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

MATH104 December 2012

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Question 06 (c)
Consider the curve $x^{2}+y^{3}-2xy=0$. 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). Approximate f(1.02) using the linear approximation of f(x) at x=1. State whether you expect this to be an overestimate, or an underestimate of f(1.02).

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
What is the formula for the linear approximation of a function? What is the a value and what is the x value?

Hint 2
How does the concavity of a function give us information about over and under estimates? What have we computed in part (a) or part (b) that can help us answer this question?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The linear approximation formula is $\displaystyle {}L(x)=f(a)+f'(a)(x-a).$ The number a is the value we are approximating around and thus in our case a=1. Since we are trying to approximate f(1.02) we take x=1.02. Using the information from part (a) we have that $\displaystyle {}f(1.02)\approx {}L(1.02)=f(1)+f'(1)(1.02-1)=1+0(0.02)=1.$ This gives the estimate that $f(1.02)=1$. From part (b), the second derivative at x=1 is negative and therefore we see that the function is concave down at this point. This tells us that points near x=1 are smaller than what the tangent line predicts and therefore we expect that our estimate is an overestimate.

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