generated from XimeraProject/ximeraNewProject
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathnewActivity.tex
159 lines (113 loc) · 5.69 KB
/
newActivity.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
150
151
152
153
154
155
156
157
158
159
\documentclass{ximera} %Use handout to supress answers???
\input{xmPreamble}
\title{Collection of Activities}
\author{Kelly Stady}
\begin{document}
\begin{abstract}
Practice Problems
\end{abstract}
\maketitle
\begin{exercise}
$2+3=\answer{5}$
\end{exercise}
\begin{exercise}
Find the limit if $f(x) = x.$
$\lim\limits_{x \to 2} f(x) = \answer{2}$
%\[
%\lim\limits_{x \to 2} f(x)
%\]
\end{exercise}
\begin{question}
The circle below has been divided into 10 equal parts. What fraction of the circle is shaded?
%\begin{image} %Makes circle huge!! scale=0.1 does not help
\hspace{0.8in} \begin{tikzpicture}
\draw[thick] (0,0) circle [radius=1.3cm];
\foreach \i in {36,72,...,360}
\draw[semithick] (0,0)--(\i:1.3cm); %The colon signifies that polar coordinates are being used. (60:.75cm) means the point that is at an angle of 60° and a distance of 0.75cm from the origin. Length of radial lines - should match radius above.
\draw[thick, fill=glaucous!60] (0,0) -- (36:1.3) arc(36:0:1.3); %To Draw Arc: \draw (x,y) arc (start:stop:radius);
\draw[thick, fill=glaucous!60] (0,0) -- (72:1.3) arc(72:36:1.3); %Colon indicates polar coordintes (angle:length)
\draw[thick, fill=glaucous!60] (0,0) -- (108:1.3) arc(108:72:1.3);
%\draw[fill=glaucous] (0,0) -- (360:1.5) arc(360:324:1.5);
\end{tikzpicture}
%\end{image}
\begin{multipleChoice}
\choice[correct]{$\displaystyle \frac{3}{10}$}
\choice{$\displaystyle \frac{1}{10}$}
\choice{$\displaystyle \frac{7}{10}$}
\end{multipleChoice}
\end{question}
%\begin{center} %% Image is found in xmPictures
%\includegraphics{missionPatch.jpg}
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
\item Calculus was discovered in the seventeenth century by
Leibniz and Newton, independently. The development of calculus was
motivated in large part by two geometric problems:
\begin{enumerate}
\item Finding areas of plane regions.
\item Finding tangent lines to curves.
\end{enumerate}
\vspace{0.1in}
\item \textbf{The Tangent Line Problem}
In algebra you learned how to find the slope of a line, which
measures the change in $y$ for a given change in $x.$ For example,
a slope of $m=-\frac{32}{1}$ indicates that when $x$ increases by
1, the value of $y$ decreases by 32. The slope of a line is also a
measure of the steepness of the line and is constant. Intuitively,
we know that the steepness of a curve is \emph{not} constant. So
how do we measure the steepness or slope of a curve at a
particular point?
\vspace{0.1in} We draw a line that touches the curve at the point.
This line is called a \textbf{tangent line}. The tangent line and
curve have this point in common, so we define the slope of the
curve at this point to be the slope of the tangent line.
\vspace{0.2in} \fbox{\parbox{6.2in}{\textbf{The slope of a curve at
a point is equal to the slope of the tangent line at that
point}.}}
\vspace{0.2in}
Below is the graph of $y=e^x$ along
with its tangent line at $x=1.2.$ To find the slope of the tangent line, we need to
know two points that lie on the line, but we only know one $(1.2,
e^{1.2}).$ In the next chapter, we will see how to find the slope
of the tangent line without finding a second point on the line.
This process involves finding a limit, which is the focus of this
section.
%with(Student[Calculus1])
%z:= plots[pointplot]([1.2, exp(1.2)], symbol = solidcircle, color = black, symbolsize = 23)
%z2:= plot([exp(x), Tangent(exp(x), x = 1.2, showpoint = true)], x = -4 .. 4, -5 .. 10, color = [black, black], linestyle = [1, 3], thickness = 3, labelfont = [COURIER, 1], axesfont = ["HELVETICA", "COMPUTER MODERN", 17])
%display({z, z2})
%Old
%with(Student[Calculus1][Tangent])
%plot([exp(x), Tangent(exp(x), x = 1.2, output = line)], x = -4 .. 4, -5 .. 10, color = [black, black], linestyle = [1, 3], thickness = 2)
\vspace{0.1in}
\hspace{0.4in}
%\includegraphics[height=2in, width=2in]{plot1s22.eps}
\vspace{0.2in}
\item The most basic use of \textbf{limits} is to
describe how a function $y=f(x)$ behaves as $x$ approaches a
particular value.
\newpage
\fbox{\parbox{6.2in}{\textbf{\fbox{\parbox{6.1in}{\textbf{Limit Notation}}}}
\vspace{0.1in}
The \textbf{limit of a function} $y=f(x)$ is the value that $f(x)$
approaches as $x$ approaches a fixed value $c.$ The value of
the limit is commonly denoted by $L$. The mathematical notation
for limits is given below.
\[ \lim_{x \rightarrow c} f(x) = L \hspace{0.5in} \small{\textrm{Read as:} \ ``\textrm{\textbf{The limit of} $f(x)$ \textrm{\textbf{as}} $x$
\textrm{\textbf{approaches}} $c$ is $L$."}} \] }}
\textbf{Remark:} This notation allows us to mathematically describe the behavior of a function around a point at
which it is undefined or discontinuous. It also allows us to describe the end behavior of the function, that is how does
the function behave as $x \rightarrow \pm \infty$.
\vspace{0.2in} \item[$\bigstar$] \textbf{Problem I:} \ Use a limit
to describe the behavior of the function $f(x)=\ln x$ as $x$
approaches $1.$
\begin{definition}\index{limits}
The \textbf{limit of a function} $f(x)$ is the value that $f(x)$
approaches as $x$ approaches a fixed value $c.$ The value of
the limit is commonly denoted by $L$. The mathematical notation
for limits is given below.
\[ \lim_{x \rightarrow c} f(x) = L \]
\end{definition}
\end{itemize}
\end{document}