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Question 03 (b)
Suppose that a function is implicitly deﬁned by an equation:
(b) If , find the linear approximation of the function at .
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!
The linear approximation of a function at a point is
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
What to do
The linear approximation of a function in a point is
We have that . To find , we must solve the equation , where for the variable .
Hence, are solution to this equations. With the knowledge,that , the value must be
We already know from part a), that . Pluging in the point gives
We know that Pluging in the point gives
Hence the linear approximation is