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\begin{document}
%MODULE 1: Optimization


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%INPUT THE INSTRUCTIONS PAGE 


%BUILD  TITLE PAGE 



\title{Linear Approximations}


\subtitle{In this module we explore how to approximate a value of a function
using the line tangent to the function at a nearby point. This method
is known as {\hilite linear approximation}. We will then explore
a related idea known as {\hilite relative rates of change}. }

\makebeamertitle

\lyxframeend{}\lyxframe{{Sample problems}}

\label{page:modf-problem} \textbf{\em Sample problems:} 
\begin{itemize}
\item Approximate the value of $\ec\sqrt[5]{2}$ to the tenths place. 
\item Suppose that blood flows along a blood vessel. The volume of blood
per unit time that flows past a given point is called the \emph{flux}
$\ec F$. Poiseulle's Law says that this value is proportional to
the fourth power of the radius $\ec R$ of the blood vessel; that
is,
\[
\ec R=kF^{4}.
\]
What is ratio of the relative change in $\ec F$ to the relative change
in $\ec R$, and how will a 5\% increase in the radius affect the
flow of blood? 
\end{itemize}
\noindent %1



\lyxframeend{}\lyxframe{{SAGE worksheets}}

For this module you will not need a SAGE worksheet, although you can
perform some of the calculations and plots yourself.


\lyxframeend{}\lyxframe{\noindent {Linear Approximations: the idea}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent Let $\ec f$ be a function, and $\ec a$ a real number.
In general, the line tangent to $\ec f$ at $\ec a$ stays \emph{very
close} to the curve defined by $f$ on a small interval.


\pause{}


For example, if $\ec f(x)=\sqrt[5]{x}$, the graph below shows both
$\ec f$ and the line $\ec L$ tangent to $\ec f$ at $\ec x=1$.


\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/linapp_surd5_onetenth} 
\end{figure}

\begin{onlyenv}%{}
<2>

\noindent The dashed red line indicates a boundary of 1/10 from $\ec f$.
It shows that $\ec L(x)-f(x)<1/10$ on the interval $\ec[0.25,2.5]$.
Thus, if you want to compute $\ec\sqrt[5]{2}$ and you don't mind
an error of less than {\ec 1/10}, you could compute $\ec L(2)$
instead.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

\noindent On the other hand, if you travel too far from $\ec x=1$,
the error is too large. If you wanted to compute $\ec\sqrt[5]{3}$,
the dashed red line lies between the tangent line and the curve $\ec f(x)$.
The linear approximation would give an error greater than 1/10.

\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Linear Approximations}}

In general, the strategy to compute a linear approximation of $\ec f(a)$
is as follows: \begin{method} 
\begin{itemize}
\item Identify the function $\ec f(x)$. 
\item Identify a point $\ec b$ such that 

\begin{itemize}
\item $\ec b$ is close to $\ec a$, and 
\item both $\ec f(b)$ and $\ec f'(b)$ are \emph{easy to compute}. 
\end{itemize}
\item Determine $\ec L(x)$, the equation of the line tangent to $\ec f$
at $\ec x=b$. 
\item Compute $\ec L(b)$; that is, substitute $\ec x=b$ into $\ec L(x)$. 
\end{itemize}
\noindent \end{method} $\ec L(b)$ is the linear approximation of
$\ec f(a)$.


\lyxframeend{}\lyxframe{{Linear Approximations} }

\noindent We can illustrate what we are doing with several graphs.
We return to approximating $\ec\sqrt[5]{2}$.

\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.3]{figures/surd5_2} 
\end{figure}

\begin{itemize}
\item The function $\ec f(x)$ is $\ec\sqrt[5]{x}$, and $\ec a=2$. 
\item For a {}``nice'' point that is easy to evaluate and close to $\ec a=2$
we use $\ec b=1$. 
\end{itemize}

\lyxframeend{}\lyxframe{Linear Approximations}

\noindent We can illustrate what we are doing with several graphs.
We return to approximating $\ec\sqrt[5]{2}$.

\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.3]{figures/surd5_2_1_and_L} 
\end{figure}

\begin{itemize}
\item The line tangent to $\ec f(x)$ at $\ec x=b$ is $\ec L(x)=\frac{1}{5}(x-1)+1$.\\
 (You should check this to make sure you can obtain it yourself.) 
\item We evaluate $\ec L(a)$, which is $\ec L(2)$, and we get the point
shown on the line. 
\end{itemize}
\noindent The estimate comes to $\ec L(2)=\frac{1}{5}(2-1)+1=1.2$.


\pause{}

\noindent We conclude that $\ec\sqrt[5]{2}\approx1.2$. This is within
one-tenth of the true value.


\lyxframeend{}\lyxframe{{Applied Problem}}

Earlier we posed the problem about blood flow: \begin{question} Suppose
that blood flows along a blood vessel. The volume of blood per unit
time that flows past a given point is called the \emph{flux} $\ec F$.
Poiseulle's Law says that this value is proportional to the fourth
power of the radius $\ec R$ of the blood vessel; that is,
\[
{\ec R=kF^{4}}.
\]
 What is the ratio of the relative change in $F$ to the relative
change in $\ec R$, and how will a 5\% increase in the radius affect
the flow of blood? \end{question} Solving this problem uses a similar
idea.


\lyxframeend{}\lyxframe{\noindent {Differentials}}

\noindent Recall the \emph{\hilite Leibniz notation} of the derivative.
For any point $\ec x$ where $\ec f$ is differentiable,
\[
{\ec f'(x)=\frac{dy}{dx}}.
\]
 Remember also that $\ec dy/dx$ was not a fraction, because $\ec dx$
and $\ec dy$ had no individual meaning. Now we give them a meaning.


\pause{}
\begin{definition}%{}
\noindent  We say that the quantities $\ec dx$ and $\ec dy$ are
\emph{\hilite differentials} when the satisfy the property 
\[
{\ec dy=f'(x)\; dx}.
\]
 Here $\ec dy$ is a dependent variable, whose quantity depends on
the values of $\ec x$ and $\ec dx$.
\end{definition}%{}

\lyxframeend{}\lyxframe{{Relative Change in $x$, $y$}}
\begin{definition}%{}
 If $\ec y$ depends on $\ec x$, the \emph{\hilite relative change}
of $\ec y$ is 
\[
{\ec\frac{dy}{f(x)}}.
\]
 
\end{definition}%{}

\pause{}

\noindent In a linear approximation, we also call relative change
the \emph{\hilite relative error}.


\lyxframeend{}\lyxframe{{Applied Problem: first part}}

We return to the applied problem. \begin{question} Given $\ec F=kR^{4}$,
find the ratio of the relative change in $\ec F$ to the relative
change in $\ec R$. \end{question}


\pause{}

\noindent Translated into symbols, we have to find
\[
{\ec\frac{\mathrm{relative\ change\ in\ }F}{\mathrm{relative\ change\ in\ }R}=\frac{\frac{dF}{F}}{\frac{dR}{R}}}.
\]



\pause{}

\noindent From the definition of a differential, we know that
\begin{align*}
\ec dF & \ec=F'\; dR\\
\ec dF & {\ec=4kR^{3}\; dR}.
\end{align*}



\pause{}

\noindent By substitution,
\[
{\ec\frac{\mathrm{relative\ change\ in\ }F}{\mathrm{relative\ change\ in\ }R}=\frac{\frac{dF}{F}}{\frac{dR}{R}}=\frac{\frac{4kR^{3}\; dR}{kR^{4}}}{\frac{dR}{R}}=\frac{4\; dR}{R}\cdot\frac{R}{dR}=4}.
\]
 The ratio of the relative rate of change of $\ec F$ to the relative
rate of change of $\ec R$ is $\ec4$.


\lyxframeend{}\lyxframe{{Applied Problem: second part}}

The second part of the problem asked, \begin{question} How will a
5\% increase in the radius affect the flow of blood? \end{question}


\pause{}

\noindent  The phrase, {}``a 5\% increase in the radius,'' tells
us that the relative rate of change of $\ec R$ is 5\%.


\pause{}

It asks for the relative rate change of change in $\ec F$. In the
first part of the problem, we learned that the ratio between the two
is 
\[
{\ec\frac{dF/F}{dR/R}=4}.
\]
 The relative change of $\ec F$ is 4\ times the relative change
of $\ec R$. Thus, the flow of blood should increase by {\hilite
20\%}.


\lyxframeend{}\lyxframe{{Conclusion}}

In this module, you have seen: 
\begin{itemize}
\item how to compute a {\hilite linear approximation} of an irrational
number by using the line tangent to a function; \begin{method} 

\begin{itemize}
\item Identify the function $\ec f(x)$. 
\item Identify a point $\ec b$ such that 

\begin{itemize}
\item $\ec b$ is close to $\ec a$, and 
\item both $\ec f(b)$ and $\ec f'(b)$ are \emph{easy to compute}. 
\end{itemize}
\item Determine $\ec L(x)$, the equation of the line tangent to $\ec f$
at $\ec x=b$. 
\item Compute $\ec L(b)$; that is, substitute $\ec x=b$ into $\ec L(x)$. 
\end{itemize}

\end{method} 

\item how a linear approximation can lose its accuracy as one moves away
from the point where the line is built; and 
\item how {\hilite differentials} are used to compute relative rates
of change. 
\end{itemize}
\noindent Later you will learn a method of approximating irrational
numbers that is usually a better choice, called \emph{Newton's Method}. 


\lyxframeend{}
\end{document}
