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12.4 Derivatives  (Page 2/18)

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Finding the average rate of change

Find the average rate of change connecting the points   ( 2 , −6 )   and   ( −1 , 5 ) .

We know the average rate of change connecting two points may be given by

AROC = f ( a + h ) − f ( a ) h .

If one point is   ( 2 , − 6 ) , or   ( 2 , f ( 2 ) ) , then   f ( 2 ) = −6.

The value   h   is the displacement from   2   to   − 1 , which equals   − 1 − 2 = −3.

For the other point,   f ( a + h )   is the y -coordinate at   a + h , which is   2 + ( −3 )   or   −1 , so   f ( a + h ) = f ( −1 ) = 5.

AROC = f ( a + h ) − f ( a ) h             = 5 − ( − 6 ) − 3             = 11 − 3             = − 11 3
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Find the average rate of change connecting the points   ( − 5 , 1.5 )   and   ( − 2.5 , 9 ) .

3

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Understanding the instantaneous rate of change

Now that we can find the average rate of change, suppose we make   h   in [link] smaller and smaller. Then   a + h   will approach   a   as   h   gets smaller, getting closer and closer to 0. Likewise, the second point   ( a + h , f ( a + h ) )   will approach the first point,   ( a , f ( a ) ) .   As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at   x = a , and the slope of the secant line will get closer and closer to the slope of the tangent at   x = a .   See [link] .

Graph of an increasing function that contains a point, P, at (a, f(a)). At the point, there is a tangent line and two secant lines where one secant line is connected to Q1 and another secant line is connected to Q2.
The connecting line between two points moves closer to being a tangent line at   x = a .

Because we are looking for the slope of the tangent at   x = a , we can think of the measure of the slope of the curve of a function   f   at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change , or the derivative of the function at   x = a .   Both can be found by finding the limit of the slope of a line connecting the point at   x = a   with a second point infinitesimally close along the curve. For a function   f   both the instantaneous rate of change of the function and the derivative of the function at   x = a   are written as   f ' ( a ) , and we can define them as a two-sided limit    that has the same value whether approached from the left or the right.

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

The expression by which the limit is found is known as the difference quotient .

Definition of instantaneous rate of change and derivative

The derivative    , or instantaneous rate of change    , of a function   f   at   x = a , is given by

f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

The expression   f ( a + h ) − f ( a ) h   is called the difference quotient.

We use the difference quotient to evaluate the limit of the rate of change of the function as   h   approaches 0.

Derivatives: interpretations and notation

The derivative    of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.

  • The derivative of a function   f ( x )   at a point   x = a   is the slope of the tangent line to the curve   f ( x )   at   x = a .   The derivative of   f ( x )   at   x = a   is written   f ′ ( a ) .
  • The derivative   f ′ ( a )   measures how the curve changes at the point   ( a , f ( a ) ) .
  • The derivative   f ′ ( a )   may be thought of as the instantaneous rate of change of the function   f ( x )   at   x = a .
  • If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time   t = a .

Questions & Answers

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.
Kate Reply
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?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
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
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
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
Saheed Reply
50 m/s due south east
Someone
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
Ramon Reply
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
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
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?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
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?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
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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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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