009C Sample Final 3, Problem 7

A curve is given in polar coordinates by

$r=1+\cos ^{2}(2\theta )$

(a) Show that the point with Cartesian coordinates  $(x,y)={\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}$  belongs to the curve.

(b) Sketch the curve.

(c) In Cartesian coordinates, find the equation of the tangent line at  ${\bigg (}{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}{\bigg )}.$

Foundations:
1. What two pieces of information do you need to write the equation of a line?
You need the slope of the line and a point on the line.
2. How do you calculate   $y'$   for a polar curve  $r=f(\theta )?$
Since   $x=r\cos(\theta ),~y=r\sin(\theta ),$  we have
$y'={\frac {dy}{dx}}={\frac {{\frac {dr}{d\theta }}\sin \theta +r\cos \theta }{{\frac {dr}{d\theta }}\cos \theta -r\sin \theta }}.$

Solution:

(a)

Step 1:

Step 2:

(b)
Insert graph

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)

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