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
and
denote the position and velocity of the car, respectively, for
h. Assuming that the position function
is differentiable, we can apply the Mean Value Theorem to conclude that, at some time
the speed of the car was exactly
Mean value theorem and velocity
If a rock is dropped from a height of 100 ft, its position
seconds after it is dropped until it hits the ground is given by the function
Determine how long it takes before the rock hits the ground.
Find the average velocity
of the rock for when the rock is released and the rock hits the ground.
Find the time
guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is
When the rock hits the ground, its position is
Solving the equation
for
we find that
Since we are only considering
the ball will hit the ground
sec after it is dropped.
The average velocity is given by
The instantaneous velocity is given by the derivative of the position function. Therefore, we need to find a time
such that
Since
is continuous over the interval
and differentiable over the interval
by the Mean Value Theorem, there is guaranteed to be a point
such that
Taking the derivative of the position function
we find that
Therefore, the equation reduces to
Solving this equation for
we have
Therefore,
sec after the rock is dropped, the instantaneous velocity equals the average velocity of the rock during its free fall:
ft/sec.
At time
sec, the velocity of the rock is equal to its average velocity from the time it is dropped until it hits the ground.
Suppose a ball is dropped from a height of 200 ft. Its position at time
is
Find the time
when the instantaneous velocity of the ball equals its average velocity.
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
for all
in some interval
then
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
be differentiable over an interval
If
for all
then
constant for all
Proof
Since
is differentiable over
must be continuous over
Suppose
is not constant for all
in
Then there exist
where
and
Choose the notation so that
Therefore,
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
