After a cleanup, algae covers 2 miles of a coastline. The length of the coastline covered after the cleanup is $\frac{1}{3}$ 1
3 of the previous length. How many miles of the coast did the algae previously cover?

The miles previously covered by algae is 6 miles

let m represent the miles of the coastline covered by algae before the clean-up  

fraction of miles covered after the clean-up = x m =

miles covered after the clean-up = 2 miles

= 2 miles

to determine the value of m, make m the subject of the formula.

To do this, take the following step :

multiply both sides of the equation by 2

m = 2 miles x 3

m = 6 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.