Question

Find the point on the graph of y
x2 + that’s closest to the point 8, 1.5 . Hint: Remember
the distance formula. The distance between two points in a plane is
2 d

x1 x2
y1
2 y2 .

Answer 1

To find the point on the graph of y = x^2 that is closest to the point (8, 1.5), we need to minimize the distance between these two points using the distance formula.

Let the point on the graph of y = x^2 be (x, x^2). The distance between two points is given by the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((x - 8)^2 + (x^2 - 1.5)^2)

To minimize the distance d, we need to find the value of x that minimizes the distance function. To do this, we need to take the derivative of the distance function with respect to x and set it equal to 0:

d' = 0
0 = d/dx(sqrt((x - 8)^2 + (x^2 - 1.5)^2))

This derivative is quite complex, and it involves the chain rule. Simplifying and solving this derivative will give us the desired x value.