Determine the slope of the graph 3(x^2+y^2)^2=100xy at the points (3,1)

Answer 1

To determine the slope of the graph at a point, we need to find the derivative of the function with respect to x and y, and then evaluate it at that point.

Taking the partial derivative with respect to x, we get:

12x(x^2+y^2) = 100y

Taking the partial derivative with respect to y, we get:

12y(x^2+y^2) = 100x

At the point (3,1), we can substitute these values to get:

12(3)(10) = 100(1) -> 360 = 100 (not true)
12(1)(10) = 100(3) -> 120 = 300 (not true)

Since these equations do not hold true at the point (3,1), we cannot find the slope of the graph at this point using the partial derivatives. We can conclude that the slope is undefined at this point.

Answer 2

3(x^2+y^2)^2=100xy
3*2(x^2+y^2)(2x + 2yy') = 100(y + xy')
at (3,1) we have
60(6+2y') = 100(1+3y')
360+120y' = 100 + 300y'
y' = 260/180 = 13/9

Answer 3

Thank you for correcting me, as an AI language model, I can sometimes make mistakes.

Using implicit differentiation, we can determine the slope at the point (3,1) as follows:

Take the derivative with respect to x:

6(x^2 + y^2)(2x + 2y(dy/dx)) + 3(2)(x^2+y^2)^2(2x + 2y(dy/dx)) = 100y + 100x(dy/dx)

Simplifying and plugging in the point (3,1), we get:

360 + 120y' = 100 + 300y'
where y' is the slope.

Solving for y', we get:

y' = 13/9

Therefore, the slope of the graph at the point (3,1) is 13/9.