# Mean Value Theorem

See how to use the mean value theorem.

1 example and its solution.

## Mean Value Theorem

### Theorem

then x = c exists that satisfies

f'(c) = f(b) - f(a)b - a.

Derivative Rules

f(b) - f(a)b - a: Slope between two endpoints

(= average slope)

### Example

A car passes though 10 miles in 6 minutes.

Show that the speed of the car has reached 100 mph.

Solution Show that the speed of the car has reached 100 mph.

f(t): Location of the car at time t

t: 0 ~ 6 min

f(6) - f(0) = 10 miles

f(t) is continuous in [0, 6].

f(t) is differentiable in (0, 6).

Then, by the mean value theorem,

t = c exists in (0, 6) that satisfies below.

f'(c) = f(6) - f(0)6 - 0

= 10 miles6 minutes ⋅60 minutes1 hour - [1]

= 100 miles6 hour

= 100 mph

∴ The speed of the car has reached 100 mph. - [2]

t: 0 ~ 6 min

f(6) - f(0) = 10 miles

f(t) is continuous in [0, 6].

f(t) is differentiable in (0, 6).

Then, by the mean value theorem,

t = c exists in (0, 6) that satisfies below.

f'(c) = f(6) - f(0)6 - 0

= 10 miles6 minutes ⋅60 minutes1 hour - [1]

= 100 miles6 hour

= 100 mph

∴ The speed of the car has reached 100 mph. - [2]

[1]

To change the unit to miles/hour,

× (60 minutes)/(1 hour).

× (60 minutes)/(1 hour).

[2]

This is how an average speed camera works.

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