Final answer:
To find the tangent lines parallel to x² = 2y - 2 for the curve y = x²/(x+1), calculate the slope of the given line, find the derivative of the curve, equate it to the line's slope, and use the point-slope form with the discovered points.
Step-by-step explanation:
To find the equations of the tangent lines to the curve y = x²/(x+1) that are parallel to the line x² = 2y - 2, we first need to find the slope of the given line. Rewriting the line's equation in slope-intercept form, we get y = (1/2)x² + 1, which means the slope is the derivative of (1/2)x², which is x. Next, we find the derivative of the given curve, which gives us the slope of the tangent lines at any point on the curve. The derivative of the curve y = x²/(x+1) is found using the quotient rule. We will set the derivative equal to the slope of the given line, x, to find the points where the tangent is parallel to the line. Once we have those points, we use the point-slope form y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point on the curve, to write the equations of the tangents.