-
Notifications
You must be signed in to change notification settings - Fork 388
/
breakGround.tex
149 lines (115 loc) · 6.22 KB
/
breakGround.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
\documentclass{ximera}
\input{../preamble.tex}
\outcome{Use limits to find the slope of the tangent line at a point.}
\outcome{Understand the definition of the derivative at a point.}
\title[Break-Ground:]{Slope of a curve}
\begin{document}
\begin{abstract}
Two young mathematicians discuss the novel idea of the ``slope of a curve.''
\end{abstract}
\maketitle
Check out this dialogue between two calculus students (based on a true
story):
\begin{dialogue}
\item[Devyn] Riley, do you remember ``slope?'
\item[Riley] Most definitely. ``Rise over run.''
\item[Devyn] You know it.
\item[Riley] ``Change in $y$ over change in $x$.''
\item[Devny] That's right.
\item[Riley] Brought to you by the letter ``$m$.''
\item[Devny] Enough! My important question is: could we define
``slope'' for a curve that's not a straight line?
\item[Riley] Well, maybe if we ``zoom in'' on a curve, it would look
like a line, and then we could call it ``the slope at that point.''
\item[Devyn] Ah! And this ``zoom in'' idea sounds like a limit!
\item[Riley] This is so awesome. We just made math!
\end{dialogue}
The concept introduced above, of the ``slope of a curve at a point,''
is in fact one of the central concepts of calculus. It will,
of course, be completely explained. Let's explore Devyn
and Riley's ideas a little more, first.
To find the ``slope of a curve at a point,'' Devyn and Riley spoke of
``zooming in'' on a curve until it looks like a line. When you zoom in
on a \textit{smooth} curve, it will eventually look like a line. This
line is called the tangent line.
\begin{image}
\begin{tikzpicture}
\begin{axis}[
domain=0:6, range=0:7,
ymin=-.2,ymax=7,
width=6in,
height=2.5in, %% Hard coded height! Moreover this effects the aspect ratio of the zoom--sort of BAD
axis lines=none,
]
\addplot [draw=none, fill=textColor!10!background] plot coordinates {(.8,1.25) (5.334,3.666)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=textColor!10!background] plot coordinates {(5.334,3.666) (6.666,3.666)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=background] plot coordinates {(1.2,1.25) (6.666,3.666)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=background] plot coordinates {(.8,1.25) (1.2,1.25)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=textColor!10!background] plot coordinates {(.8,.833) (5.334,2.334)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=textColor!10!background] plot coordinates {(5.334,2.334) (6.666,2.334)} \closedcycle; %% zoom fill
\addplot [draw=none, fill=background] plot coordinates {(.8,.833) (6.666,2.334)} \closedcycle; %% zoom fill
\addplot[very thick,penColor, smooth,domain=(0:1.833)] {-1/(x-2)};
\addplot[very thick,penColor, smooth,domain=(5.334:6.666)] {x-3}; %% 5 to 6.833
\addplot[color=penColor,fill=penColor,only marks,mark=*] coordinates{(1,1)}; %% point to be zoomed
\addplot [dashed] plot coordinates {(6,2.334) (6,3)}; %% zoom fill
\addplot [dashed] plot coordinates {(5.334,3) (6,3)}; %% zoom fill
\addplot[color=penColor,fill=penColor,only marks,mark=*] coordinates{(6,3)}; %% zoomed pt 2
\addplot [->,textColor] plot coordinates {(0,0) (0,6)}; %% axis
\addplot [->,textColor] plot coordinates {(0,0) (2,0)}; %% axis
\addplot [textColor!50!background] plot coordinates {(.8,.833) (.8,1.25)}; %% box around pt
\addplot [textColor!50!background] plot coordinates {(1.2,.833) (1.2,1.25)}; %% box around pt
\addplot [textColor!50!background] plot coordinates {(.8,1.25) (1.2,1.25)}; %% box around pt
\addplot [textColor!50!background] plot coordinates {(.8,.833) (1.2,.833)}; %% box around pt
\addplot [textColor] plot coordinates {(5.334,2.334) (5.334,3.666)}; %% zoomed box 2
\addplot [textColor] plot coordinates {(6.666,2.334) (6.666,3.666)}; %% zoomed box 2
\addplot [textColor] plot coordinates {(5.334,2.334) (6.666,2.334)}; %% zoomed box 2
\addplot [textColor] plot coordinates {(5.334,3.666) (6.666,3.666)}; %% zoomed box 2
\node at (axis cs:6,2.334) [anchor=north] {$a$};
\node at (axis cs:6.666,2.334) [anchor=north] {$a+h$};
\node at (axis cs:5.334,2.334) [anchor=north] {$a-h$};
\node at (axis cs:5.334,3) [anchor=east] {$f(a)$};
\node at (axis cs:5.334,3.666) [anchor=east] {$f(a+h)$};
\node at (axis cs:5.334,2.334) [anchor=east] {$f(a-h)$};
\node at (axis cs:2.2,0) [anchor=east] {$x$};
\node at (axis cs:0,6.6) [anchor=north] {$y$};
\end{axis}
\end{tikzpicture}
\end{image}
\begin{problem}
Which of the following approximate the slope of the ``zoomed line''?
\begin{selectAll}
\choice{$\frac{(f(a)+h) - f(a)}{(a+h)-a}$}
\choice[correct]{$\frac{f(a+h) - f(a)}{(a+h)-a}$}
\choice{$\frac{(f(a)-h) - f(a)}{(a-h)-a}$}
\choice[correct]{$\frac{f(a-h) - f(a)}{(a-h)-a}$}
\choice{$\frac{f(a) - (f(a)+h)}{a-(a+h)}$}
\choice[correct]{$\frac{f(a) - f(a+h)}{a-(a+h)}$}
\choice{$\frac{f(a) - (f(a)-h)}{a-(a-h)}$}
\choice[correct]{$\frac{f(a) - f(a-h)}{a-(a-h)}$}
\end{selectAll}
\end{problem}
\begin{problem}
Let $f(x) = 3x-1$. Zoom in on the curve around $a = -2$ so that $h
= 0.1$. Use one of the formulations in the problem above to
approximate the slope of the curve. The slope of the curve at $a =
-2$ is approximately\dots \begin{prompt}$\answer{3}$\end{prompt}
\end{problem}
\begin{problem}
Repeat the previous problem for $f(x) = x^2 - 1$, $a = 0$, and $h =
0.2$. Choose a formulation that will give you a positive answer
for the slope. The (positive) slope of the curve at $a = 0$ is
approximately\dots\begin{prompt} $\answer{0.2}$\end{prompt}
\end{problem}
\begin{problem}
Zoom in on the curve $f(x) = x^2 - 1$ near $x=0$ again. By looking
at the graph, what is your best guess for the actual slope of the
curve at zero?
\begin{multipleChoice}
\choice{impossible to say}
\choice[correct]{zero}
\choice{one}
\choice{infinity}
\end{multipleChoice}
\end{problem}
%\input{../leveledQuestions.tex}
\end{document}