What do the differences between the points (as shown on the graph) represent?

On a coordinate plane, point B is at (2, 7) and point D is at (4, 14).

The difference between the points is

Explanation:

The coordinate of point B is

The coordinate of point D is

The difference between the points can be determined using the formula,

Substituting the coordinates and , in the formula, we get,

Subtracting, we get,

Squaring the terms, we have,

Adding, we get,

Hence, the difference between the points is

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.