-
Notifications
You must be signed in to change notification settings - Fork 1
Expand file tree
/
Copy pathIntroModeling_Notes.aux
More file actions
244 lines (244 loc) · 26.3 KB
/
IntroModeling_Notes.aux
File metadata and controls
244 lines (244 loc) · 26.3 KB
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
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
\relax
\providecommand\hyper@newdestlabel[2]{}
\providecommand\zref@newlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {chapter}{\numberline {0}To the Student and the Instructor}{5}{chapter.0}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {0.1}An Inquiry Based Approach}{5}{section.0.1}}
\@writefile{toc}{\contentsline {section}{\numberline {0.2}Online Texts and Other Resources}{7}{section.0.2}}
\newlabel{pref:resources}{{0.2}{7}{Online Texts and Other Resources}{section.0.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {0.3}To the Instructor}{7}{section.0.3}}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Fundamental Notions from Calculus}{9}{chapter.1}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Sections from Active Calculus}{9}{section.1.1}}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Modeling Explorations with Calculus}{10}{section.1.2}}
\newlabel{prob:Torricelli}{{1.1}{10}{}{theorem.1.1}{}}
\newlabel{eqn:1.1.ex4}{{1.1}{10}{}{equation.1.2.1}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces Populations by year since 1982.\relax }}{14}{table.caption.2}}
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{tab:pop_table}{{1.1}{14}{Populations by year since 1982.\relax }{table.caption.2}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}An Introduction to Linear Algebra}{16}{chapter.2}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Why Linear Algebra?}{16}{section.2.1}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces A metal rod partitioned into several discrete points.\relax }}{18}{figure.caption.4}}
\newlabel{fig:10.1.rod}{{2.1}{18}{A metal rod partitioned into several discrete points.\relax }{figure.caption.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Matrix Operations and Gaussian Elimination}{20}{section.2.2}}
\newlabel{S:10.2.MatrixAlgebra}{{2.2}{20}{Matrix Operations and Gaussian Elimination}{section.2.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.3}Gaussian Elimination: A First Look At Solving Systems}{23}{section.2.3}}
\newlabel{eqn:S10.2:system1}{{2.1}{23}{}{equation.2.3.1}{}}
\newlabel{eqn:S10.2:system1b}{{2.2}{23}{}{equation.2.3.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.4}Systems of Linear Equations}{31}{section.2.4}}
\newlabel{S:10.3.Systems}{{2.4}{31}{Systems of Linear Equations}{section.2.4}{}}
\newlabel{eqn:10.3.system}{{2.3}{31}{Systems, Matrix Equations, and Vector Equations}{equation.2.4.3}{}}
\newlabel{thm:10.3.exist_unique}{{2.21}{33}{}{Item.44}{}}
\newlabel{A:10.3.1}{{2.22}{33}{}{theorem.2.22}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.2}{\ignorespaces Three possible solution sets in two spatial dimensions\relax }}{35}{figure.caption.10}}
\newlabel{fig:10.3.soln_sets}{{2.2}{35}{Three possible solution sets in two spatial dimensions\relax }{figure.caption.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.5}Linear Combinations}{38}{section.2.5}}
\newlabel{eqn:10.3.lincom}{{2.10}{38}{Linear Combinations}{equation.2.5.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.3}{\ignorespaces A graphical example of a linear combination.\relax }}{39}{figure.caption.11}}
\newlabel{fig:10.3.lincom}{{2.3}{39}{A graphical example of a linear combination.\relax }{figure.caption.11}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.6}Inverses and Determinants}{41}{section.2.6}}
\newlabel{S:10.4.InvDet}{{2.6}{41}{Inverses and Determinants}{section.2.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.6.1}Inverses}{43}{subsection.2.6.1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.6.2}Determinants}{45}{subsection.2.6.2}}
\newlabel{thm:10.4.det}{{2.42}{47}{Important Properties of Determinants}{theorem.2.42}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.4}{\ignorespaces A 2D mapping from region with non-zero area to a region non-zero area.\relax }}{50}{figure.caption.12}}
\newlabel{fig:determinant_vol_1}{{2.4}{50}{A 2D mapping from region with non-zero area to a region non-zero area.\relax }{figure.caption.12}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.5}{\ignorespaces A 2D mapping from a region non-zero area to a region with zero area.\relax }}{50}{figure.caption.13}}
\newlabel{fig:determinant_vol_2}{{2.5}{50}{A 2D mapping from a region non-zero area to a region with zero area.\relax }{figure.caption.13}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.7}Technology For Linear Algebra}{50}{section.2.7}}
\@writefile{toc}{\contentsline {section}{\numberline {2.8}The Magic Carpet Ride}{52}{section.2.8}}
\@writefile{toc}{\contentsline {section}{\numberline {2.9}Span}{54}{section.2.9}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.6}{\ignorespaces The span of $\texttt {v}_1$ and $\texttt {v}_2$ in $\mathbb {R}^3$.\relax }}{56}{figure.caption.14}}
\newlabel{fig:10.5.span}{{2.6}{56}{The span of $\bv _1$ and $\bv _2$ in $\mathbb {R}^3$.\relax }{figure.caption.14}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.10}Linear Independence, Linear Dependence, and Basis}{57}{section.2.10}}
\@writefile{toc}{\contentsline {section}{\numberline {2.11}The Column and Null Spaces of a Matrix}{62}{section.2.11}}
\@writefile{toc}{\contentsline {section}{\numberline {2.12}Modeling Explorations with Linear Algebra}{64}{section.2.12}}
\newlabel{PA:10.5}{{2.70}{64}{}{theorem.2.70}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.12.1}Input-Output Economies}{65}{subsection.2.12.1}}
\newlabel{eqn:10.5.inout1}{{2.11}{65}{Input-Output Economies}{equation.2.12.11}{}}
\newlabel{eqn:10.5.inout2}{{2.12}{65}{Input-Output Economies}{equation.2.12.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.12.2}Traffic Networks}{67}{subsection.2.12.2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.12.3}Balancing Chemical Equations}{69}{subsection.2.12.3}}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}First Order Models}{71}{chapter.3}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Birth, Death, and Immigration Exploration}{71}{section.3.1}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Models for Death, Birth, and Immigration}{73}{section.3.2}}
\newlabel{prob:bdi_models}{{3.2}{73}{}{theorem.3.2}{}}
\newlabel{prob:bdi_difference_Excel}{{3.4}{74}{}{theorem.3.4}{}}
\newlabel{prob:bdi_differential_eqns}{{3.6}{74}{}{theorem.3.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Difference Equations and Differential Equations}{76}{section.3.3}}
\newlabel{prob:bank_difference_eqn}{{3.14}{77}{}{theorem.3.14}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces Numerical solutions to $a_{n+1} = a_n + \frac {r}{12}a_n - p$ with $a_0=5000$ and $r=0.1$ and different values of $p$\relax }}{80}{figure.caption.18}}
\newlabel{fig:9.3.ex1p}{{3.1}{80}{Numerical solutions to $a_{n+1} = a_n + \frac {r}{12}a_n - p$ with $a_0=5000$ and $r=0.1$ and different values of $p$\relax }{figure.caption.18}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.2}{\ignorespaces Numerical solutions to $a_{n+1} = a_n + \frac {r}{12}a_n - p$ with $a_0=5000$ and $p=\$200$ and different values of $r$\relax }}{80}{figure.caption.19}}
\newlabel{fig:9.3.ex1r}{{3.2}{80}{Numerical solutions to $a_{n+1} = a_n + \frac {r}{12}a_n - p$ with $a_0=5000$ and $p=\$200$ and different values of $r$\relax }{figure.caption.19}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.4}Stability, Equilibria, Phase Plots, and Slope Fields}{82}{section.3.4}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.3}{\ignorespaces A plot of $\frac {dy}{dt}$ vs $y$ (top) and a phase line diagram (bottom) for the differential equation $\frac {dy}{dt}=-\frac {1}{2}(y-4)$\relax }}{83}{figure.caption.20}}
\newlabel{fig:7.2.phase}{{3.3}{83}{A plot of $\frac {dy}{dt}$ vs $y$ (top) and a phase line diagram (bottom) for the differential equation $\frac {dy}{dt}=-\frac {1}{2}(y-4)$\relax }{figure.caption.20}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.4}{\ignorespaces A slope field for the differential equation $\frac {dy}{dt} = -\frac {1}{2}(y-4)$. \relax }}{84}{figure.caption.21}}
\newlabel{fig:slope_field_1}{{3.4}{84}{A slope field for the differential equation $\frac {dy}{dt} = -\frac {1}{2}(y-4)$. \relax }{figure.caption.21}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.5}{\ignorespaces A plot of $\frac {dy}{dt}$ vs $y$ (top) and a phase line diagram (bottom) for $\frac {dy}{dt}=-\frac {1}{2}(y-4)$\relax }}{85}{figure.caption.22}}
\newlabel{fig:7.3.phase}{{3.5}{85}{A plot of $\frac {dy}{dt}$ vs $y$ (top) and a phase line diagram (bottom) for $\frac {dy}{dt}=-\frac {1}{2}(y-4)$\relax }{figure.caption.22}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.6}{\ignorespaces A slope field for the differential equation $\frac {dy}{dt} = \frac {1}{2}(y-4)$. Several solution curves are shown with different initial conditions. \relax }}{85}{figure.caption.23}}
\newlabel{fig:slope_field_2}{{3.6}{85}{A slope field for the differential equation $\frac {dy}{dt} = \frac {1}{2}(y-4)$. Several solution curves are shown with different initial conditions. \relax }{figure.caption.23}{}}
\newlabel{prob:Slope_Fields_Matching}{{3.29}{86}{}{theorem.3.29}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.7}{\ignorespaces Slope fields for Problem \ref {prob:Slope_Fields_Matching}.\relax }}{87}{figure.caption.24}}
\newlabel{fig:Slope_Fields_Matching}{{3.7}{87}{Slope fields for Problem \ref {prob:Slope_Fields_Matching}.\relax }{figure.caption.24}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.8}{\ignorespaces Four cases for phase line analysis. In each plot we see a small portion of the $\frac {dy}{dt}$ vs $y$ plot for an autonomous first order differential equation: $\frac {dy}{dt} = f(y)$.\relax }}{88}{figure.caption.25}}
\newlabel{fig:phaseline}{{3.8}{88}{Four cases for phase line analysis. In each plot we see a small portion of the $\frac {dy}{dt}$ vs $y$ plot for an autonomous first order differential equation: $\frac {dy}{dt} = f(y)$.\relax }{figure.caption.25}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.9}{\ignorespaces Phase plots and solution plots. On the left are the phase plots and on the right are coordinate axes to sketch the solution plots.\relax }}{89}{figure.caption.26}}
\newlabel{fig:phase2}{{3.9}{89}{Phase plots and solution plots. On the left are the phase plots and on the right are coordinate axes to sketch the solution plots.\relax }{figure.caption.26}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.5}Euler's Method: Numerical Solutions for Differential Equations}{90}{section.3.5}}
\newlabel{eqn:euler_discrete_ex1}{{3.2}{90}{Euler's Method: Numerical Solutions for Differential Equations}{equation.3.5.2}{}}
\newlabel{eqn:euler_continuous_ex1}{{3.3}{90}{Euler's Method: Numerical Solutions for Differential Equations}{equation.3.5.3}{}}
\newlabel{eqn:euler_continuous_approx}{{3.4}{90}{Euler's Method: Numerical Solutions for Differential Equations}{equation.3.5.4}{}}
\newlabel{eqn:euler_continuous_approx2}{{3.5}{90}{Euler's Method: Numerical Solutions for Differential Equations}{equation.3.5.5}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3.1}{\ignorespaces Excel setup for a difference equation (left) and differential equation (right).\relax }}{91}{table.caption.27}}
\newlabel{tab:euler_excel}{{3.1}{91}{Excel setup for a difference equation (left) and differential equation (right).\relax }{table.caption.27}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.10}{\ignorespaces A depiction of Euler's method with step size $h=1$ (red) and $h=0.5$ (blue).\relax }}{92}{figure.caption.28}}
\newlabel{fig:Euler}{{3.10}{92}{A depiction of Euler's method with step size $h=1$ (red) and $h=0.5$ (blue).\relax }{figure.caption.28}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.11}{\ignorespaces Graphical depiction of Euler's method. Here we use the simple differential equation $y'=y$ with $y(0) = 0.5$ and $\Delta t = 1$. The exact solution is shown in red.\relax }}{92}{figure.caption.29}}
\newlabel{fig:Euler_graphical}{{3.11}{92}{Graphical depiction of Euler's method. Here we use the simple differential equation $y'=y$ with $y(0) = 0.5$ and $\Delta t = 1$. The exact solution is shown in red.\relax }{figure.caption.29}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.6}Classifying Difference and Differential Equations}{95}{section.3.6}}
\newlabel{eqn:class1}{{3.6}{95}{}{equation.3.6.6}{}}
\newlabel{eqn:class2}{{3.7}{95}{}{equation.3.6.7}{}}
\newlabel{eqn:class3}{{3.8}{95}{}{equation.3.6.8}{}}
\newlabel{eqn:class4}{{3.9}{95}{}{equation.3.6.9}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.7}Technique: Solving $1^{st}$ Order Linear Homogeneous Equations}{97}{section.3.7}}
\newlabel{eqn:1st_order_hom_difference}{{3.19}{97}{Technique: Solving $1^{st}$ Order Linear Homogeneous Equations}{equation.3.7.19}{}}
\newlabel{eqn:1st_order_hom_differential}{{3.20}{97}{Technique: Solving $1^{st}$ Order Linear Homogeneous Equations}{equation.3.7.20}{}}
\newlabel{thm:9.6.linearhom}{{3.50}{97}{}{theorem.3.50}{}}
\newlabel{eqn:9.6.lienarhom_soln}{{3.21}{97}{}{equation.3.7.21}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.8}Technique: Solving $1^{st}$ Order Linear Nonhomogeneous Equations}{99}{section.3.8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.8.1}Solution Technique: Separation of Variables}{99}{subsection.3.8.1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.8.2}Solution Technique: Undetermined Coefficients}{104}{subsection.3.8.2}}
\newlabel{prob:undet_coeff}{{3.76}{107}{}{theorem.3.76}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.9}Sensitivity Analysis}{111}{section.3.9}}
\newlabel{prob:sens1}{{3.84}{112}{}{theorem.3.84}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.10}Modeling Explorations with Difference and Differential Equations}{116}{section.3.10}}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Second Order Models}{127}{chapter.4}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}Modeling Oscillations}{127}{section.4.1}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces A mass and spring oscillating system.\relax }}{127}{figure.caption.30}}
\newlabel{fig:7.8.mass_spring}{{4.1}{127}{A mass and spring oscillating system.\relax }{figure.caption.30}{}}
\newlabel{eqn:7.8.mass_spring_basic}{{4.1}{127}{}{equation.4.1.1}{}}
\newlabel{eqn:7.8.mass_spring1}{{4.2}{128}{Modeling Oscillations}{equation.4.1.2}{}}
\newlabel{eqn:7.8.mass_spring2}{{4.3}{128}{Modeling Oscillations}{equation.4.1.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.2}Homogeneous Linear $2^{nd}$ Order Differential Equations}{130}{section.4.2}}
\newlabel{eqn:7.8.second_order_hom}{{4.4}{130}{Homogeneous Linear $2^{nd}$ Order Differential Equations}{equation.4.2.4}{}}
\newlabel{eqn:7.8.charpoly1}{{4.5}{130}{Homogeneous Linear $2^{nd}$ Order Differential Equations}{equation.4.2.5}{}}
\newlabel{eqn:7.8.char_poly}{{4.6}{130}{}{equation.4.2.6}{}}
\newlabel{thm:7.8.second_hom_soln}{{4.4}{131}{}{theorem.4.4}{}}
\newlabel{ex:7.8.ex1}{{4.5}{131}{}{theorem.4.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces Several solutions to $y''+4y'+3y=0$ shown in example \ref {ex:7.8.ex2}.\relax }}{132}{figure.caption.31}}
\newlabel{fig:7.8.ex1}{{4.2}{132}{Several solutions to $y''+4y'+3y=0$ shown in example \ref {ex:7.8.ex2}.\relax }{figure.caption.31}{}}
\newlabel{ex:7.8.ex2}{{4.6}{132}{}{theorem.4.6}{}}
\newlabel{ex:7.8.ex3}{{4.7}{132}{}{theorem.4.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces Several solutions to $y''+y'+y=0$ shown in example \ref {ex:7.8.ex2}. Notice that this equation models an underdamped oscillator where some oscillations occur.\relax }}{133}{figure.caption.32}}
\newlabel{fig:7.8.ex2}{{4.3}{133}{Several solutions to $y''+y'+y=0$ shown in example \ref {ex:7.8.ex2}. Notice that this equation models an underdamped oscillator where some oscillations occur.\relax }{figure.caption.32}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces Several solutions to $y''+6y'+9y=0$ shown in example \ref {ex:7.8.ex3}. This is a model for a critically damped oscillator where no oscillations can occur..\relax }}{133}{figure.caption.33}}
\newlabel{fig:7.8.ex3}{{4.4}{133}{Several solutions to $y''+6y'+9y=0$ shown in example \ref {ex:7.8.ex3}. This is a model for a critically damped oscillator where no oscillations can occur..\relax }{figure.caption.33}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Forced Oscillations}{137}{section.4.3}}
\@writefile{toc}{\contentsline {section}{\numberline {4.4}Energy in Mass Spring Systems -- A Lab Exploration}{139}{section.4.4}}
\newlabel{eqn:NewtonSecond}{{4.7}{139}{Background}{equation.4.4.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.5}{\ignorespaces A mass-spring oscillating system connected to a rigid body above with mass $m$. The coordinate system uses $y=0$ as the rest position of the mass with $y>0$ indicating positions above equilibrium and $y<0$ indicating position below equilibrium.\relax }}{139}{figure.caption.36}}
\newlabel{fig:3-10-SpringMass}{{4.5}{139}{A mass-spring oscillating system connected to a rigid body above with mass $m$. The coordinate system uses $y=0$ as the rest position of the mass with $y>0$ indicating positions above equilibrium and $y<0$ indicating position below equilibrium.\relax }{figure.caption.36}{}}
\newlabel{eqn:mass-spring-forces}{{4.8}{140}{Background}{equation.4.4.8}{}}
\newlabel{eqn:mass-spring}{{4.9}{140}{Background}{equation.4.4.9}{}}
\newlabel{eqn:mass-spring2}{{4.10}{140}{Background}{equation.4.4.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.5}Modeling Explorations with $2^{nd}$ Order Differential Equations}{143}{section.4.5}}
\@writefile{toc}{\contentsline {chapter}{\numberline {5}Systems of Difference and Differential Equations}{147}{chapter.5}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}Spread of Disease}{147}{section.5.1}}
\newlabel{prob:SIR_outbreak_simulation}{{5.2}{148}{}{theorem.5.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.2}Spreading a Juicy Rumor}{149}{section.5.2}}
\@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces Table to organize transfer between compartments. Fill in the blanks.\relax }}{149}{table.caption.38}}
\newlabel{tab:hir_table}{{5.1}{149}{Table to organize transfer between compartments. Fill in the blanks.\relax }{table.caption.38}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.2}{\ignorespaces Organization table for how the populations might transfer people. Fill in the blanks.\relax }}{150}{table.caption.39}}
\newlabel{tab:hir_transfer}{{5.2}{150}{Organization table for how the populations might transfer people. Fill in the blanks.\relax }{table.caption.39}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces A graphical representation of how the sub-populations trade people. Fill in the blanks.\relax }}{150}{figure.caption.40}}
\newlabel{fig:hir_graph}{{5.1}{150}{A graphical representation of how the sub-populations trade people. Fill in the blanks.\relax }{figure.caption.40}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}The H1N1 Virus}{152}{section.5.3}}
\@writefile{toc}{\contentsline {section}{\numberline {5.4}Writing Systems of Difference Equations}{153}{section.5.4}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces Phase plane showing the bear population dynamics\relax }}{154}{figure.caption.41}}
\newlabel{fig:9.10.ex1bears2}{{5.2}{154}{Phase plane showing the bear population dynamics\relax }{figure.caption.41}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces Phase plane showing the bear population dynamics\relax }}{155}{figure.caption.42}}
\newlabel{fig:9.10.ex1bears2_poach}{{5.3}{155}{Phase plane showing the bear population dynamics\relax }{figure.caption.42}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.4}{\ignorespaces Numerical visualization of the voting problem. The equilibrium state appears to be $D_n \to 1200$, $R_n \to 975$, and $T_n \to 825$; hence giving the democrats a clear advantage.\relax }}{156}{figure.caption.43}}
\newlabel{fig:9.10.ex2_voting}{{5.4}{156}{Numerical visualization of the voting problem. The equilibrium state appears to be $D_n \to 1200$, $R_n \to 975$, and $T_n \to 825$; hence giving the democrats a clear advantage.\relax }{figure.caption.43}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.5}Linear Systems}{158}{section.5.5}}
\newlabel{sec:linear_systems}{{5.5}{158}{Linear Systems}{section.5.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.6}The Eigenvalue / Eigenvector Problem}{160}{section.5.6}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.6.1}Geometry of Eigenvectors and Eigenvalues}{161}{subsection.5.6.1}}
\newlabel{thm:10.6.eig1}{{5.23}{161}{}{theorem.5.23}{}}
\newlabel{thm:10.6.ev2}{{5.26}{163}{}{theorem.5.26}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.6.2}The Eigenvalue Eigenvector Problem}{164}{subsection.5.6.2}}
\newlabel{eqn:10.6.evev1}{{5.1}{165}{The Eigenvalue Eigenvector Problem}{equation.5.6.1}{}}
\newlabel{thm:eigen_process}{{5.30}{165}{}{Item.303}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.6.3}Technology for the Eigenvalue Eigenvector Problem}{169}{subsection.5.6.3}}
\@writefile{toc}{\contentsline {section}{\numberline {5.7}Markov Chains}{171}{section.5.7}}
\newlabel{prob:bull_bear}{{5.35}{171}{}{theorem.5.35}{}}
\newlabel{def:10.7.markov}{{5.36}{171}{}{theorem.5.36}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.3}{\ignorespaces Table showing the evolution of a Markov chain.\relax }}{172}{table.caption.44}}
\newlabel{tab:10.7.markov}{{5.3}{172}{Table showing the evolution of a Markov chain.\relax }{table.caption.44}{}}
\newlabel{thm:10.7.markov_state}{{5.37}{172}{}{theorem.5.37}{}}
\newlabel{thm:10.7.stochastic_eig}{{5.39}{173}{}{theorem.5.39}{}}
\newlabel{thm:10.7.eig_one}{{5.40}{174}{}{theorem.5.40}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.8}Analysis of Linear Systems}{176}{section.5.8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.8.1}Homogeneous Systems of Linear Difference Equations}{176}{subsection.5.8.1}}
\@writefile{lot}{\contentsline {table}{\numberline {5.4}{\ignorespaces Tabular solution to a homogeneous system of linear difference equations.\relax }}{176}{table.caption.45}}
\newlabel{tab:10.8.linear_hom}{{5.4}{176}{Tabular solution to a homogeneous system of linear difference equations.\relax }{table.caption.45}{}}
\newlabel{thm:10.8.linear_hom}{{5.44}{176}{}{theorem.5.44}{}}
\newlabel{thm:linear_homog_soln}{{5.45}{177}{}{theorem.5.45}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.5}{\ignorespaces Evil empire customers and Rebel Alliance customers over a 200 week time period.\relax }}{178}{figure.caption.46}}
\newlabel{fig:10.8.empire1}{{5.5}{178}{Evil empire customers and Rebel Alliance customers over a 200 week time period.\relax }{figure.caption.46}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.8.2}Analysis of Equilibrium Behavior}{178}{subsection.5.8.2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.8.3}Non Homogeneous Systems of Linear Difference Equations}{180}{subsection.5.8.3}}
\newlabel{thm:10.8.gen_soln_nonhom}{{5.51}{181}{}{theorem.5.51}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.9}Modeling Explorations with Systems}{183}{section.5.9}}
\@writefile{lot}{\contentsline {table}{\numberline {5.5}{\ignorespaces Bedridden Boys Data\relax }}{186}{table.caption.47}}
\newlabel{tab:bedridden_boys}{{5.5}{186}{Bedridden Boys Data\relax }{table.caption.47}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.6}{\ignorespaces Ibuprofen Data\relax }}{187}{table.caption.48}}
\newlabel{tab:ibuprofen_data}{{5.6}{187}{Ibuprofen Data\relax }{table.caption.48}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Infinite Series}{188}{chapter.6}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Sections from Active Calculus}{188}{section.6.1}}
\@writefile{toc}{\contentsline {chapter}{Appendices}{189}{section*.49}}
\@writefile{toc}{\contentsline {chapter}{\numberline {A}MATLAB Basics}{190}{Appendix.a.A}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{app:MATLAB}{{A}{190}{MATLAB Basics}{Appendix.a.A}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.1}Vectors and Matrices}{190}{section.a.A.1}}
\@writefile{toc}{\contentsline {section}{\numberline {A.2}Looping}{192}{section.a.A.2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.1}For Loops}{192}{subsection.a.A.2.1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.2.2}The While Loop}{193}{subsection.a.A.2.2}}
\@writefile{toc}{\contentsline {section}{\numberline {A.3}Conditional Statements}{194}{section.a.A.3}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.3.1}If Statements}{194}{subsection.a.A.3.1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A.3.2}Case-Switch Statements}{195}{subsection.a.A.3.2}}
\@writefile{toc}{\contentsline {section}{\numberline {A.4}Functions}{195}{section.a.A.4}}
\@writefile{toc}{\contentsline {section}{\numberline {A.5}Plotting}{196}{section.a.A.5}}
\@writefile{toc}{\contentsline {section}{\numberline {A.6}Animations}{197}{section.a.A.6}}