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.
If a function
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is increasing on
$\text{\hspace{0.17em}}(a,b)\text{\hspace{0.17em}}$ and decreasing on
$\text{\hspace{0.17em}}(b,c),\text{\hspace{0.17em}}$ then what can be said about the local extremum of
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}(a,c)?\text{\hspace{0.17em}}$
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,
$\text{\hspace{0.17em}}y={x}^{2},\text{\hspace{0.17em}}$ in terms of increasing and decreasing intervals?
Algebraic
For the following exercises, find the average rate of change of each function on the interval specified for real numbers
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}h.$
$f\left(x\right)=4{x}^{2}-7\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[1,\text{}b]$
$4\left(b+1\right)$
$g\left(x\right)=2{x}^{2}-9\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}\left[4,\text{}b\right]$
$p\left(x\right)=3x+4\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[2,\text{}2+h]$
3
$k\left(x\right)=4x-2\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[3,\text{}3+h]$
$f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[x,x+h]$
$4x+2h$
$g\left(x\right)=3{x}^{2}-2\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[x,x+h]$
$a\left(t\right)=\frac{1}{t+4}\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[9,9+h]$
$\frac{-1}{13\left(13+h\right)}$
$b\left(x\right)=\frac{1}{x+3}\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[1,1+h]$
$j\left(x\right)=3{x}^{3}\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[1,1+h]$
$3{h}^{2}+9h+9$
$r\left(t\right)=4{t}^{3}\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[2,2+h]$
$\frac{f\left(x+h\right)-f\left(x\right)}{h}\text{\hspace{0.17em}}$ given
$\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}-3x\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}[x,x+h]$
$4x+2h-3$
Graphical
For the following exercises, consider the graph of
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in
[link] .
Estimate the average rate of change from
$\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ to
$\text{\hspace{0.17em}}x=4.$
Estimate the average rate of change from
$\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ to
$\text{\hspace{0.17em}}x=5.$
$\frac{4}{3}$
For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.
increasing on
$\text{\hspace{0.17em}}\left(-\infty ,-2.5\right)\cup \left(1,\infty \right),\text{\hspace{0.17em}}$ decreasing on
$\text{\hspace{0.17em}}(-2.5,\text{}1)$
increasing on
$\text{\hspace{0.17em}}\left(-\infty ,1\right)\cup \left(3,4\right),\text{\hspace{0.17em}}$ decreasing on
$\text{\hspace{0.17em}}\left(1,3\right)\cup \left(4,\infty \right)$
For the following exercises, consider the graph shown in
[link] .
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?