Mean Value Theorem

How to use the mean value theorem to solve its application problem: theorem, 1 example, and its solution.

Theorem

Theorem

If y = f(x) is continuous in the interval [a, b],
and
if y = f'(x) is differentiable in the interval (a, b),

then x = c exists
that satisfies
f'(c) = [f(b) - f(a)]/[b - a].

This is the mean value theorem.

Example

Example

Solution

Set f(t) as the location of the car at time t.

It says
the car passes through 10 miles in 6 minutes.

So set b - a = 6 min.
And set f(b) - f(a) = 10 miles.

f(t) is the location of the car.

The car doesn't warp or teleport.

So f(t), the location, is continuous
in the time interval [a, b].

And f(t) is differentiable
in the time interval (a, b).

Then, by the mean value theorem,
t = c exists in the time interval (a, b)
that satisfies
f'(c) = [f(b) - f(a)]/[b - a].

b - a = 6 min
f(b) - f(a) = 10 miles

So [f(b) - f(a)]/[b - a] = [10 miles]/[6 min].

To change the unit to mph (miles/hr),
multiply [60 min]/[1 hr].

Change Unit (Measure)

Cancel the denominator 6
and reduce the numerator 60 to, 60/6, 10.

Cancel the min units.

Then you get 100 miles/hr = 100 mph.

t = c exists in the time interval (a, b)
that satisfies
f'(c) = 100 mph.

So the speed of the car has reached 100mph.

So, by using the mean value theorem,
you can prove that
the speed of the car has reached 100mph.