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Question 01 (b)
Consider the function
b) Find the equation of the tangent plane to the graph at the point where .
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Remember that all vectors lying in the tangent plane to are perpendicular to the normal vector to the surface at the same point. I.e., let be a point on the surface of . If is the normal vector to at this point, then
gives the equation of the tangent plane to the surface at the point .
In this case, the normal vector, , is determined by the gradient of function :
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
The equation of the tangent plane is given by evaluating
where and is a normal vector to the surface at . The normal vector at the point is given by
We simplify the above equation (although it is not necessary) and get the tangent plane to at :
We can also get the tangent plane by using a linear Taylor approximation. Recall for a function we have the linear approximation at a point is
We are given that we want to approximate and so and . We have
Since we have
as the tangent plane approximation at the desired point.