009C Sample Final 3, Problem 7

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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  

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     for a polar curve  

       Since     we have




Step 1:  
First, we need to convert this Cartesian point into polar.
We have
Also, we have
Now, this point in polar is  
Step 2:  
Now, we plug in    into our polar equation.
We get
So, the point    belongs to the curve.
9CSF3 7.jpg


Step 1:  




we have


Step 2:  
Now, recall from part (a) that the given point in polar coordinates is  
Therefore, the slope of the tangent line at this point is
Therefore, the equation of the tangent line at the point    is

Final Answer:  
    (a)     See above.
    (b)     See above.

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