Science:Math Exam Resources/Courses/MATH104/December 2012/Question 06 (a)
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Question 06 (a) 

Consider the curve . Assume that the point (x,y) = (1,1) lies on the curve, and that nearby points on the curve satisfy y=f(x) for some function of f(x). Find and . 
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. 
Hint 1 

To find , note that the curve passes through the point . When , what is ? 
Hint 2 

Note that ƒ'(1) is equal to at the point . How can you find ? 
Hint 3 

Use implicit differentiation to find and evaluate at . 
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.

Solution 

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(Explanation: We are given that the curve passes through the point . Near this point, the curve is the graph of a function, . When , .) is the same thing as evaluated at the point . In order to find this, we use implicit differentiate the equation with respect to .
In order to solve for , we first factorize,
Then subtract from both sides and divide by to obtain
When
Hence, . 