To estimate M'(10), M'(60), and M'(90), we need to evaluate the derivative function M'(x) at those values of x.
To calculate the derivative of M(x), we use the rules of differentiation. In this case, the function M(x) is given as M(x) = 1 / (3,600x + x^2).
Now let's find M'(x):
M'(x) = [d/dx (1)] / (3,600x+x^2) + 1 / [d/dx (3,600x+x^2)]
= 0 / (3,600x+x^2) + 1 / (3,600 + 2x)
Now we can substitute the values x = 10, x = 60, and x = 90 into the derivative function M'(x) to estimate M'(10), M'(60), and M'(90).
M'(10) = 1 / (3,600 + 2*10)
= 1 / (3,600 + 20)
= 1 / 3,620
≈ 0.000276243
M'(60) = 1 / (3,600 + 2*60)
= 1 / (3,600 + 120)
= 1 / 3,720
≈ 0.000268817
M'(90) = 1 / (3,600 + 2*90)
= 1 / (3,600 + 180)
= 1 / 3,780
≈ 0.000264901
Therefore, the estimated values for M'(10), M'(60), and M'(90) are approximately 0.000276243 mpg/mph, 0.000268817 mpg/mph, and 0.000264901 mpg/mph, respectively.