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Calculus CW #1/HW.tex

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+\documentclass[12pt]{book}
+
+%These tell TeX which packages to use.
+\usepackage{array,epsfig}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsxtra}
+\usepackage{amsthm}
+\usepackage{mathrsfs}
+\usepackage{color}
+\usepackage{enumitem}
+%Here I define some theorem styles and shortcut commands for symbols I use often
+\theoremstyle{definition}
+\newtheorem{defn}{Definition}
+\newtheorem{thm}{Theorem}
+\newtheorem{cor}{Corollary}
+\newtheorem*{rmk}{Remark}
+\newtheorem{lem}{Lemma}
+\newtheorem*{joke}{Joke}
+\newtheorem{ex}{Example}
+\newtheorem*{soln}{Solution}
+\newtheorem{prop}{Proposition}
+
+\newcommand{\lra}{\longrightarrow}
+\newcommand{\ra}{\rightarrow}
+\newcommand{\surj}{\twoheadrightarrow}
+\newcommand{\graph}{\mathrm{graph}}
+\newcommand{\bb}[1]{\mathbb{#1}}
+\newcommand{\Z}{\bb{Z}}
+\newcommand{\Q}{\bb{Q}}
+\newcommand{\R}{\bb{R}}
+\newcommand{\C}{\bb{C}}
+\newcommand{\N}{\bb{N}}
+\newcommand{\M}{\mathbf{M}}
+\newcommand{\m}{\mathbf{m}}
+\newcommand{\MM}{\mathscr{M}}
+\newcommand{\HH}{\mathscr{H}}
+\newcommand{\Om}{\Omega}
+\newcommand{\Ho}{\in\HH(\Om)}
+\newcommand{\bd}{\partial}
+\newcommand{\del}{\partial}
+\newcommand{\bardel}{\overline\partial}
+\newcommand{\textdf}[1]{\textbf{\textsf{#1}}\index{#1}}
+\newcommand{\img}{\mathrm{img}}
+\newcommand{\ip}[2]{\left\langle{#1},{#2}\right\rangle}
+\newcommand{\inter}[1]{\mathrm{int}{#1}}
+\newcommand{\exter}[1]{\mathrm{ext}{#1}}
+\newcommand{\cl}[1]{\mathrm{cl}{#1}}
+\newcommand{\ds}{\displaystyle}
+\newcommand{\vol}{\mathrm{vol}}
+\newcommand{\cnt}{\mathrm{ct}}
+\newcommand{\osc}{\mathrm{osc}}
+\newcommand{\LL}{\mathbf{L}}
+\newcommand{\UU}{\mathbf{U}}
+\newcommand{\support}{\mathrm{support}}
+\newcommand{\AND}{\;\wedge\;}
+\newcommand{\OR}{\;\vee\;}
+\newcommand{\Oset}{\varnothing}
+\newcommand{\st}{\ni}
+\newcommand{\wh}{\widehat}
+
+%Pagination stuff.
+\setlength{\topmargin}{-0.75in}
+\setlength{\oddsidemargin}{0in}
+\setlength{\evensidemargin}{0in}
+\setlength{\textheight}{9.in}
+\setlength{\textwidth}{6.5in}
+\pagestyle{empty}
+\begin{document}
+\begin{center}
+{\Large Math 1041-012 \hspace{0.5cm} Classwork \#1}
+\end{center}
+\vspace{0.2 cm}
+\subsection*{Solve each of the following for $x$.}
+\begin{enumerate}
+\item $11x=2x^2+12$\\[2pt]
+\item $4x^2=8x$\\[2pt]
+\item $\frac{x}{x-2}-3=0$\\[2pt]
+\end{enumerate}
+\subsection*{Find $\frac{f(x+h)-f(x)}{h}$, $h\neq 0$ for the following functions. Simplify Fully!}
+\begin{enumerate}[resume]
+    \item $f(x)=\frac{1}{x+3}$\\[16pt]
+    \item $f(x)= -3x^2+4x+6$\\[16pt]
+\end{enumerate}
+\subsection*{Simplify the following expressions}
+\begin{enumerate}[resume]
+    \item $(e^{5x})^2$
+    \item $7^{4\log_7 3}$
+    \item $e^{\ln 2}-3\ln \sqrt[3]{e}$
+    \item $\log_9 4+\log_9\left(\frac{81}{4}\right)$
+\end{enumerate}
+\subsection*{Solve the following equations}
+\begin{enumerate}[resume]
+    \item $\ln x-\ln(x+3)=-1$\\[4pt]
+    \item $e^{7-4x}=6$\\[4pt]
+    \item $\ln (x^2-1)=3$\\[4pt]
+    \item $e^{2x}-3e^x+2=0$\\[4pt]
+\end{enumerate}
+\subsection*{Find the exact values of each of the following expressions.}
+\begin{enumerate}[resume]
+    \item $\sin\frac{7\pi}{3}$\\[4pt]
+    \item $2\sin\frac{\pi}{6}+4\cos\frac{\pi}{3}$\\[4pt]
+    \item $\tan\frac{2\pi}{3}\cot\frac{\pi}{3}$\\[4pt]
+    \item $\csc\frac{3\pi}{4}$\\[4pt]
+\end{enumerate}
+\subsection*{Find all solutions to the following equations in the interval $[0,2\pi)$.}
+\begin{enumerate}[resume]
+    \item $2\cos x-1=0$\\[8pt]
+    \item $2\sin^2x=1$\\[8pt]
+    \item $\sin 2x=\cos x$\\[8pt]
+    \item $2+\cos 2x=3\cos x$
+\end{enumerate}
+\end{document}

Calculus CW #2/HW.tex

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+\documentclass[10pt]{book}
+
+%These tell TeX which packages to use.
+\usepackage{array,epsfig}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsxtra}
+\usepackage{amsthm}
+\usepackage{mathrsfs}
+\usepackage{color}
+\usepackage{enumitem}
+
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.6}
+
+\pgfplotsset{soldot/.style={color=black,only marks,mark=*}} \pgfplotsset{holdot/.style={color=black,fill=white,only marks,mark=*}}
+
+%Here I define some theorem styles and shortcut commands for symbols I use often
+\theoremstyle{definition}
+\newtheorem{defn}{Definition}
+\newtheorem{thm}{Theorem}
+\newtheorem{cor}{Corollary}
+\newtheorem*{rmk}{Remark}
+\newtheorem{lem}{Lemma}
+\newtheorem*{joke}{Joke}
+\newtheorem{ex}{Example}
+\newtheorem*{soln}{Solution}
+\newtheorem{prop}{Proposition}
+
+\newcommand{\lra}{\longrightarrow}
+\newcommand{\ra}{\rightarrow}
+\newcommand{\surj}{\twoheadrightarrow}
+\newcommand{\graph}{\mathrm{graph}}
+\newcommand{\bb}[1]{\mathbb{#1}}
+\newcommand{\Z}{\bb{Z}}
+\newcommand{\Q}{\bb{Q}}
+\newcommand{\R}{\bb{R}}
+\newcommand{\C}{\bb{C}}
+\newcommand{\N}{\bb{N}}
+\newcommand{\M}{\mathbf{M}}
+\newcommand{\m}{\mathbf{m}}
+\newcommand{\MM}{\mathscr{M}}
+\newcommand{\HH}{\mathscr{H}}
+\newcommand{\Om}{\Omega}
+\newcommand{\Ho}{\in\HH(\Om)}
+\newcommand{\bd}{\partial}
+\newcommand{\del}{\partial}
+\newcommand{\bardel}{\overline\partial}
+\newcommand{\textdf}[1]{\textbf{\textsf{#1}}\index{#1}}
+\newcommand{\img}{\mathrm{img}}
+\newcommand{\ip}[2]{\left\langle{#1},{#2}\right\rangle}
+\newcommand{\inter}[1]{\mathrm{int}{#1}}
+\newcommand{\exter}[1]{\mathrm{ext}{#1}}
+\newcommand{\cl}[1]{\mathrm{cl}{#1}}
+\newcommand{\ds}{\displaystyle}
+\newcommand{\vol}{\mathrm{vol}}
+\newcommand{\cnt}{\mathrm{ct}}
+\newcommand{\osc}{\mathrm{osc}}
+\newcommand{\LL}{\mathbf{L}}
+\newcommand{\UU}{\mathbf{U}}
+\newcommand{\support}{\mathrm{support}}
+\newcommand{\AND}{\;\wedge\;}
+\newcommand{\OR}{\;\vee\;}
+\newcommand{\Oset}{\varnothing}
+\newcommand{\st}{\ni}
+\newcommand{\wh}{\widehat}
+%Pagination stuff.
+\setlength{\topmargin}{-0.75in}
+\setlength{\oddsidemargin}{0in}
+\setlength{\evensidemargin}{0in}
+\setlength{\textheight}{9.in}
+\setlength{\textwidth}{6.5in}
+\pagestyle{empty}
+\begin{document}
+\begin{center}
+{\Large Math 1041-012 \hspace{0.5cm} Classwork \#2}
+\end{center}
+\vspace{0.2 cm}
+\subsection*{Problem 1}The displacement (in feet) of a certain particle moving in a straight line is given by $s(t)=\frac{t^3}{6}$, where $t$ is measured in seconds. Find the average velocity over the following time intervals:
+\begin{enumerate}[label=(\alph*)]
+    \item $[1,3]$\\[8pt]
+    \item $[1,2]$\\[8pt]
+    \item $[1,1.5]$\\[8pt]
+    \item $[1,1.1]$
+\end{enumerate}
+\subsection*{Problem 2}
+Use the graph of $g(x)$ below to find the values.
+\begin{figure}[h]
+\centering
+\includegraphics[width=4cm]{piecewise1.png}
+\end{figure}
+\begin{alignat*}{3}
+&\lim_{x\rightarrow 0^+}g(x)=\hspace{2cm} &&  \lim_{x\rightarrow 0^-}g(x)=\hspace{2cm} && \lim_{x\rightarrow 0}g(x)=\\[8pt]
+&\lim_{x\rightarrow 2^+}g(x)=\hspace{2cm} &&  \lim_{x\rightarrow 2^-}g(x)=\hspace{2cm} && \lim_{x\rightarrow 2}g(x)=\\[8pt]
+&\lim_{x\rightarrow 4^+}g(x)=\hspace{2cm} &&  \lim_{x\rightarrow 4^-}g(x)=\hspace{2cm} && \lim_{x\rightarrow 4}g(x)=
+\end{alignat*}
+\subsection*{Problem 3}
+Determine the infinite limits.
+\begin{enumerate}[label=(\roman*)]
+    \item $\displaystyle\lim_{x\rightarrow 5^+}\frac{2x}{x-5}$
+    \item $\displaystyle\lim_{x\rightarrow 2^+}\ln(x^2-4)$
+    \item $\displaystyle\lim_{x\rightarrow 0^+}\ln(1-\cos{x})$
+    \item $\displaystyle\lim_{x\rightarrow -2^-}\frac{-4x}{x+2}$
+\end{enumerate}
+\subsection*{Problem 4}
+Below is the plot of $f(x)$, evaluate the limits.
+\begin{center}
+\begin{tikzpicture}
+\begin{axis}[
+  axis x line=middle, axis y line=middle,
+  ymin=-4, ymax=6, ytick={-3,...,6}, ylabel=$y$,
+  xmin=-6, xmax=11, xtick={-6,...,10}, xlabel=$x$,
+  axis line style={<->}, width=12cm,height=8cm
+]
+\addplot[domain=-2:2,black] {x*x-1};
+\addplot[domain=2:6,black] {5-x};
+\addplot[<-,domain=-6:-2,black] {-3};
+\addplot[->,domain=6:10,black]{1/(x-6)};
+\draw[dashed] (axis cs:6,6) -- (axis cs:6,-5);
+\addplot[holdot] coordinates{(-2,-3)(-2,3)(2,3)};
+\addplot[soldot] coordinates{(-2,1)(2,4)(6,-1)};
+\end{axis}
+\end{tikzpicture}
+\end{center}
+\begin{alignat*}{3}
+&\lim_{x\rightarrow -2^-}f(x)=\hspace{2cm} &&  \lim_{x\rightarrow -2^+}f(x)=\hspace{2cm} && \lim_{x\rightarrow -2}f(x)=\\[8pt]
+&\lim_{x\rightarrow 2^-}f(x)=\hspace{2cm} &&  \lim_{x\rightarrow 2^+}f(x)=\hspace{2cm} && \lim_{x\rightarrow 2}f(x)=\\[8pt]
+&\lim_{x\rightarrow 6^+}f(x)=\hspace{2cm} &&  \lim_{x\rightarrow 6^-}f(x)=\hspace{2cm} && \lim_{x\rightarrow 0}f(x)=\\[8pt]
+&f(-2)= && f(2)= &&f(6)=
+\end{alignat*}
+\subsection*{Problem 5}
+Evaluate each of the limits.
+\begin{enumerate}[label=(\alph*)]
+    \item $\displaystyle \lim_{x\rightarrow 5}\frac{x^2-25}{5-x}$\\[8pt]
+    \item $\displaystyle \lim_{h\rightarrow 0}\frac{(4+h)^2-16}{h}$\\[8pt]
+    \item $\displaystyle \lim_{x\rightarrow 9}\frac{\frac{1}{x}-\frac{1}{9}}{x-9}$\\[8pt]
+    \item $\displaystyle \lim_{x\rightarrow 7}\frac{\sqrt{5x+1}-6}{x-7}$
+\end{enumerate}
+\end{document}

Calculus CW #3/HW.tex

@@ -0,0 +1,106 @@
+\documentclass[10pt]{book}
+
+%These tell TeX which packages to use.
+\usepackage{array,epsfig}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsxtra}
+\usepackage{amsthm}
+\usepackage{mathrsfs}
+\usepackage{color}
+\usepackage{enumitem}
+
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.6}
+
+\pgfplotsset{soldot/.style={color=black,only marks,mark=*}} \pgfplotsset{holdot/.style={color=black,fill=white,only marks,mark=*}}
+
+%Here I define some theorem styles and shortcut commands for symbols I use often
+\theoremstyle{definition}
+\newtheorem{defn}{Definition}
+\newtheorem{thm}{Theorem}
+\newtheorem{cor}{Corollary}
+\newtheorem*{rmk}{Remark}
+\newtheorem{lem}{Lemma}
+\newtheorem*{joke}{Joke}
+\newtheorem{ex}{Example}
+\newtheorem*{soln}{Solution}
+\newtheorem{prop}{Proposition}
+
+\newcommand{\lra}{\longrightarrow}
+\newcommand{\ra}{\rightarrow}
+\newcommand{\surj}{\twoheadrightarrow}
+\newcommand{\graph}{\mathrm{graph}}
+\newcommand{\bb}[1]{\mathbb{#1}}
+\newcommand{\Z}{\bb{Z}}
+\newcommand{\Q}{\bb{Q}}
+\newcommand{\R}{\bb{R}}
+\newcommand{\C}{\bb{C}}
+\newcommand{\N}{\bb{N}}
+\newcommand{\M}{\mathbf{M}}
+\newcommand{\m}{\mathbf{m}}
+\newcommand{\MM}{\mathscr{M}}
+\newcommand{\HH}{\mathscr{H}}
+\newcommand{\Om}{\Omega}
+\newcommand{\Ho}{\in\HH(\Om)}
+\newcommand{\bd}{\partial}
+\newcommand{\del}{\partial}
+\newcommand{\bardel}{\overline\partial}
+\newcommand{\textdf}[1]{\textbf{\textsf{#1}}\index{#1}}
+\newcommand{\img}{\mathrm{img}}
+\newcommand{\ip}[2]{\left\langle{#1},{#2}\right\rangle}
+\newcommand{\inter}[1]{\mathrm{int}{#1}}
+\newcommand{\exter}[1]{\mathrm{ext}{#1}}
+\newcommand{\cl}[1]{\mathrm{cl}{#1}}
+\newcommand{\ds}{\displaystyle}
+\newcommand{\vol}{\mathrm{vol}}
+\newcommand{\cnt}{\mathrm{ct}}
+\newcommand{\osc}{\mathrm{osc}}
+\newcommand{\LL}{\mathbf{L}}
+\newcommand{\UU}{\mathbf{U}}
+\newcommand{\support}{\mathrm{support}}
+\newcommand{\AND}{\;\wedge\;}
+\newcommand{\OR}{\;\vee\;}
+\newcommand{\Oset}{\varnothing}
+\newcommand{\st}{\ni}
+\newcommand{\wh}{\widehat}
+%Pagination stuff.
+\setlength{\topmargin}{-0.75in}
+\setlength{\oddsidemargin}{0in}
+\setlength{\evensidemargin}{0in}
+\setlength{\textheight}{9.in}
+\setlength{\textwidth}{6.5in}
+\pagestyle{empty}
+\begin{document}
+\begin{center}
+{\Large Math 1041-012 \hspace{0.5cm} Classwork Finding Limits}
+\end{center}
+\vspace{0.2 cm}
+\subsection*{Problem 1}
+Evaluate each of the limits.
+\begin{enumerate}[label=(\alph*)]
+    \item $\displaystyle \lim_{x\rightarrow 3}\frac{3x^2+4x-1}{x}$\vspace{2.5cm}
+    \item $\displaystyle \lim_{x\rightarrow 5}\frac{x^2-25}{5-x}$ \vspace{2.5cm}
+    \item $\displaystyle \lim_{h\rightarrow 0}\frac{(4+h)^2-16}{h}$\vspace{2.5cm}
+    \item $\displaystyle \lim_{x\rightarrow 9}\frac{\frac{1}{x}-\frac{1}{9}}{x-9}$\vspace{2.5cm}
+    \item $\displaystyle \lim_{x\rightarrow 7}\frac{\sqrt{5x+1}-6}{x-7}$\vspace{2.5cm}
+    \item $\displaystyle\lim_{x\rightarrow 3}(2x+|x-3|)$
+\end{enumerate}
+\pagebreak
+\subsection*{Problem 2}
+If
+\[
+f(x)=\begin{cases}
+\sqrt{x-4} & x>4\\
+8-2x & x<4
+\end{cases}
+\]
+determine if $\lim_{x\rightarrow 4}$ exists.
+\vspace{5cm}
+\subsection*{Problem 3}
+Evaluate the limit.
+\[
+\lim_{x\rightarrow 1}\frac{x^3-1}{x^2-1}
+\]
+\end{document}