Let
be a function and
be in its domain. If
is differentiable at
then
is continuous at
Proof
If
is differentiable at
then
exists and
We want to show that
is continuous at
by showing that
Thus,
Therefore, since
is defined and
we conclude that
is continuous at
□
We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function
This function is continuous everywhere; however,
is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For
The function
is continuous at
but is not differentiable at
Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function
Thus
does not exist. A quick look at the graph of
clarifies the situation. The function has a vertical tangent line at
(
[link] ).
The function
has a vertical tangent at
It is continuous at
but is not differentiable at
The function
also has a derivative that exhibits interesting behavior at
We see that
This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero (
[link] ).
The function
is not differentiable at
In summary:
We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable.
We saw that
failed to be differentiable at
because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at
From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point.
As we saw in the example of
a function fails to be differentiable at a point where there is a vertical tangent line.
As we saw with
a function may fail to be differentiable at a point in more complicated ways as well.
A piecewise function that is continuous and differentiable
A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line (
[link] ). The function that describes the track is to have the form
where
and
are in inches. For the car to move smoothly along the track, the function
must be both continuous and differentiable at
Find values of
and
that make
both continuous and differentiable.
For the car to move smoothly along the track, the function must be both continuous and differentiable.
For the function to be continuous at
Thus, since
and
we must have
Equivalently, we have
For the function to be differentiable at
must exist. Since
is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:
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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