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Notebook[{
Cell[CellGroupData[{Cell[TextData[
"Investigating by Graphing \n(Who's In Charge?) "], "Title",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["By Shirley Treadway"], "Subtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["Initialization Cells"], "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontSize->10],
Cell[TextData["Needs[\"Graphics`Colors`\"]"], "Input",
InitializationCell->True,
AspectRatioFixed->True]}, Open]],
Cell[TextData[
"\"The use of technology in instruction should alter both the teaching and \
learning of mathematics.\" - NCTM Standards"], "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["Introduction"], "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"This project will develop three lessons which will investigate and compare \
the graphs of different functions. The objective of the first lesson is to \
help the students become familiar with the properties of odd and even \
functions and also to see what happens when a function is negative. The \
second lesson takes polynomial functions of the same degree and compares \
their graphs as x gets very large. The third lesson looks at the growth of \
the square root function, the sin function, the natural log function, a \
polynomial function, and last of all the exponential function. Students will \
be given an \"Investigation Sheet\" on which to record the discoveries they \
make on the computer."], "Text",
Evaluatable->False,
AspectRatioFixed->True]}, Open]],
Cell[CellGroupData[{Cell[TextData["Exploring Polynomial Functions"], "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["Lesson 1"], "Subtitle",
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Cell[TextData[{
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"We will begin our study of polynomials by investigating \nthe shape and \
direction of certain curves using the \ngraphing capabilities of the \
software, ",
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FontSlant->"Italic"],
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}], "Text",
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Cell[TextData[{
StyleBox["\nFirst we must learn some \"simple\" ",
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StyleBox["Mathematica",
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"\ncommands for graphing. Once we learn these it\nwill be easy to just copy \
and revise them for\neach curve that we wish to investigate. Let us look at \
the \nexample below. It should graph the curve y = x\[Currency].\n\n\n ",
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}], "Text",
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Cell[TextData[{
StyleBox["l. First we must define the function so that ",
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" understands what we are wishing to graph. To do that use the \"f[x",
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StyleBox["_",
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FontFamily->"Courier",
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"]=\" after we use the Clear[f,x] command that makes sure that we haven't \
left some other value for f or x in the machine. Let's try it and see what \
happens. ",
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}], "Text",
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Cell[TextData["Clear[f,x]\nf[x_]= x^2\n"], "Input",
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Cell[TextData[{
StyleBox[
"Notice that the machine now knows that the curve we want is \
y=x\[Currency]. So we are ready to begin our plotting of this curve. we will \
name it \"aplot\". The first letter needs to be lower case because if it is a \
capital letter the machine will think it might be a ",
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StyleBox["Mathematica",
Evaluatable->False,
AspectRatioFixed->True,
FontSlant->"Italic"],
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" programming command. The {x,-100,100} tell us what our domain is and \
the others add a little flourish to our graph. Let's give it a try and take a \
look!\n",
Evaluatable->False,
AspectRatioFixed->True]
}], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"aplot=Plot[f[x],{x,-100,100},\n \
PlotStyle->{{Thickness[0.01],Maroon}},\n \
AxesLabel->{\"x\",\"f(x)\"}];\n "], "Input",
AspectRatioFixed->True],
Cell[TextData[
"\n*On your Investigation sheet, please make a sketch of this curve. \n\n2. \
Now we would like to sketch y = x\:2039 and make a sketch of it on our \
Investigation Sheet. To begin with we will need to clear and redefine--that's \
easy but if you have forgot, just scroll back up to find out how!\n\n "],
"Text",
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Cell[TextData["Clear[f,x]\nf[x_] = x^3"], "Input",
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Cell[TextData[
" Okay!! It looks like our machine knows that we are wanting a new function \
and that is good. Now we would like to graph it for\nthe same domain and \
with the same pretty Maroon color. We could retype that entire mess from \
above or we could use this super technology and just copy and revise. Here is \
how you do that---First scroll back up to our commands for graphing \"aplot\" \
and highlight all of that cell. Now pull down the edit menu and highlight \
\"copy\". Then move to where we want our new graph and click the mouse. Now \
go back to the edit menu and click on paste. You should have the very same \
command that we used to plot our first graph!!\n"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
" But we really don't want the same graph--but remember that we redefined \
f(x) so maybe we will be okay--let's take a look and see if we have the same \
curve."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"aplot=Plot[f[x],{x,-100,100},\n \
PlotStyle->{{Thickness[0.01],Maroon}},\n \
AxesLabel->{\"x\",\"f(x)\"}];"], "Input",
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Cell[TextData[
"Would you look at that \"hugger\"!!! This machine may make us look smart \
yet!! When we renamed f(x) the machine understood and it graphed the new \
function. Before we get too excited we had better return to our investigation \
sheet and sketch a graph. "], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"We are about to proceed--we want to continue our investigation by looking at \
y= x\:203a. Remember to first redefine your function and then copy. "], "Text",\
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Cell[TextData["Remember to record on your Investigaton Sheet."], "Input",
AspectRatioFixed->True,
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Cell[TextData[
"This is getting interesting, right? This one looks a little like another of \
our graphs doesn't it? Let us continue. We may become technological giants, \
yet."], "Text",
Evaluatable->False,
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Cell[TextData[
"3. Repeat this procedure for y = x\[FiLigature] through y = x\:2215\:201a \
and be sure to record each sketch on your investigation sheet. \n\n4. Now \
let's see what conclusions we can draw from our sketches. \n Record \
the answer to the following questions on your investigations sheet.\n A. \
Are the sketches of some of these similar?\n B. Do some seem to fit in the \
same group?"], "Text",
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Cell[TextData[{
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" C. Why might we call some of these \"even\" functions and\n some \
\"odd\"?\n D. How do they differ?\n E. Describe each group in your own \
words. \n5. Now let's get busy and check out what happens if these \
functions need an attitude adjustment--that is that they are negative. Let's \
graph y = -x\[Currency] through y = -x\:2215\:201a and record these. Remember \
to use our Mighty Mac and ",
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StyleBox["Mathematica",
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FontSlant->"Italic"],
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" to make this happen quick and easy. Record these sketches and compare \
these to their positive cousins from our work above. \n ",
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}], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Bonus time: Just to see if you really are becoming a technological giant \
let's try something. As you know one of these days we may consolidate and it \
will not be so important to be Maroon--in fact we might find orange and blue \
to be as attractive at RHS as it is at my old alma mater, U of I. So try to \
regraph your favorite curve making it orange instead of maroon."], "Text",
Evaluatable->False,
AspectRatioFixed->True]}, Open]]}, Open]],
Cell[CellGroupData[{Cell[TextData[
"Exploring Polynomials of the Same Power And Dominance"], "Section",
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Cell[CellGroupData[{Cell[TextData["Lesson 2"], "Subtitle",
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Cell[CellGroupData[{Cell[TextData[
"\n\nThe task of this lesson is to investigate what happens to certain \
functions as x gets very, very large. Hopefully this will bring to mind the \
famous mathematical term \"limit\". We also will see why it is so helpful to \
move to the computer to make these discoveries. \n "], "Text",
Evaluatable->False,
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Cell[TextData[
"l. We will graph three functions together on the same\n axis and then \
change our domain \"until x gets very large\" to see if we might make a good \
conclusion about what part of the\nexpression for each curve is most \
important. The commands that tell the mighty Mac what to do are a little bit \
more complicated but I'll bet we can handle it! Just as before we will need \
to tell the machine that we want to start with a clean slate--so let's begin \
by clearing f and x and g and h and j because we will need to see at least \
three functions at the same time. Then we will need to to define our \
functions. Let's begin with three functions that all are quadratic, but some \
have more baggage than others.\n"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["\nClear[f,x,g,h,j]"], "Input",
AspectRatioFixed->True],
Cell[TextData["f[x_]= x^2\ng[x_]= x^2 - 5 x\nh[x_]= x^ 2 + 8x -25"], "Input",
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Cell[TextData[
"Now our machine knows what three functions we are wanting to see--poor thing \
it turned these around--it obviously missed the Algebra I lecture about \
descending order-- or maybe it has decided that this very important matter \
really isn't so important after all. Whatever--let's get to the graph!!"],
"Text",
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AspectRatioFixed->True],
Cell[TextData[
"dominantplot=Plot[{f[x],g[x],h[x]},{x,-10,10}, \n PlotStyle->{Red, \
Blue, Green,},\n AxesLabel->{\"x\",\"f(x)\"}];"], "Input",
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Cell[TextData[
" 2. These folks look a lot alike but let's see what happens as we look at \
them from a much \"wider\" persepective--in math jargon---let's change the \
domain and see what happens. Remember how we just copied before --well this \
time it is even easier. As soon as you make a sketch on your Investigation \
Sheet, take the cursor and go back to the commands before the graph. We \
learned in lesson 1 that {x,-10,10} determines the domain of our graph. \
Remembering that we are headed to infinity(to find the limit we want to go \
big time!) so let's increase this to {x,-100,100} by just going up there and \
changing it."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"\n3. Son of a gun as Dr. Uhl says these huggers are \"sharing ink\". Just \
for fun let's try {x,-1000,1000}.\n"], "Text",
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Cell[TextData[
"And this time they really \"share ink\"--that is if you did your graph \
correctly. But the big question is --who wins??? That is which one of the \
original curves do the others seem to mimick?If you agree that it is y = x\
\[Currency] then we can say that x\[Currency] is the dominant term--that is \
that in g(x) and h(x) the x\[Currency] term dominates the other terms. So as \
x gets really large all three of these graphs look like the graph of what? "],
"Text",
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Cell[TextData[
"4. Now let's repeat this for three new functions, y = x\:2039 , y = x\:2039 \
- 3x\[Currency] + 5x - 4 y = x\:2039 - 6x . Don't forget to clear x,f,g, and \
h. Then define the three functions. \n"], "Text",
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Cell[TextData[
"Clear[x,f,g,h]\nf[x_]= x^3\ng[x_] = x^3 - 3x^2 + 5x - 4 \nh[x_] = x^3 - 6x"],
"Input",
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Cell[TextData[
"5. Let's continue by copying our graphing commands from before and repeat as \
we did before---That is begin with a domain of -10 to 10 , then 100 to -100, \
and last -1000 to 1000."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"\n\n6. As a result of the exercise above, please record on your \
Investigation Sheet what term seems to be dominant in each of these \
functions."], "Text",
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Cell[TextData[{
StyleBox[
"Note: In exercise 5 would we have had to go as far as the last domain to \
learn what we wanted to know? In Calculus and ",
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AspectRatioFixed->True],
StyleBox["Mathematica",
Evaluatable->False,
AspectRatioFixed->True,
FontSlant->"Italic"],
StyleBox[
" we learn a term, \"global scale\" that stems from this situation. What \
do you think the meaning of \"global scale\" might be?",
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}], "Text",
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Cell[TextData[
"7. Continue this with the following sets of functions (our Mighty Mac will \
not be afraid of these even though all would agree they are a bit messy):\n \
A. y = x\:203a, y = x\:203a - x\[Currency] + 8 , y = x\:203a - 7x\:2039 + 2 \
x\[Currency] -5 + 7\n \n \:ffff B. y = x\[DoubleDagger], y = x\
\[DoubleDagger]-5x\:203a + 13x \:2039 -26x\[Currency] + 34 x - 728 , y = x\
\[DoubleDagger]-5x\[FlLigature] + 3x\[FiLigature] + 8x\:203a -9x\:2039+11x\
\[Currency]+27x - 95"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"8. Now let's try some conclusions.\n A. In each of the functions in \
exercise 7, what term seems \n to be dominant?\n B. If y = x \
\[FiLigature]- 4x\:203a + 6x\:2039 -4x\[Currency] + 8x - 3 , what is the \n \
dominant term?\n \.03C. Go to your Investigation Sheet and circle \
the dominant \n term of each function listed there.\n \n D. \
In your own words please tell me what this would mean \n in terms of \
the limit of these functions as x gets very \n large.\nBonus: Just for \
fun, you may go back to any one of the sets of curves above and add \
another(j(x)) to the set and begin the graph---feel free to be creative with \
your new function.--Is the end result still the same?Make a sketch on your \
Investigation Sheet. \:ffff\n \n \n "], "Text",
Evaluatable->False,
AspectRatioFixed->True]}, Open]]}, Open]]}, Open]]}, Open]],
Cell[CellGroupData[{Cell[TextData[
"Exploring the Growth of Different Functions"], "Section",
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Cell[TextData["\n\n\nLesson 3"], "Subtitle",
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Cell[TextData[
"Here we go again with another mathematical investigation. This time we are \
going to become familiar with other functions that are not necessarily \
polynomial. Some of these are much more \"dominant\" than others. Lesson 4 \
will be an investigation of the comparative \"dominance\" of these fellows by \
speculation and investigation with our Mighty Mac, but for now we just would \
like to know each when we see it."], "Text",
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Cell[TextData[{
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"l. Let's begin by looking at a function that is the square root of x.\n \
Remember how we use ",
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" to graph. Because square root is involved be sure to think carefully \
about the domain of x.\n Also just for fun let's call this plot oblongplot.",
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}], "Text",
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Cell[TextData[
"2. Let's look to see if this is what happens if we alter it a little. How \
about y = 3\[EAcute]x ?"], "Text",
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Cell[TextData[
"3. Not much change is there? Because our Mac is so easy to use, we might \
just make that coefficient 100. Now we can feel pretty\ncomfortable about the \
nature of this fella!!! In your own words, describe the \"growth\" of this \
curve on your Investigation Sheet. Include in your answer why you think this \
seems like a logical picture for this plot. (As we make comparisons with \
other functions, we'll see why we need to be able to put this in words--also \
we can guess why we call it oblongplot.)\n\n"], "Text",
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Cell[TextData[{
StyleBox[
"4. Next out in the forest we have our friend \"log\". Let's graph y = \
log[x]. Our first plot was oblongplot, now let's make this one palestineplot \
and call it g(x) instead of f(x), so these will be different. If you don't \
remember how to put this in computer, let me know. For our friendly software \
package, ",
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StyleBox["Mathematica",
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" Log means the natural log or what we usually refer to as \"ln\". Also \
you will need to watch the domain---think a minute about what we know about \
the log function and what it can have as a domain. If you happen to get \"red \
lined\" the problem is probably with the domain. Describe it's growth on your \
Investigations Sheet. \n\n ",
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}], "Text",
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Cell[TextData[
"5. Now let's plot log, and square root on the same axis---just to be able to \
see the difference in growth(remember that has to do with what happens as x \
gets very big or maybe the word limit)."], "Text",
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Cell[TextData[{
StyleBox[
"Good for us! We have the functions in. Now let's try to graph them. This \
time instead of rewriting the plot command and doing both, we can use a new \
command, Show. If we tell our machine to Show{oblongplot, palestineplot}, \
this will give us both graphs and we'll be in good shape. You will need to \
go back and activate each of the two plots and then use Show---a little quick \
trick from the world of ",
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StyleBox["Mathematica",
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".. On your Investigations Sheet, please record a sketch of each of these \
and also record which one appears to be dominant at this point.",
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}], "Text",
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Cell[TextData[
"6. We might try hutsonvilleplot which happens to be a the sin function from \
the world of trigonometry. It now becomes h(x), Let's plot these three \
together. Since their colors are orange and block, let's try orange for h(x). \
Record your observations about the growth of each of these three.\n\n7. Just \
a minute and let's think where we are---we've looked at logs, square roots, \
and now, let's look at some power growth and we'll call it robinsonplot. \
Let's plot a very friendly polynomial of your choosing, which now becomes \
j(x). "], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Whew, we knew that robinson was good, but that growth is the best yet. \
Record a sketch of it on your Investigation Sheet and indicate how its growth \
compares with the rest of the county."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"8. Remember exponential growth--our old friend from the national debt. This \
time your task is to plot some real number raised to the x power(7\
\[CapitalUGrave] would be fine--and let's call it crawfordplot). Give it the \
color of your choice--just not one that we've already used. Sketch the graph \
of 7\[CapitalUGrave] and see how it grows. Describe its growth on your \
Investigation Sheet.\n9. For our grand finale--we will plot each town and the \
whole county on our map(graph). The domain again is very important, because \
of some of the functions we need to be careful--experiment but be sure to \
look at x from 1 to 10. Remember we need to look at x as time goes on to get \
a true picture. Write a short paragraph describing and contrasting the graphs \
of all. Be sure to include how the picture changes as x gets larger. For a \
bonus point or two, explain the significance of the naming of these graphs."],
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" Investigation Sheet \n Lesson 1 "],
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StyleBox["l. Record a sketch of y = x",
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"\[Currency].\n\n\n2. Record a sketch of y = x\:2039 and y = x\:203a.\n\n\n\
\n3. Now record each of the functions for y = x\[FiLigature] through x\:2215\
\:201a.\n\n\n\n\n\n",
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"4. A. Are the sketches of some similar?\n\n B. Do some seem to fit in the \
same group?\n \n C. Why might we call one group \"odd\" and the other \
\"even\"?\n \n D. How do they differ?\n \n E. Describe each group in your \
own words.\n \n \n5. Record the sketches of the \"negative functions\" and \
compare to \n the positives functions.\n \n BONUS: \n "], "Text",
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"2. Make a sketch for each of three domains.\n\n\n\n\n3. A. Which one is \
mimicked by the others?\n\n B. So as x gets really big all three looks like \
the graph of which one?\n \n4. Computer Activity \n5. Repeat exercise \
two for the functions in exercise 4. \n\n6. Repeat exercise three for the \
function in exercise 4. "], "Text",
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Cell[TextData[
"7A. Repeat exercises 2 and 3 for these functions.\n \n \n \n B. Repeat \
exercises 2 and 3 for these functions. \n"], "Text",
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Cell[TextData[
"8. A. In each of the functions in exercise 7, what term seems to be \
dominant?"], "Text",
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