Final answer:
The maximum vertical distance between the line y=x and the parabola y=x^2 for -2
Step-by-step explanation:
To find the maximum vertical distance between the line y=x and the parabola y=x^2 for -2<x<3, we need to determine the points of intersection between the line and the parabola. Setting the equations equal to each other, we have: x=x^2. Rearranging the equation, we get: x^2 - x = 0. Factoring out x, we have: x(x - 1) = 0. So, the points of intersection are (0,0) and (1,1).
Now, we can calculate the maximum vertical distance by evaluating the y-values of the line and the parabola at the points of intersection. For the line y=x, at (0,0) and (1,1), the y-values are both 0 and 1 respectively. For the parabola y=x^2, at (0,0) and (1,1), the y-values are both 0 and 1 respectively.
Therefore, the maximum vertical distance between the line y=x and the parabola y=x^2 for -2<x<3 is 1 unit.