Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (h)

MATH100 December 2011

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Question 01 (h)

Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question. Find the slope of the tangent line to the curve y=f(x) at the point (0,2), where the function y=f(x) satisfies the equation $\displaystyle x^{2}+xy+y^{2}=4$

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!

Hint
Is y=ƒ(x) implicit or explicit? What method of differentiation do you need to use?

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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. In order to find the slope of the tangent line, we need to take the derivative. However, because this expression is not written in the form y = f(x), we will have to use implicit differentiation with respect to x. Thus we implicitly differentiate the expression with respect to x. Note that the term x2 can be differentiated directly, the term xy requires the product rule, and the term y2 requires the chain rule. $2x+y+x{\frac {dy}{dx}}+2y{\frac {dy}{dx}}=0$ Since we are finding a specific derivative at the point (0,2), we can go ahead and plug in the values x=0 and y=2 to get: $2(0)+(2)+(0){\frac {dy}{dx}}+2(2){\frac {dy}{dx}}=0$ $2+4{\frac {dy}{dx}}=0$ If we solve the above expression for the derivative dy/dx we get: ${\frac {dy}{dx}}={\frac {-2}{4}}={\frac {-1}{2}}$ So the slope of the tangent line is -1/2.

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