# Mean Value Theorem

How to solve the mean value theorem problems: theorem, examples, and their solutions.

## Theorem

If *f*(*x*) is continuous in [*a*, *b*]

and differentiable in (*a*, *b*),

then *c* exists in (*a*, *b*)

that satisfies *f*'(*c*) = [*f*(*b*) - *f*(*a*)] / [*b* - *a*].

In other words,

if the graph satisfies these two hypotheses,

no matter how the graph looks like,

there's always a point

whose tangent's slope (blue) is equal to

the slope of the green line.

## Example 1

*f*(*x*) = ln *x**f*'(*x*) = 1/*x*

So *f*'(*c*) = 1/*c*.

Derivative of ln *x*

*f*'(*c*) = 1/*c**f*(*x*) = ln *x*

So 1/*c* = [ln *e* - ln 1]/[*e* - 1].

So *c* = *e* - 1.

This means

the slope of the tangent line at *x* = *e* - 1, *f*'(*e* - 1),

is equal to

the slope of the green line, [ln *e* - ln 1]/[*e* - 1].

## Example 2

Draw a coordinate plane

whose *x* is time (hours)

and whose *y* is distance (miles).

Set *b* - *a*

as the amount of time to pass two cameras,

which is 6 minutes = 0.1 hour.

Then *f*(*b*) - *f*(*a*) is the distance between the cameras,

10 miles.

Then, by the mean value theorem,

there's a moment *c* that satisfies*f*'(*c*) = (10 miles)/(0.1 hour) = 100 mph.

So, a moment *c* exists between two cameras

that satisfies *f*'(*c*) = 100 mph.