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Cell[CellGroupData[{
Cell["Calculus: The Language of Change", "Title"],
Cell[TextData[{
StyleBox["Exercise Set 9.1\n",
FontVariations->{"Underline"->True}],
"Graphs from Formulas"
}], "Subtitle"],
Cell["by John Robeson", "Subsubtitle"],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 1) Three Points Are Not Enough", "Subtitle"]], "Subtitle"],
Cell["\<\
This exercise shows you a simple reason why points alone are not enough.\
\>", "Text"],
Cell["\<\
(a) The cartesian pairs (-2,-2), (0,0), (2,2) are recorded in the table on \
page 147. Plot these points.\
\>", "Text"],
Cell["\<\
(b) Show that the graph of y = x contains all three points from part (a) and \
sketch the graph.\
\>", "Text"],
Cell[TextData[{
"(c) Show that the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
" contains all three points from part (a). Can you sketch it without more \
points?"
}], "Text"],
Cell[TextData[{
"(d) Find a point on the graph of y = x that is not on the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
" and plot it."
}], "Text"],
Cell[TextData[{
"(e) Use the graphing program in ",
StyleBox["aComputerIntro",
FontWeight->"Bold"],
" to plot both functions on the same graph."
}], "Text"],
Cell[CellGroupData[{
Cell["(a)", "Section"],
Cell["\<\
(a) The cartesian pairs (-2,-2), (0,0), (2,2) are recorded in the table on \
page 147. Plot these points.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (a)", "Subsection"],
Cell["Evaluate the cell below.", "Text"],
Cell[BoxData[
\(plot\ = \
ListPlot[{{\(-2\), \(-2\)}, \ {0, 0}, \ {2, 2}}, \n\ \ \ \ \ \ \
Prolog\ -> \ AbsolutePointSize[5]]\)], "Input"]
}, Open ]]
}, Closed]],
Cell[CellGroupData[{
Cell["(b)", "Section"],
Cell["\<\
(b) Show that the graph of y = x contains all three points from part (a) and \
sketch the graph.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (b)", "Subsection"],
Cell["\<\
y = x means that every y coordinate is equal to its x coordinate. Since this \
is the case for all three of our points in part (a), we know that they will \
all be on the graph of y = x.
Now evaluate the cell below to see the graphs.\
\>", "Text"],
Cell[BoxData[
\(Clear[x, y]; \ny\ = x; \n
line\ = \
Plot[y, {x, \(-3\), 3}, AxesLabel -> {"\", "\"},
PlotRange -> {\(-3\), 3}, PlotStyle -> {RGBColor[1, 0, 0]}]; \n
Show[line, plot\ , Prolog\ -> \ AbsolutePointSize[5]]; \)], "Input"]
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Cell[TextData[{
"(c) Show that the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
" contains all three points from part (a). Can you sketch it without more \
points?"
}], "Text"],
Cell[CellGroupData[{
Cell["Solution to (c)", "Subsection"],
Cell[TextData[{
"For x = -2, y = ",
Cell[BoxData[
\(\((\(-2\))\)\^3 - 3\ \((\(-2\))\)\)]],
" = -8 + 6 = -2 \n\nFor x = 0, y = ",
Cell[BoxData[
\(0\^3 - 3 \((0)\)\)]],
" = 0\n\nFor x = 2, y = ",
Cell[BoxData[
\(2\^3 - 3\ \((2)\)\)]],
" = 8 - 6 = 2"
}], "Text"],
Cell[TextData[{
"So all three of our points are also on the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
". However, since this function is not linear, these three points do not \
give us enough information to know how they are connected.\n"
}], "Text"]
}, Open ]]
}, Closed]],
Cell[CellGroupData[{
Cell["(d)", "Section"],
Cell[TextData[{
"(d) Find a point on the graph of y = x that is not on the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
" and plot it."
}], "Text"],
Cell[CellGroupData[{
Cell["Solution to (d)", "Subsection"],
Cell[TextData[{
"The point (1, 1) is on the graph of y = x, but when x = 1, the value of y \
for y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
" is y = ",
Cell[BoxData[
\(1\^3 - 3\ \((1)\)\)]],
" = 1 - 3 = -2. So the point (1, 1) is not on the graph of y = ",
Cell[BoxData[
\(x\^3 - 3\ x\)]],
"."
}], "Text"],
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\(Clear[x, y]; \ny\ = \ x\^3 - 3\ x; \n
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\ncurve\ = \
Plot[y, {x, \(-3\), 3}, AxesLabel -> {"\", "\"},
PlotStyle -> {RGBColor[1, 0, 0]}]; \n
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Cell[TextData[{
"(e) Use the graphing program in ",
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FontWeight->"Bold"],
" to plot both functions on the same graph."
}], "Text"],
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Cell["Solution to (e)", "Subsection"],
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Cell["", "Text"]
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Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 2) Find Equations", "Subtitle"]], "Subtitle"],
Cell["Find the equations of the lines shown in Figure 9.3", "Text"],
Cell[CellGroupData[{
Cell[" Solution", "Subsection"],
Cell[TextData[{
"All three lines in the first graph have the same slope, which is",
Cell[BoxData[
\(1\/2\)]],
". This is determined by examining the middle line and noticing that it \
passes through the points (0,0) and (1, ",
Cell[BoxData[
\(1\/2\)]],
"). The y-intercepts are (top to bottom) 1, 0 and -1.5 so the equations of \
the three lines are"
}], "Text"],
Cell[TextData[{
"\t\ty = ",
Cell[BoxData[
\(1\/2\)]],
"x + 1\n\n\n\t\ty = ",
Cell[BoxData[
\(1\/2\)]],
"x \n\t\t\n\t\t\nand\t\ty = ",
Cell[BoxData[
\(1\/2\)]],
"x - 1.5\n"
}], "Text"],
Cell[TextData[{
"The lines in the second graph have slopes of -",
Cell[BoxData[
\(1\/4\)]],
" and +",
Cell[BoxData[
\(2\/3\)]],
". Their y-intercepts are 1 and -2 respectively. So their equations are"
}], "Text"],
Cell[TextData[{
"\t\ty = -",
Cell[BoxData[
\(1\/4\)]],
"x + 1\n\n\nand\t\ty = ",
Cell[BoxData[
\(2\/3\)]],
"x - 2\n"
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 3) Scale of the Plot", "Subtitle"]], "Subtitle"],
Cell[TextData[{
"Use the computer to make several plots on different scales. First, replot\
\n\t\t",
Cell[BoxData[
\(y = x\^5 + 4\ x\^4 + x\)]],
"\nat the four scales described in Example 9.1 above, but leave the default \
\"Ticks\" on so that the computer puts the scales on the plots. This will \
show you more clearly what we described in the previous paragraph.\n\nSecond, \
plot the functions below on the different suggested scales:\n\ny =",
Cell[BoxData[
\(x\^x\)]],
"\t0 \[Precedes] x \[Precedes] 2\tand\t0 \[Precedes] x \[Precedes] 5\n\n",
Cell[BoxData[
\(y = 3\ x\^4 - 4\ x\^3 - 36\ x\^2\)]],
"\t-10 \[Precedes] x \[Precedes] 10 and\t-3 \[Precedes] x \[Precedes] 5\n\n\
Explain why the pairs of graphs appear to be different even though they are \
of the same function."
}], "Text"],
Cell[CellGroupData[{
Cell[" Solution to the first part", "Subsection"],
Cell["Evaluate the cell below.", "Text"],
Cell[BoxData[{
\(\(Clear[x, f]; \)\),
\(f[x_] := x\^5 + 4\ x\^4 + x\),
\(scale = 100; \n
\(large = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Red}]; \n
scale = 10; \n
medium = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Blue}]; \n
scale = 1; \n
small = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {DarkGreen}]; \n
scale = 0.1; \n
tiny = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Brown}]; \n
Show[GraphicsArray[{{large, medium}, {small, tiny}}]]; \)\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[" Solution to the second part", "Subsection"],
Cell["Evaluate the cells below.", "Text"],
Cell[BoxData[{
\(\(Clear[x, f]; \)\),
\(f[x_] := x\^x\),
\(plot1 = Plot[f[x], {x, 0, 2}, PlotStyle -> {Red}]; \n
plot2 = Plot[f[x], {x, 0, 5}, PlotStyle -> {Blue}]; \n\)}], "Input"],
Cell["\<\
Both graphs begin at (0,0) and appear to get very large as x gets large. But \
the first graph is negative for x less than 1and the second graph appears to \
be positive for all values of x. The vertical scale is deceiving us. Notice \
that the range of the second graph is 0 to 250 whereas the range of the first \
is only -1 to 2. So the second graph is also negative from 0 to 1, but -1 on \
the much larger scale cannot be perceived.\
\>", "Text"],
Cell["For the second function evaluate the cell below.", "Text"],
Cell[BoxData[{
\(Clear[x, f]; \ng[x_] := 3\ x\^4 - 4\ x\^3 - 36\ x\^2\),
\(plot3 = Plot[g[x], {x, \(-10\), 10}, PlotStyle -> {Red}]; \n
plot4 = Plot[g[x], {x, \(-3\), 5}, PlotStyle -> {Blue}]; \n\)}], "Input"],
Cell["These two graphs actually appear to be similar.", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 4) Which is the Graph?", "Subtitle"]], "Subtitle"],
Cell[TextData[{
"Which of the graphs shown on pages 148 and 149 is the graph of\n\n\t\ty = \
",
Cell[BoxData[
\(3\ x\^5 - 5\ x\^3\)]],
" ?\n\t\t\nWhich shape features of the incorrect ones makes each one \
wrong?"
}], "Text"],
Cell[CellGroupData[{
Cell[" Solution", "Subsection"],
Cell["\<\
Evaluate the cell below to see four graphs of our same function plotted on \
different scales.\
\>", "Text"],
Cell[BoxData[{
\(\(Clear[x, f]; \)\),
\(f[x_] := 3\ x\^5 - 5\ x\^3\),
\(scale = 100; \n
large = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Red}]; \n
scale = 10; \n
medium = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Blue}]; \n
scale = 2; \n
small = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {DarkGreen}]; \n
scale = 0.1; \n
tiny = Plot[f[x], {x, \(-scale\), scale}, PlotStyle -> {Brown}]; \n
Show[GraphicsArray[{{large, medium}, {small, tiny}}]]; \)}], "Input"],
Cell["", "Text"],
Cell["\<\
Graph (a) appears incorrect since it is symmetrical with respect to the y \
axis.
Graph (b) appears correct since it is similar to the above dark green graph.
Graphs (c) and (d) appear incorrect since they do not have a \"flat\" spot at \
the origin. Since y'[0] = 0, the correct graph must have a slope of zero at \
the origin.\
\>", "Text"]
}, Closed]]
}, Closed]]
}, Open ]]
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