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Question
Thank you for taking my question. My name is John, I am homeschooling my son in Calculus this year, but I myself am having some issues with an upcoming lesson material. The workbook isn't helping much, I was hoping some good examples would help. Could you solve the two problems below and then briefly state your steps?

#1
What is the slope of the tangent to the curve 3y^2-2x^2=6-2xy at the point (3,2)

#2
Write the equation of the tangent line to the curve of x^2+y^2=16 at (3,sqrt(7))

Thanks,
I appreciate your assistance with this (if two problems are too much, just answer one of them; I understand)
John

#1 The slope is found by taking the derivative.
Differentiating the equation gives 6yy' - 4x = - 2y - 2xy'.
Solving for y' gives ...
First, put y' on one side, giving 6yy' + 2xy' = 4x - 2y.
Next, factor out y', giving y'(6y + 2x) = 4x - 2y.
Finally, divide by the multiple, giving y' = (4x-2y)/(6y+2x).
Put in (3,2) for (x,y), giving y' = (12-4)/(12+6) = 8/18 = 4/9.

#2 Note that the point given is on the curve, since putting in (3,sqrt(7)) gives 9 + 7 = 16.
The slope would be the derivative at that point.  That would be found by differentiation the equation, giving 2x + 2yy' = 0.  It can be seen that y' = -x/y.
Put (3,sqrt(7)) in for (x,y) to get the answer.

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#### Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.