| title | layout |
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Rates of Change and Behavior of Graphs |
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Gasoline costs have experienced some wild fluctuations over the last several decades. [link]1{: data-type="footnote-link"} lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
<tr>
<td data-align="center"><strong><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>C</mi><mrow><mo>(</mo>
<mi>y</mi>
<mo>)</mo></mrow></mrow>
</math>
| y | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
| 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.
The price change per year is a rate of change{: data-type="term"} because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in [link] did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change{: data-type="term"} over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.
The Greek letter Δ
(delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in y
divided by the change in x.
” Occasionally we write Δf
instead of Δy,
which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
- A population of rats increasing by 40 rats per week
- A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
- A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
- The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
- The amount of money in a college account decreasing by $4,000 per quarter
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
-
Calculate the difference y 2 − y 1 =Δy.
-
Calculate the difference x 2 − x 1 =Δx.
-
Find the ratio Δy Δx . {: type="1"}
per year.
shown in [link], find the average rate of change on the interval [ −1,2 ].
{: #Figure_01_03_001}
[link] shows g( −1 )=4.
At t=2,
the graph shows g( 2 )=1.
{: #Figure_01_03_002}
The horizontal change Δt=3
is shown by the red arrow, and the vertical change Δg(t)=−3
is shown by the turquoise arrow. The output changes by –3 while the input changes by 3, giving an average rate of change of
we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as ( x 1 , y 1 )
and ( x 2 , y 2 ).
<tr>
<td data-align="center"><strong><em>D</em>(<em>t</em>) (miles)</strong></td>
<td data-align="center">10</td>
<td data-align="center">55</td>
<td data-align="center">90</td>
<td data-align="center">153</td>
<td data-align="center">214</td>
<td data-align="center">240</td>
<td data-align="center">292</td>
<td data-align="center">300</td>
</tr>
</tbody></table>
on the interval [2, 4].
on the interval [1, 9].
measured in newtons, between two charged particles can be related to the distance between the particles d,
in centimeters, by the formula F( d )= 2 d 2 .
Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.
on the interval [2,6].
newton per centimeter.
on the interval [0, a].
The answer will be an expression involving a.
between t=0
and any other point t=a.
For example, on the interval [0,5],
the average rate of change would be 5+3=8.
on the interval [5, a].
As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. [link] shows examples of increasing and decreasing intervals on a function.
{: #Figure_01_03_004}
While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum{: data-type="term"}. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a local minimum{: data-type="term"}. The plural form is “local minima.” Together, local maxima and minima are called local extrema{: data-type="term"}, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.
Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
For the function whose graph is shown in [link], the local maximum is 16, and it occurs at x=−2.
The local minimum is −16
and it occurs at x=2.
{: #Figure_01_03_015}
To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. [link] illustrates these ideas for a local maximum.
{: #Figure_01_03_005}
These observations lead us to a formal definition of local extrema.
is an increasing function{: data-type="term"} on an open interval if f( b )>f( a )
for every a
, b
interval where b>a.
A function f
is a decreasing function{: data-type="term"} on an open interval if f( b )<f( a )
for every a
, b
interval where b>a.
A function f
has a local maximum at a point b
in an open interval (a,c)
if f(b)
is greater than or equal to f(x)
for every point x
(x
does not equal b
) in the interval. Likewise, f
has a local minimum at a point b
in (a,c)
if f(b)
is less than or equal to f(x)
for every x
( x
does not equal b
) in the interval.
in [link], identify the intervals on which the function appears to be increasing.
{: #Figure_01_03_006}
to t=3
and from t=4
on.
In interval notation{: data-type="term" .no-emphasis}, we would say the function appears to be increasing on the interval (1,3) and the interval (4,∞).
, t=3
, and t=4
. These points are the local extrema (two minima and a maximum).
Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.
and x=3,
and a mirror-image high point, or local maximum, somewhere between x=−3
and x=−2.
{: #Figure_01_03_007}
{: #Figure_01_03_008}
Based on these estimates, the function is increasing on the interval (−∞,−2.449)
and (2.449,∞).
Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at ± 6 ,
but determining this requires calculus.)
to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.
and the local minimum occurs at (5,−80).
The function is increasing on (−∞,−1)∪(5,∞)
and decreasing on (−1,5).
whose graph is shown in [link], find all local maxima and minima.
{: #Figure_01_03_011}
The graph attains a local maximum at x=1
because it is the highest point in an open interval around x=1.
The local maximum is the y
-coordinate at x=1,
which is 2.
The graph attains a local minimum at x=−1
because it is the lowest point in an open interval around x=−1.
The local minimum is the y-coordinate at x=−1,
which is −2.
We will now return to our toolkit functions and discuss their graphical behavior in [link], [link], and [link].
{: #Figure_01_03_012}
{: #Figure_01_03_016}
{: #Figure_01_03_017}
There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The y-
coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.
To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See [link].
{: #Figure_01_03_014}
Not every function has an absolute maximum or minimum value. The toolkit function f( x )= x 3
is one such function.
at x=c
is f( c )
where f( c )≥f( x )
for all x
in the domain of f.
The absolute minimum{: data-type="term"} of f
at x=d
is f( d )
where f( d )≤f( x )
for all x
in the domain of f.
shown in [link], find all absolute maxima and minima.
{: #Figure_01_03_013}
The graph attains an absolute maximum in two locations, x=−2
and x=2,
because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at x=−2
and x=2,
which is 16.
The graph attains an absolute minimum at x=3,
because it is the lowest point on the domain of the function’s graph. The absolute minimum is the y-coordinate at x=3,
which is −10.
| t (hours) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Average rate of change |
Δy
Δx
=
f(
x
2
|
- 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].
is increasing on (a,b)
and decreasing on (b,c),
then what can be said about the local extremum of f
on (a,c)?
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.
on [1, b]
on [ 4, b ]
on [2, 2+h]
on [3, 3+h]
on [x,x+h]
on [x,x+h]
on [9,9+h]
on [1,1+h]
on [1,1+h]
on [2,2+h]
given f( x )=2 x 2 −3x
on [x,x+h]
For the following exercises, consider the graph of f
shown in [link].
{: #Figure_01_03_201}
to x=4.
to x=5.
For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.
decreasing on (−2.5, 1)
decreasing on ( 1,3 )∪( 4,∞ )
For the following exercises, consider the graph shown in [link].
{: #Figure_01_03_206}
has a local maximum or a local minimum.
local minimum: (3, −60)
For the following exercises, consider the graph in [link].
{: #Figure_01_03_207}
absolute minimum at approximately (−7.5, −220)
| Year | Sales (millions of dollars) | | 1998 | 201 | | 1999 | 219 | | 2000 | 233 | | 2001 | 243 | | 2002 | 249 | | 2003 | 251 | | 2004 | 249 | | 2005 | 243 | | 2006 | 233 | {: #Table_01_03_03 summary="The first column is labeled, “Year”, and the second column is labeled, “Sales (millions of dollars)”. Reading the columns as ordered pairs, we have: (1998, 201), (1999, 219), (2000, 233), (2001, 243), (2002, 249), (2003, 251), (2004, 243), (2005, 243) and (2006, 233)."}
| Year | Population (thousands) | | 2000 | 87 | | 2001 | 84 | | 2002 | 83 | | 2003 | 80 | | 2004 | 77 | | 2005 | 76 | | 2006 | 78 | | 2007 | 81 | | 2008 | 85 | {: #Table_01_03_04 summary="Two columns and ten rows. The first column is labeled, “Year”, and the second column is labeled, “Population (thousands)”. Reading the columns as ordered pairs, we have: (2000, 87), (2001, 84), (2002, 83), (2003, 80), (2004, 77), (2005, 76), (2006, 78), (2007, 81), and (2008, 85)."}
For the following exercises, find the average rate of change of each function on the interval specified.
on [1, 5]
on [−2,4]
on [−4,2]
on [−3,3]
on [1, 3]
on [−3,1]
on [−1,3]
For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.
decreasing on (−∞, 3),
increasing on (3, ∞)
decreasing on (−3,−2),
increasing on (−2, ∞)
local minima at (−3.25,−47)
and (2.1,−32),
decreasing on (−∞,−3.25)
and (−0.5, 2.1),
increasing on (−3.25, −0.5)
and (2.1, ∞)
is shown in [link].
{: #Figure_01_03_213}
Based on the calculator screen shot, the point (1.333, 5.185)
is which of the following?
- a relative (local) maximum of the function
- the vertex of the function
- the absolute maximum of the function
- a zero of the function {: type="A"}
Find a number c
such that the average rate of change of the function f
on the interval (1,c)
is − 1 4 .
. Find the number b
such that the average rate of change of f
on the interval (2,b)
is − 1 10 .
where t
is in seconds and d( t )
is in feet. If an object is dropped from a certain height, find the average velocity of the object from t=1
to t=2.
days.
{: #Figure_01_03_214}
Use the graph to estimate the average decay rate from t=5
to t=15.
- {: data-type="footnote-ref" #footnote1} 1{: data-type="footnote-ref-link"} http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. {: data-list-type="bulleted" data-bullet-style="none"}
absolute maximum : the greatest value of a function over an interval ^
absolute minimum : the lowest value of a function over an interval ^
average rate of change : the difference in the output values of a function found for two values of the input divided by the difference between the inputs ^
decreasing function : a function is decreasing in some open interval if f( b )<f( a )
for any two input values a
and b
in the given interval where b>a ^
increasing function : a function is increasing in some open interval if f( b )>f( a )
for any two input values a
and b
in the given interval where b>a ^
local extrema : collectively, all of a function's local maxima and minima ^
local maximum : a value of the input where a function changes from increasing to decreasing as the input value increases. ^
local minimum : a value of the input where a function changes from decreasing to increasing as the input value increases. ^
rate of change : the change of an output quantity relative to the change of the input quantity