45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)​​

Verified step by step guidance

1. First, substitute the given point (1, 1) into the equation of the curve x³ + y³ = 2xy to verify if it satisfies the equation.
2. Substitute x = 1 and y = 1 into the equation: (1)³ + (1)³ = 2(1)(1).
3. Calculate the left side of the equation: 1 + 1 = 2.
4. Calculate the right side of the equation: 2 * 1 * 1 = 2.
5. Since both sides of the equation are equal, the point (1, 1) lies on the curve.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In the context of the equation x³ + y³ = 2xy, we differentiate both sides with respect to x, treating y as a function of x. This allows us to find the derivative dy/dx, which is essential for determining the slope of the tangent line at a specific point.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. To find the equation of the tangent line, we use the point-slope form, which requires the slope (found via differentiation) and the coordinates of the point on the curve.

Verifying Points on Curves

Verifying that a point lies on a curve involves substituting the coordinates of the point into the equation of the curve. If the left-hand side equals the right-hand side after substitution, the point is confirmed to be on the curve. In this case, substituting (1, 1) into the equation x³ + y³ = 2xy helps establish that the point is indeed on the curve before proceeding to find the tangent line.