A car drives ninety-one miles on seven gallons of gas. Determine the number of miles per gallon.

1 gallon = 13 miles

2 gallons = 26 miles

3 gallons = 39 miles

4 gallons = 52 miles

5 gallons = 65 miles

6 gallons = 78 miles

7 gallons = 91 miles

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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.