<< Chapter < Page Chapter >> Page >

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

Verbal

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

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

Got questions? Get instant answers now!

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 ) ?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Algebraic

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 )

Got questions? Get instant answers now!

g ( x ) = 2 x 2 9 on [ 4 ,   b ]

Got questions? Get instant answers now!

p ( x ) = 3 x + 4 on [ 2 ,   2 + h ]

3

Got questions? Get instant answers now!

k ( x ) = 4 x 2 on [ 3 ,   3 + h ]

Got questions? Get instant answers now!

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

4 x + 2 h

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1 13 ( 13 + h )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

3 h 2 + 9 h + 9

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4 x + 2 h 3

Got questions? Get instant answers now!

Graphical

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.

Got questions? Get instant answers now!

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

4 3

Got questions? Get instant answers now!

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

Graph of a cubic function.

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

Got questions? Get instant answers now!
Graph of a reciprocal function.

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

Got questions? Get instant answers now!

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

Questions & Answers

can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
Practice Key Terms 9

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask