MATH110 April 2012
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The curve below is called a lemniscate; it is given by the equation
Find the equation of the tangent line at the point and sketch the tangent line.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
In order to find the slope of the tangent line you will need to use implicit differentiation.
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To find the equation of the tangent line we need a point and a slope. We already have the point, so it remains to find the slope of the tangent line. We will do so by using implicit differentiation.
Implicitly differentiating gives:
We could solve for and then plug in the point to find the slope. However, it might be easier to plug in the point first and then solve for , which is what I will do here:
Using the point slope formula, with slope and point , we get:
as our tangent line.
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