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\title[]{MA140-Engineering Calculus}
\author[ Lecture 1 ]{\large{ Lecture 1} 
\and %\\ \small{Supervisor: Dr. John Burns}
\\ 
 \includegraphics[scale=0.0]{Figures/lec14.jpg}}
%\date[June 2-4, 2016]{June 2-4, 2016}
%\institute[Uppsala University]{ }


\begin{document}
\begin{frame}
\titlepage
\end{frame}





\section{MA140}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Introduction}
\begin{frame}
\begin{itemize}
\item {\textbf {\textcolor{red}{Lectures:}}} 10 am, ENG-G018, Tuesday, Wednesday, Thursday
\item {\textbf {\textcolor{red}{Lecturer:}}} Dr. Adib Makrooni \\ \hspace{1.5cm} Office: ADB-2003 \\ email: mohammadadib.makrooni@nuigalway.ie
\item {\textbf {\textcolor{red}{Tutorials:}}} details will be announced later, normally they start two weeks after the first lecture
\item {\textbf {\textcolor{red}{Supporting centre:}}} SUMS, visit the homepage to see the timetables: http://www.maths.nuigalway.ie/sums/
\item {\textbf {\textcolor{red}{Text books:}}} Modern Engineering Mathematics by Glyn James,\\
Thomas' Calculus or any basic calculus text book 
\end{itemize}
\end{frame}
\subsection{Numbers}
\begin{frame}{Lecture 1}
In this lecture we review real and complex numbers.
\begin{itemize}
\item Natural Numbers=$\mathbb{N}=\{1,2,3,\cdots\}$\\
$1+3=4$, $4$ is a natural number or $4$ belongs to the set of natural numbers $\simeq 4\in \mathbb{N}$\\
$$1-1=0, \quad 0\notin \mathbb{N}$$
\item Whole Numbers=$\mathbb{N}_{0}=\{0,1,2,\cdots\}$ so we see that $2-2=0\in \mathbb{N}_{0}$ but $1-3=-2\notin \mathbb{N}_{0}$\\
$\mathbb{N}$ is a subset of $\mathbb{N}_{0} \quad \longleftrightarrow \quad \mathbb{N}\subset \mathbb{N}_{0}$ \\
Remark: 
\begin{itemize}
\item $A\subset B$ it means for all elements in $A$ or for any element in $A$ like $x$ then $x$ is in $B \quad \longleftrightarrow \quad (\forall x\in A \quad \rightarrow x\in B)$\\
So $\forall$ means ''for all''
\item $A\not\subset B$ it means there exists an element in $A$ like $x$ which is not in 
$B \quad \longleftrightarrow \quad (\exists x\in A \quad .s.t \quad x\notin B)$ 
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}

\begin{itemize}
\item Integers=$\mathbb{Z}=\{\cdots,-2,-1,0,1,2,\cdots\}$\\
$1+(-4)=-3\in \mathbb{Z}$ and $1-5=-4\in \mathbb{Z}$ so the addition and subtraction of two integers is again an integer but what about the division?\\
$\displaystyle{ \frac{4}{2}}=2\in \mathbb{Z}$, $\displaystyle(\frac{4}{3})\notin \mathbb{Z}$
\item Rational Numbers=$\mathbb{Q}=\{\displaystyle{\frac{a}{b}} \quad | \quad a,b\in \mathbb{Z} \quad and \quad b\neq 0 \}$\\ for example $\displaystyle{\frac{-3}{4}}=-0.75000...$ or $\displaystyle{\frac{1}{3}}=-0.33333...$\\ But $\sqrt{2}\notin \mathbb{Q}$
\end{itemize}
\end{frame}
\begin{frame}
\begin{itemize}
\item Real Numbers=$\mathbb{R}$, the set of real numbers includes all the rational numbers and numbers like $\sqrt{2}, \pi=3.14..., 0.11236$, these numbers are called irrational numbers so \\real numbers=(rational numbers)$\cup$(irrational numbers)\\
$\mathbb{R}=\mathbb{Q} \cup  (\mathbb{R}\backslash \mathbb{Q})$
\begin{center}
			\includegraphics[scale=0.1]{Figures/lec11.jpg}
		\end{center}
		$$\mathbb{N}\subset \mathbb{N}_{0} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$
		\item Complex Numbers=$\mathbb{C}$\\
		If $c\in \mathbb{C}$, we can write $c=a+ib, \quad a,b\in \mathbb{R} \quad and \quad i=\sqrt{-1}$  $$\mathbb{N}\subset \mathbb{N}_{0} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$
		 
\end{itemize}
\end{frame}
\subsection{Functions}
\begin{frame}
We represent a function in one of the two ways:$$f:x\rightarrow y \quad or \quad y=f(x)$$
\begin{itemize}
\item $x$ is called the independent variable and $y$ is called the dependent variable.
\item when we write $y=f(x)$, $"x"$ is known as the \textit{argument} of the function.
\item here $x$ is in the set $X$ and the set $X$ is called the \textit{domain} of the function
\item  and $y$ is in the set $Y$, the set $Y$ is called the \textit{codomain}
\end{itemize} 
\begin{definition}
A \textit{function} from a set $X$ to a set $Y$ is a rule that assigns a \textcolor{red}{unique} (single) element $f(x)\in Y$ to each element $x\in X$.
\end{definition}

\end{frame}
\begin{frame}

{\textbf {\textcolor{red}{Note:}}} $f$ associates each value of $x$ in $X$ with exactly one value of $y$ in $Y$. It means we can not have different outputs for the same input.
\begin{center}
			\includegraphics[scale=0.6]{Figures/lec12.jpg}
		\end{center}
\end{frame}
\begin{frame}
\begin{itemize}

\item when $y=f(x)$, $y$ is said to be the \textit{image} of $x$ under $f$
\item the set of all images is called the \textit{image set} or \textit{range} of $f$
\end{itemize}


{\textbf {\textcolor{red}{Note:}}} It is not necessary for all elements of the codomain set $Y$ to be images under $f$
\begin{center}
			\includegraphics[scale=0.6]{Figures/lec13.jpg}
		\end{center}

\end{frame}
\begin{frame}
\begin{example}
Identify the domain, codomain and range of 
\begin{itemize}
\item[(a)] $f(x)=3x^{2}+1$
\item[(b)] $f(x)=\sqrt{(x+4)(3-x)}$
\end{itemize}
\end{example}
{\textcolor{red}{solution (a):}}\\
{\textcolor{blue}{$f(x)$ can be evaluated for all $x\in \mathbb{R}$, so domain=$ \mathbb{R}$.\\ The lowest value $f(x)$ can take is $1$ (when $x=0$) so range=$[1,\infty]$. \\ We could write this as $\{y \enspace | \enspace  y\geq 1 \enspace , \enspace y\in \mathbb{R}\}$.\\We could take the codomain as $\mathbb{R}$ as it contains the range.}}\\

\end{frame}
\begin{frame}

{\textcolor{red}{solution (b):}}\\
{\textcolor{blue}{The domain is $[-4,3]$, outside this range the function is not real valued i.e. it involves $\sqrt{-1}$.\\ The function takes value $0$ at $x=-4$ and $x=3$ and takes $7/2$, its highest value at $x=-1/2$. Therefore its range is $[0,7/2]$. we can take the codomain to be $\mathbb{R}$ (as it contains the range)}}

\end{frame}






















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