To find the point on the graph of y = x^2 - 1 that is closest to the point (8, 1.5), we need to minimize the distance between the two points.
Let's denote the point on the graph as (x, x^2 - 1). The distance between the points (8, 1.5) and (x, x^2 - 1) is given by the formula:
d = sqrt((x - 8)^2 + (x^2 - 1 - 1.5)^2)
= sqrt((x - 8)^2 + (x^2 - 2.5)^2)
To minimize the distance, we need to find the value of x that minimizes the function d. We can do this by taking the derivative of d with respect to x and setting it equal to 0:
d' = (1/2) * (2(x - 8) + 2(x^2 - 2.5)(2x))
d' = 2(x - 8) + 4x(x^2 - 2.5)
d' = 2x - 16 + 4x^3 - 10x
d' = 4x^3 + 2x - 10x - 16
d' = 4x^3 - 8x - 16
Setting d' equal to 0:
4x^3 - 8x - 16 = 0
x^3 - 2x - 4 = 0
We can solve this cubic equation using numerical methods or approximations to find the value of x that minimizes the distance. This value of x will give us the x-coordinate of the point on the graph that is closest to (8, 1.5).
Find the point on the graph of y � x2 1 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 .
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