e'cos 2t, e sin 2t,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(10) Let r(t) = (e'cos 2t, e sin 2t, e?t )
a) The curve lies on a quadric surface. Give the equation of the surface and sketch its graph.
Describe the graph of the curve.
b) Find parametric equations for the tangent line to the curve at t = 0.

Expert Solution

part a

Given parametric equation is

$r (t) = <e^{t} \cos 2 t, e^{t} \sin 2 t, e^{2 t}>$ (i)

Let us consider

$x = e^{t} \cos 2 t y = e^{t} \sin 2 t z = e^{2 t}$

On squaring x and y and subtracting their sum from z,

$x^{2} + y^{2} - z = e^{2 t} \cos^{2} t + e^{2 t} \sin^{2} t - e^{2 t} = 0$

So, the equation of the quadratic surface is $x^{2} + y^{2} = z$ . The surface is an elliptic paraboloid

Graph of the surface is shown below

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