Difference between revisions of "009C Sample Final 3, Problem 7"

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<span class="exam">A curve is given in polar coordinates by  
 
<span class="exam">A curve is given in polar coordinates by  
  
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::<math>r=1+\cos^2(2\theta)</math>
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::<math>r=1+\cos^2(2\theta).</math>
  
 
<span class="exam">(a) Show that the point with Cartesian coordinates &nbsp;<math style="vertical-align: -15px">(x,y)=\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg)</math>&nbsp; belongs to the curve.
 
<span class="exam">(a) Show that the point with Cartesian coordinates &nbsp;<math style="vertical-align: -15px">(x,y)=\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg)</math>&nbsp; belongs to the curve.
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<span class="exam">(c) In Cartesian coordinates, find the equation of the tangent line at &nbsp;<math style="vertical-align: -15px">\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg).</math>
 
<span class="exam">(c) In Cartesian coordinates, find the equation of the tangent line at &nbsp;<math style="vertical-align: -15px">\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg).</math>
  
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!Foundations: &nbsp;
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[[009C Sample Final 3, Problem 7 Solution|'''<u>Solution</u>''']]
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|'''1.''' What two pieces of information do you need to write the equation of a line?
 
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&nbsp; &nbsp; &nbsp; &nbsp;You need the slope of the line and a point on the line.
 
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|'''2.''' How do you calculate &nbsp; <math style="vertical-align: -5px">y'</math> &nbsp; for a polar curve &nbsp;<math style="vertical-align: -5px">r=f(\theta)?</math>
 
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&nbsp; &nbsp; &nbsp; &nbsp;Since &nbsp; <math style="vertical-align: -5px">x=r\cos(\theta),~y=r\sin(\theta),</math>&nbsp; we have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>y'=\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta+r\cos\theta}{\frac{dr}{d\theta}\cos\theta-r\sin\theta}.</math>
 
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'''Solution:'''
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[[009C Sample Final 3, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''(a)'''
 
  
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!Final Answer: &nbsp;
 
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|&nbsp;&nbsp; '''(a)'''
 
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|&nbsp;&nbsp; '''(b)'''
 
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[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 16:25, 3 December 2017

A curve is given in polar coordinates by

(a) Show that the point with Cartesian coordinates    belongs to the curve.

(b) Sketch the curve.

(c) In Cartesian coordinates, find the equation of the tangent line at  

Solution


Detailed Solution


Return to Sample Exam

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