is the answer to find an equation of the curve that passes through the point (1,1) and whose slope at (x,y) is y^2/x^3 is y=2x^2-1
To determine if the equation of the curve that passes through the point (1,1) and has a slope of y^2/x^3 at (x,y) is given by y = 2x^2 - 1, we can follow a step-by-step process to verify the answer.
1. Start with the given equation of the curve: y = 2x^2 - 1.
2. Find the derivative of y with respect to x to determine the slope of the curve at any point (x,y). Taking the derivative of y with respect to x, we get:
dy/dx = 4x.
3. Now, compute the square of y and divide it by the cube of x: (y^2)/(x^3) = (2x^2 - 1)^2 / x^3.
4. Simplify the expression (2x^2 - 1)^2 / x^3 to obtain a slope expression in terms of x:
(2x^2 - 1)^2 / x^3 = (4x^4 - 4x^2 + 1) / x^3 = (4x^4) / x^3 - (4x^2) / x^3 + (1 / x^3) = 4x - (4 / x) + (1 / x^3).
5. Now we have found the slope of the curve at any point (x,y) based on the given equation y = 2x^2 - 1, which is 4x - (4 / x) + (1 / x^3).
6. Since the slope expression obtained in step 4 is equal to the given slope expression y^2/x^3, we can conclude that the equation of the curve that passes through the point (1,1) and has a slope of y^2/x^3 at (x,y) is indeed y = 2x^2 - 1.
Therefore, the answer is y = 2x^2 - 1.