Identify all discontinuities for the following functions as either a jump or a removable discontinuity.
Notice that the function is defined everywhere except at
Thus,
does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as
approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at
Condition 2 is satisfied because
Notice that the function is a
piecewise function , and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining the left- and right-hand limits as
approaches 2.
Left-hand limit:
The left-hand limit exists.
Right-hand limit:
The right-hand limit exists. But
So,
does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at
Recognizing continuous and discontinuous real-number functions
Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of
as
approaches
is the same as the value of
when
So
There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.
Examples of continuous functions
The following functions are continuous everywhere:
Polynomial functions
Ex:
Exponential functions
Ex:
Sine functions
Ex:
Cosine functions
Ex:
The following functions are continuous everywhere they are defined on their domain:
Logarithmic functions
Ex:
,
Tangent functions
Ex:
is an integer
Rational functions
Ex:
Given a function
determine if the function is continuous at
Check Condition 1:
exists.
Check Condition 2:
exists at
Check Condition 3:
If all three conditions are satisfied, the function is continuous at
If any one of the conditions is not satisfied, the function is not continuous at
Determining whether a piecewise function is continuous at a given number
Determine whether the function
is continuous at
To determine if the function
is continuous at
we will determine if the three conditions of continuity are satisfied at
.
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches 1.
Left-hand limit:
Right-hand limit:
Because
does not exist.
There is no need to proceed further. Condition 2 fails at
If any of the conditions of continuity are not satisfied at
the function
is not continuous at
Condition 1: Does
exist?
Condition 2: Does
exist?
To the left of
to the right of
We need to evaluate the left- and right-hand limits as
approaches
Left-hand limit:
Right-hand limit:
Because
exists,
Condition 3: Is
Because all three conditions of continuity are satisfied at
the function
is continuous at
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
Got questions? Join the online conversation and get instant answers!