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\title[]{MA140-Engineering Calculus}
\author[ Lecture 2 ]{\large{ Lecture 2} 
\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{Polynomials}
\begin{frame}
A polynomial is a function of the form:
$$y=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$
where $a_{n},a_{n-1},\cdots,a_{0}$ are constants.\\
These constants are called the \textit{coefficients } of the polynomial. \\
The number $n$ is the degree of the polynomial.
\begin{example}
$y=x,$ is a \textit{linear} polynomial
\end{example}
\begin{center}
			\includegraphics[scale=0.2]{Figures/lec15.png}
		\end{center}
\end{frame}
\begin{frame}
\begin{example}
$y=x^{2}-4x+3$ is a \textit{quadratic} polynomial.
\end{example}
\begin{center}
			\includegraphics[scale=0.3]{Figures/lec16.png}
		\end{center}
		\end{frame}
		\begin{frame}
\begin{example}
$y=x^{3}-4x^{2}+x+6$ is a \textit{cubic} polynomial.
\end{example}
\begin{center}
			\includegraphics[scale=0.3]{Figures/lec17.png}
		\end{center}
\end{frame}
\begin{frame}

{\textbf {\textcolor{red}{General facts on polynomial sketching:}}}\\
A polynomial of degree $n$ has up to $n-1$ bends.
\begin{example}
a typical fourth degree polynomial has $3$ bends. for example $y=x^{4}-2x^{2}+x$

\end{example} 
\begin{center}
			\includegraphics[scale=0.3]{Figures/lec18.png}
		\end{center}
		\end{frame}
		\begin{frame}
		\begin{example}
the fourth degree polynomial $y=x^{4}-2x$ has only one bend.
\end{example} 
\begin{center}
			\includegraphics[scale=0.3]{Figures/lec19.png}
		\end{center}
		\end{frame}
		\begin{frame}
		
{\textbf {\textcolor{red}{Find the intercepts:}}}\\
The $y-$intercept can be found by letting $x=0$.\\
The $x-$intercepts are the roots (or zeros).\\
{\textbf {\textcolor{red}{Note:}}} you do not have to use the same scale on the $x$ and $y$ axis.

\begin{example}
$y=x^{3}-4x^{2}+x+6$
\end{example}
\textcolor{blue}{$x=0 \enspace \Rightarrow \enspace y=6$ ($y-$intercept)\\
(The constant coefficient=$-$ product of the roots if the coefficient of the highest power=$1$)\\
By trial roots are $x=-1,2,3$}
\end{frame}
\begin{frame}
\begin{center}
			\includegraphics[scale=0.3]{Figures/lec20.png}
		\end{center}
\end{frame}
\subsection{Rational Functions}
\begin{frame}
\begin{definition}
Rational functions have the general form $$f(x)=\frac{p(x)}{q(x)}$$
where $p(x)$ and $q(x)$ are polynomials.
\end{definition}
\begin{itemize}
\item \textbf{IF} degree of $p(x) <$ degree of $q(x)$, then $f(x)$ is a strictly proper rational function. 
\item \textbf{IF} degree of $p(x) =$ degree of $q(x)$, then $f(x)$ is a proper rational function. 
\item \textbf{IF} degree of $p(x) >$ degree of $q(x)$, then $f(x)$ is an improper rational function. 
\end{itemize}

\end{frame}
\begin{frame}
An improper or proper rational function can be expressed in terms of a strictly proper rational function
\begin{example}
Express $f(x)=\displaystyle{\frac{3x^{4}+2x^{3}-5x^{2}+6x-7}{x^{2}-2x+3}}$ in terms of a strictly proper rational function
\end{example}
{\textcolor{blue}{$f(x)=3x^{2}+8x+2-\displaystyle{\frac{14x+13}{x^{2}-2x+3}}$}}
\end{frame}






















\end{document}