Calculus-Sec-10-3-Solution


  10.3    Polar Coordinates                           by SGLee - HSKim - JHLee

                                                                                                              http://youtu.be/4hoVKvk8dq0

1-4. Plot the point whose polar coordinates are given. Then, find the Cartesian coordinates of the point.

 

 1.  = 

            http://matrix.skku.ac.kr/cal-lab/cal-9-3-1.html 

            http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html 

 

 







2.  = 

 

 




3.  = 

 

 

       .




4.  =

 

 

      .




5-8. The Cartesian coordinates are given.

       Find two other pairs of polar coordinates of the point, one with   and other .

 

5. 

 

      , and .

      Since  is in the fourth quadrant, the polar coordinates are

      ⑴  : , ⑵  : .




6. 

 

 

      , and .

      Since  is in the second quadrant, the polar coordinates are

      ⑴  : , ⑵  : .




7. 

 

      , and .

      Since  is in the fourth quadrant, the polar coordinates are

      ⑴  : , ⑵  : .




8. 

 

      , and .

      Since  is in the second quadrant, the polar coordinates are

      ⑴  : , ⑵  : .




9. Find a formula for the distance between the points with polar coordinates  and .

 

 

      Let 

      Let  be the e distance between  and .

         




10-11. Find a polar equation for the curve represented by the given Cartesian equation.

 

10. 

 

      .




 11. 

 

      .




 12. 

 

      .




13-14. Find a Cartesian equation for the curve represented by the given polar equation.

 

13. 

 

      ,

      a circle of radius  centered at .

      The first two equations are actually equivalent since

      .

      But  gives the point  when .

      Thus, the single equation  is equivalent to the compound condition ().




14. 

 

       .




 15. 

 

 

       




16-25. Sketch the curve with the given polar equation.

 

16.            




 17. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-17.html 

 

 

       .




 18. 

              http://matrix.skku.ac.kr/cal-lab/cal-9-3-18.html 

 

 




 19. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-19.html 

 

 




20. 

      http://matrix.skku.ac.kr/cal-lab/cal-9-3-20.html 

 

 




 21. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-21.html 

 

 




 22. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-22.html 

             http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html 

 

 




23. 

      http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html

 

 




24. 




25. 




 26. Show that the polar curve  (called a conchoid) has the line 

             as a vertical asymptote by showing that . Use this fact to sketch the conchoid.

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-26.html 

 

       .

      Now, ,

      so .

      Also, , so

      .

      Therefore,  is a vertical asymptote.







 27. Show that the curve  (also a conchoid) has the line  

             as a horizontal asymptote by showing that . Use this fact to help sketch conchoid.

             http://matrix.skku.ac.kr/cal-lab/cal-9-3-27.html 

 

       .

       Now, ,

       so .

       Also, ,

       so .

       Therefore,  is a horizontal asymptote.




28. Show that the curve  (called a cissoid of Diocles) has the line  

     as a vertical asymptote. Show also that the curve lies entirely within the vertical strip .

     Use these facts to sketch the cissoid.




29-32. Find the slope of the tangent line at the given point.

 

29. 

 

       .




30. 

 

      .




31. 

 

 

      .




32. 




33-36. Find the points on the given curve where the tangent line is horizontal or vertical.

 

33. 

 

      .

     .

     So the tangent is vertical at  and .

     . So the tangent is horizontal at  and .




34. 




35. 




 36. 




37. Show that the polar equation  represents a circle and find its center and radius.

 

      .

      Then the polar equation  represents a circle and its center : , radius :.




                                                      

 

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