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.
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. What is the slope of the tangent line of a parametric curve? 
The slope is 
Solution:
(a) 

(b)
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. 