Describe the significance of the Mean Value Theorem.
State three important consequences of the Mean Value Theorem.
The
Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.
Rolle’s theorem
Informally,
Rolle’s theorem states that if the outputs of a differentiable function
are equal at the endpoints of an interval, then there must be an interior point
where
[link] illustrates this theorem.
Rolle’s theorem
Let
be a continuous function over the closed interval
and differentiable over the open interval
such that
There then exists at least one
such that
Proof
Let
We consider three cases:
for all
There exists
such that
There exists
such that
Case 1: If
for all
then
for all
Case 2: Since
is a continuous function over the closed, bounded interval
by the extreme value theorem, it has an absolute maximum. Also, since there is a point
such that
the absolute maximum is greater than
Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point
Because
has a maximum at an interior point
and
is differentiable at
by Fermat’s theorem,
Case 3: The case when there exists a point
such that
is analogous to case 2, with maximum replaced by minimum.
□
An important point about Rolle’s theorem is that the differentiability of the function
is critical. If
is not differentiable, even at a single point, the result may not hold. For example, the function
is continuous over
and
but
for any
as shown in the following figure.
Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points
where
Using rolle’s theorem
For each of the following functions, verify that the function satisfies the criteria stated in Rolle’s theorem and find all values
in the given interval where
over
over
Since
is a polynomial, it is continuous and differentiable everywhere. In addition,
Therefore,
satisfies the criteria of Rolle’s theorem. We conclude that there exists at least one value
such that
Since
we see that
implies
as shown in the following graph.
As in part a.
is a polynomial and therefore is continuous and differentiable everywhere. Also,
That said,
satisfies the criteria of Rolle’s theorem. Differentiating, we find that
Therefore,
when
Both points are in the interval
and, therefore, both points satisfy the conclusion of Rolle’s theorem as shown in the following graph.
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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