Mean Value Theorem

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].

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

For the given function, find the value of c that satisfies the mean value theorem in [1, e]. y = ln x

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

A car passed thwo average speed cameras in 6 minutes. The distance between two cameras are 10 miles. Show that the speed of the car ever reached 100mph.

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.