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Access this online resource for additional instruction and practice with rates of change.

Key equations

Average rate of change Δ y Δ x = f ( x 2 ) f ( x 1 ) x 2 x 1

Key concepts

  • A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See [link] .
  • Identifying points that mark the interval on a graph can be used to find the average rate of change. See [link] .
  • Comparing pairs of input and output values in a table can also be used to find the average rate of change. See [link] .
  • An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See [link] and [link] .
  • The average rate of change can sometimes be determined as an expression. See [link] .
  • A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See [link] .
  • A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
  • A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
  • Minima and maxima are also called extrema.
  • We can find local extrema from a graph. See [link] and [link] .
  • The highest and lowest points on a graph indicate the maxima and minima. See [link] .

Section exercises


Can the average rate of change of a function be constant?

Yes, the average rate of change of all linear functions is constant.

If a function f is increasing on ( a , b ) and decreasing on ( b , c ) , then what can be said about the local extremum of f on ( a , c ) ?

How are the absolute maximum and minimum similar to and different from the local extrema?

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

How does the graph of the absolute value function compare to the graph of the quadratic function, y = x 2 , in terms of increasing and decreasing intervals?


For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h .

f ( x ) = 4 x 2 7 on [ 1 ,   b ]

4 ( b + 1 )

g ( x ) = 2 x 2 9 on [ 4 ,   b ]

p ( x ) = 3 x + 4 on [ 2 ,   2 + h ]


k ( x ) = 4 x 2 on [ 3 ,   3 + h ]

f ( x ) = 2 x 2 + 1 on [ x , x + h ]

4 x + 2 h

g ( x ) = 3 x 2 2 on [ x , x + h ]

a ( t ) = 1 t + 4 on [ 9 , 9 + h ]

1 13 ( 13 + h )

b ( x ) = 1 x + 3 on [ 1 , 1 + h ]

j ( x ) = 3 x 3 on [ 1 , 1 + h ]

3 h 2 + 9 h + 9

r ( t ) = 4 t 3 on [ 2 , 2 + h ]

f ( x + h ) f ( x ) h given f ( x ) = 2 x 2 3 x on [ x , x + h ]

4 x + 2 h 3


For the following exercises, consider the graph of f shown in [link] .

Graph of a polynomial.

Estimate the average rate of change from x = 1 to x = 4.

Estimate the average rate of change from x = 2 to x = 5.

4 3

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

Graph of an absolute function.
Graph of a cubic function.

increasing on ( , 2.5 ) ( 1 , ) , decreasing on ( 2.5 ,   1 )

Graph of a cubic function.
Graph of a reciprocal function.

increasing on ( , 1 ) ( 3 , 4 ) , decreasing on ( 1 , 3 ) ( 4 , )

For the following exercises, consider the graph shown in [link] .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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