Advanced Math/Calculus

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Expert: - 7/7/2010


Question
"Find the equations of the tangent line and the normal line to the curve (x^2+y^2)^2 =(x-y)^2 at the point (1, -1) on the curve."

Answer
It says that (1,-1) is on the curve.  To check and make sure, we put (1,-1) in as (x,y),
and the result is 4 = 4, so that's OK.

Differentiating the equation as is involves the chain rule.
We have different functions of x and y that are squared on both sides,
and the derivative of h²(x) is 2h(x)h'(x).
The derivative of x²+y² is 2x + 2yy' and the derivative of x-y is 1-y'.

Applying this to the problem gives 2(x²+y²)(2x+2yy') = 2(x-y)(1-y').

Both sides have a leading 2, and they cancel, giving (x²+y²)(2x+2yy') = (x-y)(1-y').

We need to solve for y', so first we multiply out, giving
2x³ + 2xy² + 2x²yy' + 2y³y' = x - y - xy' + y'.

Add the negative of two terms at the end on the right that have y' to both sides,
so add xy' - y' to both sides, so all the y' terms are on the left side of the equation.

Add the negative of the first two terms on the left side of the equation,
so add -2x³ - 2xy² to both sides, putting all terms with no y' on the right side of the equation.

Factor y' out of each of the terms on the left, so we have y'f(x,y) = g(x,y),
where f(x,y) is all of the terms multiplied by y' and g(x,y) is the other stuff.
That is, f(x,y) = 2x²y + 2y³ + x - 1 and g(x,y) = x - y - 2x³ - 2xy²

Lastly, to get y', note that it is y' can be found by dividing by f(x,y), so y' = g(x,y)/f(x,y).

Put in (1,-1) for (x,y)/g(x,y) and this will give the value of y'.

The equation to use for a line is y - y0 = m(x - x0), where (x0,y0) = (1,-1) and m = slope,
which is the value of the derivative at (1,-1).  Remember that the derivative is the slope.

If needed, this equation can be converted to y = mx - m*x0 + y0, since (x0,y0) = (1,-1),
y = mx - m - 1, and wwe just found m last paragraph.

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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 even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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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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