Consequences of the MVT

Example Consequences of the MVT

Example 1: Suppose that you drive 100 miles in 2 hours. Then there was some time during your trip at which you were going 50 MPH.

Discussion: Let s(t) be the distance traveled in miles at t hours after the start of your trip. Then s(0) = 0 and s(2) = 100. By the MVT, assuming that s(t) is continuous on [0,2] and differentiable on (0,2), there is a time t between 0 and 2 such that

100 miles - 0 miles

s'(t) =
= 50 MPH.

2 hours - 0 hours

Since the derivative of s(t) is speed, this is exactly the result we need. The differentiability assumption is satisfied by physical functions such as s(t).

Now YOU do one: Suppose you are driving in New Jersey (hey! it could happen!) where there are toll booths and that

Two toll booths A and B are 30 miles apart.
The Speed Limit is 65 MPH.
You pay your toll at toll booth A at 1:32 pm and at toll booth B at 1:56 pm.

Do you deserve a speeding ticket? Why or why not?

Example 2: Suppose the rate at which a well produces water varies between 2 and 10 gallons per minute. If you run the well for an hour, can conclude that the total water pumped is between 120 and 600 gallons?

Discussion: Let W(t) represent the total amount of water which has been pumped after t minutes. Assume that W is differentiable (a valid assumption). Then W'(t) is the instantaneous change in the amount of water pumped, which is the same thing as the rate at which water is being produced by the well.

We are interested in W(60). By the MVT there is a linear function L(t) with L(0) = W(0) = 0, with L(60) = W(60), and with L'(t) = W'(c) for some c between 0 and 60. Thus

W(60) = L(60) = 0 + W'(c) (60 - 0) = W'(c)^.60.

By hypothesis, 2 W'(c) 10, so W(60) is between 120 and 600.

Now YOU do one: Mary lived in a small cabin in which the only electric appliances were two lights with 100 watt bulbs and a computer system which consumed a maximum of 450 watts per hour. One month her electric bill showed a usage of 500 kilowatt-hours. She went to the power company and claimed someone was stealing her power.

Was her claim correct? Why or why not?

Example 3: Recall that the first derivative test says that if the derivative of a function is positive on an interval then that function is increasing on that interval.

Discussion: Suppose that f '(x) is defined and positive for every point of an interval (a,b) BUT that f is not increasing on (a,b). Then, by the definition of "increasing", there are two points x and y with a < x < y < b such that f(x) f(y). Applying the MVT to the interval [x,y], there is a point c in this interval where

	f(y) - f(x)
f '(c) =
	y - x

which is NOT POSITIVE because y - x > 0 and f(y) - f(x)

0. This is a contradiction.

Note: The first derivative test also applies to intervals such as [a,b), (a, b], and [a,b] where the function is differentiable on (a,b) and continuous on the larger interval, as well as to the case where f '(x) < 0 and the function is decreasing. The hypotheses of the MVT are strong enough to handle these cases.

Example 4: Suppose that f(x) is a function which is continuous on the interval [0,5] and differentiable on the interval (0,5). Suppose further that |f '(x)| < 2 for all x in (0,5) and that f(0) = 0. Then |f(5)| < 10.

Discussion: By the MVT there is a linear function L(x) = f(0) + m(x - 0) such that L(5) = f(5) and m = f '(c) for some c in the interval (0,5). By hypothesis, |f '(c)| < 2 and f(0) = 0. Hence

|f(5)| = |L(5)| = |m(5 - 0)| = 5|f '(c)| < 5*2 = 10.

On the next page we state and prove a special case of the Mean Value Theorem and the derive the full MVT.

/Stage7/Lesson/consequences.html

COVER CQ DIRECTORY HUB GLOSSARY