<< Chapter < Page Chapter >> Page >

One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let s ( t ) and v ( t ) denote the position and velocity of the car, respectively, for 0 t 1 h. Assuming that the position function s ( t ) is differentiable, we can apply the Mean Value Theorem to conclude that, at some time c ( 0 , 1 ) , the speed of the car was exactly

v ( c ) = s ( c ) = s ( 1 ) s ( 0 ) 1 0 = 45 mph .

Mean value theorem and velocity

If a rock is dropped from a height of 100 ft, its position t seconds after it is dropped until it hits the ground is given by the function s ( t ) = −16 t 2 + 100 .

  1. Determine how long it takes before the rock hits the ground.
  2. Find the average velocity v avg of the rock for when the rock is released and the rock hits the ground.
  3. Find the time t guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is v avg .
  1. When the rock hits the ground, its position is s ( t ) = 0 . Solving the equation −16 t 2 + 100 = 0 for t , we find that t = ± 5 2 sec . Since we are only considering t 0 , the ball will hit the ground 5 2 sec after it is dropped.
  2. The average velocity is given by
    v avg = s ( 5 / 2 ) s ( 0 ) 5 / 2 0 = 1 100 5 / 2 = −40 ft/sec .
  3. The instantaneous velocity is given by the derivative of the position function. Therefore, we need to find a time t such that v ( t ) = s ( t ) = v avg = −40 ft/sec . Since s ( t ) is continuous over the interval [ 0 , 5 / 2 ] and differentiable over the interval ( 0 , 5 / 2 ) , by the Mean Value Theorem, there is guaranteed to be a point c ( 0 , 5 / 2 ) such that
    s ( c ) = s ( 5 / 2 ) s ( 0 ) 5 / 2 0 = −40 .

    Taking the derivative of the position function s ( t ) , we find that s ( t ) = −32 t . Therefore, the equation reduces to s ( c ) = −32 c = −40 . Solving this equation for c , we have c = 5 4 . Therefore, 5 4 sec after the rock is dropped, the instantaneous velocity equals the average velocity of the rock during its free fall: −40 ft/sec.
    The function s(t) = −16t2 + 100 is graphed from (0, 100) to (5/2, 0). There is a secant line drawn from (0, 100) to (5/2, 0). At the point corresponding to x = 5/4, there is a tangent line that is drawn, and this line is parallel to the secant line.
    At time t = 5 / 4 sec, the velocity of the rock is equal to its average velocity from the time it is dropped until it hits the ground.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Suppose a ball is dropped from a height of 200 ft. Its position at time t is s ( t ) = −16 t 2 + 200 . Find the time t when the instantaneous velocity of the ball equals its average velocity.

5 2 2 sec

Got questions? Get instant answers now!

Corollaries of the mean value theorem

Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.

At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if f ( x ) = 0 for all x in some interval I , then f ( x ) is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly.

Corollary 1: functions with a derivative of zero

Let f be differentiable over an interval I . If f ( x ) = 0 for all x I , then f ( x ) = constant for all x I .

Proof

Since f is differentiable over I , f must be continuous over I . Suppose f ( x ) is not constant for all x in I . Then there exist a , b I , where a b and f ( a ) f ( b ) . Choose the notation so that a < b . Therefore,

Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask