\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.9in} \pagestyle{fancy} \lhead{\color{blue} \Large Math 207} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2012} \cfoot{} \chead{\LARGE Definitions, Theorems} \rfoot{\small Last revised: July 22, 2012} \rhead{\large page \thepage } \textwidth 7.5in \textheight 9.5in \setlength{\headheight}{29pt} \setlength{\parindent}{0pt} \begin{document} \begin{center} {\Large Continuity\bigskip }\bigskip \end{center} Definition: \qquad (\textit{Interior point}) \ A function $y=f\left( x\right) $ is continuous at an interior point $c$ of its domain if $% \lim\limits_{x\rightarrow c}f\left( x\right) =f\left( c\right) $. \qquad \qquad \qquad \qquad (Endpoint) \ A function $y=f\left( x\right) $ is continuous at a left endpoint $a$ or is continuous at a right endpoint $b$ of its domain if \begin{equation*} \lim\limits_{x\rightarrow a^{+}}f\left( x\right) =f\left( a\right) \text{ \ \ or \ \ }\lim\limits_{x\rightarrow b^{-}}f\left( x\right) =f\left( b\right) \text{, \ respectively.} \end{equation*} \bigskip Theorem: \ Properties of continuous functions? \bigskip Theorem: \ If $f$ is continuous at $c$ and $g$ is continuous at $f\left( c\right) $, then $g\circ f$ is continuous at $c$. \bigskip Theorem: \ If $g$ is continuous at the point $b$ and $\lim\limits_{x% \rightarrow c}f\left( x\right) =b$, then \begin{equation*} \lim\limits_{x\rightarrow c}g\left( f\left( x\right) \right) =g\left( b\right) =g\left( \lim\limits_{x\rightarrow c}f\left( x\right) \right) \end{equation*}% \bigskip Theorem: \ (\textbf{The Intemediate value Theorem for Continuous Functions}) \ If $f$ is continuous on a closed interval $\left[ a,b\right] $ and if $% y_{0}$ is any value between $f\left( a\right) $ and $f\left( b\right) $, then $y_{0}=f\left( c\right) $ for some $c$ in $\left[ a,b\right] $.\bigskip \begin{center} \bigskip {\Large Maximum-minimum theorems\bigskip } \end{center} Theorem: (\textbf{Extreme Value Theorem}) If $f$ is continuous on a closed interval $\left[ a,b\right] $, then $f$ attains an absolute maximum value $% f\left( c\right) $ and an absolute minimum value $f\left( d\right) $ at some numbers $c$ and $d$ in $\left[ a,b\right] $.\bigskip Definition: A critical number of a function $f$ is a number $c$ in its domain such that either $f^{\prime }\left( c\right) =0$ or $f^{\prime }\left( c\right) $ does not exist.\bigskip Theorem: (Fermat) \ If $f$ has a local maximum or minimum at $c$, then $c$ is a critical number of $f$.\bigskip Closed interval method: \ To find absolute extrema of a continuous function $% f$ on a closed interval $\left[ a,b\right] $.\newline 1) \ Find the values of $f$ at the critical numbers of $f$ in $\left[ a,b% \right] .$\newline 2) \ Find the values of $f$ at the endpoints of the interval.\newline 3) \ The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.\bigskip \bigskip \begin{center} {\Large The Mean Value Theorem\bigskip } \end{center} Theorem: (Rolle) Suppose that $f$ is continuous on $\left[ a,b\right] $ and differentiable on $\left( a,b\right) $. \ If $f\left( a\right) =f\left( b\right) ,$ then there exists $c$ in $\left( a,b\right) $ with $f^{\prime }\left( c\right) =0$.\bigskip Theorem: (Mean Value Theorem) \ Suppose that$f$ is continuous on $\left[ a,b% \right] $ and differentiable on $\left( a,b\right) $. \ Then there exists $c$ in $\left( a,b\right) $ with $f^{\prime }\left( c\right) =\dfrac{f\left( b\right) -f\left( a\right) }{b-a}$.\bigskip Corollary: If $f^{\prime }\left( x\right) =0$ on an interval $\left( a,b\right) ,$ then $f\left( x\right) =c$ on $\left( a,b\right) $ for a constant $c$.\bigskip Theorem: \ If $f^{\prime }\left( x\right) =g^{\prime }\left( x\right) $ on $% \left( a,b\right) $, then there exists $c$ with $f\left( x\right) =g\left( x\right) +c$ on $\left( a,b\right) $.\bigskip \bigskip \bigskip \pagebreak \begin{center} {\Large The Fundamental Theorem\bigskip } \end{center} Theorem: (\textbf{Mean value Theorem for Definite Integrals}) \ If $f$ is continuous on $\left[ a,b\right] $, then there exists $c$ in $\left[ a,b% \right] $ such that% \begin{equation*} f\left( c\right) =\dfrac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) dx \end{equation*}% \bigskip Theorem: (\textbf{Fundamental Theorem of Calculus, Part 1}) \ If $f$ is continuous on $\left[ a,b\right] $, then $F\left( x\right) =\dint\limits_{a}^{x}f\left( t\right) dt$ is continuous on $\left[ a,b\right] $, \ differentiable on $\left( a,b\right) $, and its derivative is $f\left( x\right) $:% \begin{equation*} F^{\prime }\left( x\right) =\dfrac{d}{dx}\dint\limits_{a}^{x}f\left( t\right) dt=f\left( x\right) \end{equation*}% \bigskip Theorem: (\textbf{Fundamental Theorem of Calculus, Part 2}) \ If $f$ is continuous on $\left[ a,b\right] $, and $F$ is any antiderivative of $f$ on $\left[ a,b\right] $, then% \begin{equation*} \dint\limits_{a}^{b}f\left( x\right) dx=F\left( b\right) -F\left( a\right) \end{equation*}% \bigskip Theorem: (Net Change Theorem) \ The net change in a function $% F\left( x\right) $ over an interval $\left[ a,b\right] $ is the integral of its rate of change:% \begin{equation*} F\left( b\right) -F\left( a\right) =\dint\limits_{a}^{b}F^{\prime }\left( x\right) dx \end{equation*}% \bigskip \pagebreak \end{document}