find the point on the parabola (y=x^2) that is closest to the point (0.75,0)
To find the point on the parabola y = x^2 that is closest to the point (0.75, 0), we need to find the minimum distance between these two points.
Step 1: Start by calculating the distance formula between two points:
d = sqrt((x₂ - x₁)^2 + (y₂ - y₁)^2)
In this case, the points are (x₁, y₁) = (0.75, 0) and (x₂, y₂) which we need to find.
Step 2: Substitute the value of y from the equation of the parabola, y = x^2, into the distance formula:
d = sqrt((x₂ - 0.75)^2 + (x₂^2 - 0)^2)
Step 3: Minimize the distance by finding the minimum value of d. To do this, take the derivative of d with respect to x and set it equal to zero, then solve for x:
d' = 0
By simplifying the equation, we get:
0 = 2x₂ - 1 + 2x₂^3
Step 4: Solve the equation from step 3 to find the x-coordinate of the point on the parabola that is closest to (0.75, 0). This can be done numerically using methods such as bisection, Newton-Raphson, or calculus.
Step 5: Once you have the x-coordinate, substitute it back into the equation of the parabola, y = x^2, to find the corresponding y-coordinate.
After following these steps, you will find the point on the parabola (y = x^2) that is closest to the point (0.75, 0).