Science:Math Exam Resources/Courses/MATH102/December 2011/Question 07 (i)

MATH102 December 2011

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Question 07 (i)
A particle is moving in the (x,y)-plane along the curve $y=x^{2}$. As it passes through the point P = (2, 4), its y coordinate changes at a rate of 8 units/sec. i) Find $dx/dt$ at the instant the particle is at the point P.

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
Remember that the rate of change of a quantity $z$ is just the derivative of that quantity with respect to time, (i.e: dz/dt).

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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. The question is asking us to find the rate of change of the variable x. (i.e: We want to solve for dx/dt). Using implicit differentiation and the equation $y=x^{2}$ we will derive a value for these quantities as follows: ${\begin{aligned}{\frac {d}{dt}}y&={\frac {d}{dt}}x^{2}\\{\frac {dy}{dt}}&=2x{\frac {dx}{dt}}\quad \rightarrow \quad {\frac {dx}{dt}}={\frac {1}{2x}}{\frac {dy}{dt}}\end{aligned}}$ From the information about the rate of change of y at the point P, we make the appropriate substitutions in the equation to determine dx/dt: ${\frac {dx}{dt}}={\frac {1}{2(2)}}8=2\quad \rightarrow \quad {\color {blue}{\frac {dx}{dt}}=2}$

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