Given the parametrized curve C in the xy plane.C: r(t) = t²i+ (−t)j, 0≤t≤33(1) Find an equation for the tangent line to the curve at the point (4,2/3) on the curve.(2) Find the length of the curve.

Given, parameterized curve, (1) Given, So,

Answered: Given the parametrized curve C in the…

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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Question

Given the parametrized curve C in the xy plane.
C: r(t) = t²i+ (−t)j, 0≤t≤3
3
(1) Find an equation for the tangent line to the curve at the point (4,2/3) on the curve.
(2) Find the length of the curve.

Expert Solution

Step 1: Explanation

Given, parameterized curve,

$C colon r left parenthesis t right parenthesis equals t squared i plus open parentheses t cubed over 3 minus t close parentheses j comma space 0 less or equal than t less or equal than 3$

(1) Given, $r left parenthesis t right parenthesis equals t squared i plus open parentheses t cubed over 3 minus t close parentheses j equals x left parenthesis t right parenthesis i plus y left parenthesis t right parenthesis j$

So, $x left parenthesis t right parenthesis equals t squared comma y left parenthesis t right parenthesis equals open parentheses t cubed over 3 minus t close parentheses$

$fraction numerator d x left parenthesis t right parenthesis over denominator d t end fraction equals fraction numerator d over denominator d t end fraction left parenthesis t squared right parenthesis equals 2 t fraction numerator d y left parenthesis t right parenthesis over denominator d t end fraction equals fraction numerator d over denominator d t end fraction open parentheses t cubed over 3 minus t close parentheses equals t squared minus 1$

$fraction numerator d y over denominator d x end fraction equals fraction numerator t squared minus 1 over denominator 2 t end fraction$