Craig is doing at a constant speed of 60 miles per hour after driving 3 hours his odometer reads 265 miles. write a linear function d that represents the miles driven after h hours. what does the odometer read after 7 hours of continuous driving?

f(x)

y-265=60(x-3)

y-265=60x-180

y =  60x+85

f(x)=60x+85

f(7)=60(7)+85

f(7)=420+85

f(7) = 505        

   505 miles

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.