Set the equation of the tangent equal to the equation of the curve. If the tangent is for a shape, treat the shape as a curve and only look at the boundary of that shape. For example, if your shape’s boundary curve is given as y =x^2 - 2x + 4 and the tangent given as y = 2x, create the new relation x^2 - 2x + 4 = 2x.
Use algebra to solve for one of the variables (usually x or y). In the example, move all terms to the left hand side of the equation. Thus x^2 - 2x + 4 = 2x becomes x^2 – 4x + 4 = 0. Factoring the equation yields (x - 2)^2 = 0. The solution is then x = 2.
Plug this value into the equation for the tangent and solve for the other variable. In the example, plug x = 2 into y = 2x to find y = 4.
Write the solution as a point. The example shows the point of intersection is at the coordinate (2, 4)
Find the tangent to the curve through calculus. For example, if the curve’s function is y = x^3 - 3x^2 + x - 1, taking the derivative gives y’ = 3x^2 - 6x + 1. This is the slope of the tangent.
Plug in the value of the point at which you wish to evaluate the rate of change. For the example, you may be interested in the rate of change when x is 1. In this case, let x = 1.
Perform the operations in the function to yield a number representing the rate of change. In the example, plugging in x = 1 gives y’ = 3*1^2 + -6*1 + 1. The solution then is y’ = -2, which indicates a negative rate of change.