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\begin_body

\begin_layout Standard
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%MODULE 1: Optimization
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%USER DEFINED COMMANDS
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\end_layout

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\begin_inset FormulaMacro
\newcommand{\ds}{{\displaystyle }}
\end_inset

 
\begin_inset FormulaMacro
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\begin_inset FormulaMacro
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\begin_inset FormulaMacro
\newcommand{\xp}{\mbox{\Large\sl e} }
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%INPUT THE INSTRUCTIONS PAGE 
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\begin_layout Plain Layout

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\begin_layout Standard
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%BUILD  TITLE PAGE 
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\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Title
Linear Approximations
\end_layout

\begin_layout 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 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 linear approximation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 We will then explore a related idea known as 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 relative rates of change
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Sample problems
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "page:modf-problem"

\end_inset

 
\series bold

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
em
\end_layout

\end_inset

 Sample problems:
\series default
 
\end_layout

\begin_layout Itemize
Approximate the value of 
\begin_inset Formula $\ec\sqrt[5]{2}$
\end_inset

 to the tenths place.
 
\end_layout

\begin_layout Itemize
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 on
flux
\emph default
 
\begin_inset Formula $\ec F$
\end_inset

.
 Poiseulle's Law says that this value is proportional to the fourth power
 of the radius 
\begin_inset Formula $\ec R$
\end_inset

 of the blood vessel; that is,
\begin_inset Formula 
\[
\ec R=kF^{4}.
\]

\end_inset

What is ratio of the relative change in 
\begin_inset Formula $\ec F$
\end_inset

 to the relative change in 
\begin_inset Formula $\ec R$
\end_inset

, and how will a 5% increase in the radius affect the flow of blood? 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
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\begin_layout Plain Layout

%1
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

SAGE worksheets
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

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

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Linear Approximations: the idea
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $\ec f$
\end_inset

 be a function, and 
\begin_inset Formula $\ec a$
\end_inset

 a real number.
 In general, the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec a$
\end_inset

 stays 
\emph on
very close
\emph default
 to the curve defined by 
\begin_inset Formula $f$
\end_inset

 on a small interval.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
For example, if 
\begin_inset Formula $\ec f(x)=\sqrt[5]{x}$
\end_inset

, the graph below shows both 
\begin_inset Formula $\ec f$
\end_inset

 and the line 
\begin_inset Formula $\ec L$
\end_inset

 tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=1$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/linapp_surd5_onetenth.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
The dashed red line indicates a boundary of 1/10 from 
\begin_inset Formula $\ec f$
\end_inset

.
 It shows that 
\begin_inset Formula $\ec L(x)-f(x)<1/10$
\end_inset

 on the interval 
\begin_inset Formula $\ec[0.25,2.5]$
\end_inset

.
 Thus, if you want to compute 
\begin_inset Formula $\ec\sqrt[5]{2}$
\end_inset

 and you don't mind an error of less than 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset

 1/10
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, you could compute 
\begin_inset Formula $\ec L(2)$
\end_inset

 instead.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
On the other hand, if you travel too far from 
\begin_inset Formula $\ec x=1$
\end_inset

, the error is too large.
 If you wanted to compute 
\begin_inset Formula $\ec\sqrt[5]{3}$
\end_inset

, the dashed red line lies between the tangent line and the curve 
\begin_inset Formula $\ec f(x)$
\end_inset

.
 The linear approximation would give an error greater than 1/10.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Linear Approximations
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In general, the strategy to compute a linear approximation of 
\begin_inset Formula $\ec f(a)$
\end_inset

 is as follows: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Identify the function 
\begin_inset Formula $\ec f(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Identify a point 
\begin_inset Formula $\ec b$
\end_inset

 such that 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec b$
\end_inset

 is close to 
\begin_inset Formula $\ec a$
\end_inset

, and 
\end_layout

\begin_layout Itemize
both 
\begin_inset Formula $\ec f(b)$
\end_inset

 and 
\begin_inset Formula $\ec f'(b)$
\end_inset

 are 
\emph on
easy to compute
\emph default
.
 
\end_layout

\end_deeper
\begin_layout Itemize
Determine 
\begin_inset Formula $\ec L(x)$
\end_inset

, the equation of the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=b$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Compute 
\begin_inset Formula $\ec L(b)$
\end_inset

; that is, substitute 
\begin_inset Formula $\ec x=b$
\end_inset

 into 
\begin_inset Formula $\ec L(x)$
\end_inset

.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset

 
\begin_inset Formula $\ec L(b)$
\end_inset

 is the linear approximation of 
\begin_inset Formula $\ec f(a)$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Linear Approximations
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

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

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/surd5_2.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The function 
\begin_inset Formula $\ec f(x)$
\end_inset

 is 
\begin_inset Formula $\ec\sqrt[5]{x}$
\end_inset

, and 
\begin_inset Formula $\ec a=2$
\end_inset

.
 
\end_layout

\begin_layout Itemize
For a 
\begin_inset Quotes eld
\end_inset

nice
\begin_inset Quotes erd
\end_inset

 point that is easy to evaluate and close to 
\begin_inset Formula $\ec a=2$
\end_inset

 we use 
\begin_inset Formula $\ec b=1$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
Linear Approximations
\end_layout

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

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/surd5_2_1_and_L.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The line tangent to 
\begin_inset Formula $\ec f(x)$
\end_inset

 at 
\begin_inset Formula $\ec x=b$
\end_inset

 is 
\begin_inset Formula $\ec L(x)=\frac{1}{5}(x-1)+1$
\end_inset

.
\begin_inset Newline newline
\end_inset

 (You should check this to make sure you can obtain it yourself.) 
\end_layout

\begin_layout Itemize
We evaluate 
\begin_inset Formula $\ec L(a)$
\end_inset

, which is 
\begin_inset Formula $\ec L(2)$
\end_inset

, and we get the point shown on the line.
 
\end_layout

\begin_layout Standard
\noindent
The estimate comes to 
\begin_inset Formula $\ec L(2)=\frac{1}{5}(2-1)+1=1.2$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
We conclude that 
\begin_inset Formula $\ec\sqrt[5]{2}\approx1.2$
\end_inset

.
 This is within one-tenth of the true value.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Applied Problem
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Earlier we posed the problem about blood flow: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 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 on
flux
\emph default
 
\begin_inset Formula $\ec F$
\end_inset

.
 Poiseulle's Law says that this value is proportional to the fourth power
 of the radius 
\begin_inset Formula $\ec R$
\end_inset

 of the blood vessel; that is,
\begin_inset Formula 
\[
{\ec R=kF^{4}}.
\]

\end_inset

 What is the ratio of the relative change in 
\begin_inset Formula $F$
\end_inset

 to the relative change in 
\begin_inset Formula $\ec R$
\end_inset

, and how will a 5% increase in the radius affect the flow of blood? 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 Solving this problem uses a similar idea.
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Differentials
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Recall the 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 Leibniz notation
\emph default
 of the derivative.
 For any point 
\begin_inset Formula $\ec x$
\end_inset

 where 
\begin_inset Formula $\ec f$
\end_inset

 is differentiable,
\begin_inset Formula 
\[
{\ec f'(x)=\frac{dy}{dx}}.
\]

\end_inset

 Remember also that 
\begin_inset Formula $\ec dy/dx$
\end_inset

 was not a fraction, because 
\begin_inset Formula $\ec dx$
\end_inset

 and 
\begin_inset Formula $\ec dy$
\end_inset

 had no individual meaning.
 Now we give them a meaning.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Definition
\noindent
 We say that the quantities 
\begin_inset Formula $\ec dx$
\end_inset

 and 
\begin_inset Formula $\ec dy$
\end_inset

 are 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 differentials
\emph default
 when the satisfy the property 
\begin_inset Formula 
\[
{\ec dy=f'(x)\; dx}.
\]

\end_inset

 Here 
\begin_inset Formula $\ec dy$
\end_inset

 is a dependent variable, whose quantity depends on the values of 
\begin_inset Formula $\ec x$
\end_inset

 and 
\begin_inset Formula $\ec dx$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Relative Change in 
\begin_inset Formula $x$
\end_inset

, 
\begin_inset Formula $y$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Definition
 If 
\begin_inset Formula $\ec y$
\end_inset

 depends on 
\begin_inset Formula $\ec x$
\end_inset

, the 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 relative change
\emph default
 of 
\begin_inset Formula $\ec y$
\end_inset

 is 
\begin_inset Formula 
\[
{\ec\frac{dy}{f(x)}}.
\]

\end_inset

 
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
In a linear approximation, we also call relative change the 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 relative error
\emph default
.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Applied Problem: first part
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We return to the applied problem.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Given 
\begin_inset Formula $\ec F=kR^{4}$
\end_inset

, find the ratio of the relative change in 
\begin_inset Formula $\ec F$
\end_inset

 to the relative change in 
\begin_inset Formula $\ec R$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

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

\end_inset


\end_layout

\begin_layout Pause

\end_layout

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

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
By substitution,
\begin_inset Formula 
\[
{\ec\frac{\mathrm{relative\ change\ in\ }F}{\mathrm{relative\ change\ in\ }R}=\frac{\frac{dF}{F}}{\frac{dR}{R}}=\frac{\frac{4kR^{3}\; dR}{kR^{4}}}{\frac{dR}{R}}=\frac{4\; dR}{R}\cdot\frac{R}{dR}=4}.
\]

\end_inset

 The ratio of the relative rate of change of 
\begin_inset Formula $\ec F$
\end_inset

 to the relative rate of change of 
\begin_inset Formula $\ec R$
\end_inset

 is 
\begin_inset Formula $\ec4$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Applied Problem: second part
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The second part of the problem asked, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 How will a 5% increase in the radius affect the flow of blood? 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
 The phrase, 
\begin_inset Quotes eld
\end_inset

a 5% increase in the radius,
\begin_inset Quotes erd
\end_inset

 tells us that the relative rate of change of 
\begin_inset Formula $\ec R$
\end_inset

 is 5%.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
It asks for the relative rate change of change in 
\begin_inset Formula $\ec F$
\end_inset

.
 In the first part of the problem, we learned that the ratio between the
 two is 
\begin_inset Formula 
\[
{\ec\frac{dF/F}{dR/R}=4}.
\]

\end_inset

 The relative change of 
\begin_inset Formula $\ec F$
\end_inset

 is 4
\begin_inset space \space{}
\end_inset

times the relative change of 
\begin_inset Formula $\ec R$
\end_inset

.
 Thus, the flow of blood should increase by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 20%
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In this module, you have seen: 
\end_layout

\begin_layout Itemize
how to compute a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 linear approximation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 of an irrational number by using the line tangent to a function; 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_deeper
\begin_layout Itemize
Identify the function 
\begin_inset Formula $\ec f(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Identify a point 
\begin_inset Formula $\ec b$
\end_inset

 such that 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec b$
\end_inset

 is close to 
\begin_inset Formula $\ec a$
\end_inset

, and 
\end_layout

\begin_layout Itemize
both 
\begin_inset Formula $\ec f(b)$
\end_inset

 and 
\begin_inset Formula $\ec f'(b)$
\end_inset

 are 
\emph on
easy to compute
\emph default
.
 
\end_layout

\end_deeper
\begin_layout Itemize
Determine 
\begin_inset Formula $\ec L(x)$
\end_inset

, the equation of the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

 at 
\begin_inset Formula $\ec x=b$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Compute 
\begin_inset Formula $\ec L(b)$
\end_inset

; that is, substitute 
\begin_inset Formula $\ec x=b$
\end_inset

 into 
\begin_inset Formula $\ec L(x)$
\end_inset

.
 
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\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset

 
\end_layout

\end_deeper
\begin_layout Itemize
how a linear approximation can lose its accuracy as one moves away from
 the point where the line is built; and 
\end_layout

\begin_layout Itemize
how 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 differentials
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 are used to compute relative rates of change.
 
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\begin_layout Standard
\noindent
Later you will learn a method of approximating irrational numbers that is
 usually a better choice, called 
\emph on
Newton's Method
\emph default
.
 
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\end_body
\end_document
