009C Sample Final 3, Problem 10

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A curve is described parametrically by

(a) Sketch the curve for  

(b) Find the equation of the tangent line to the curve at the origin.

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. What is the slope of the tangent line of a parametric curve?

       The slope is  


9CSF3 10 GP.png


Step 1:  
First, we need to find the slope of the tangent line.
Since     and     we have


Step 2:  
Now, the origin corresponds to    and  
This gives us two equations. When we solve for    we get  
Plugging in    into
we see that    is undefined at  
So, there is no tangent line at the origin.

Final Answer:  
    (a)    See above
    (b)    There is no tangent line at the origin.

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