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%MODULE 1: Epsilon-Delta limits
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\begin_layout Title
An intuitive notion of derivatives
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In this lesson you begin to study a topic called the 
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{
\backslash
hilite 
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\emph on
derivative
\emph default

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}
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.
 In Calculus, the derivative is a tool used to describe the 
\emph on
rate of change
\emph default
 of a function.
 
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A Sample problem
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}
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An electronic instrument makes a sound by vibrating a speaker's voice coil
 according to the function 
\begin_inset Formula $\ec s(x)=v\sin(x)$
\end_inset

, where
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\begin_layout Itemize
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 is a constant; 
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\begin_inset Formula $\ec x$
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 is the time in seconds; and 
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\begin_layout Itemize
\begin_inset Formula $\ec s(x)$
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 is the distance in mm that the coil has moved from equilibrium.
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\noindent
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\backslash
begin{question}
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What is the rate of change of the position of the coil at time 
\begin_inset Formula $\ec x=1$
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second when 
\begin_inset Formula $\ec v=1$
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 and when 
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?
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SAGE worksheets
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In this lab you could use a blank SAGE worksheet, although you can get by
 with just a calculator, too.
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\noindent
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{
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Two kinds of rates of change
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\noindent
We said that the derivative is the rate of change.
 There are actually two different rates of change that you need to know,
 and only one of them is the derivative.
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<2-3>
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\begin_layout Definition
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[
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(Average Rate of Change)
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]
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The 
\emph on
average
\emph default
 rate of change between 
\emph on
two points
\emph default
 
\begin_inset Formula $\ec x=a$
\end_inset

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

 is the slope of the line connecting the points 
\begin_inset Formula $\ec(a,f(a))$
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 and 
\begin_inset Formula $\ec(b,f(b))$
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.
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<3>
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\begin_deeper
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In the case of the speaker (
\begin_inset Formula $\ec s(x)=\sin(x)$
\end_inset

), the average rate of change between 
\begin_inset Formula $\ec x=0$
\end_inset

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

 is 
\begin_inset Formula 
\[
\ec\frac{\sin(1)-\sin(0)}{1-0}\approx0.84.
\]

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The 
\emph on
instantaneous
\emph default
 rate of change at 
\emph on
one point
\emph default
 
\begin_inset Formula $\ec x=b$
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 is the slope of the line 
\emph on
tangent to
\emph default
 the curve at 
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.
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In the case of the speaker (
\begin_inset Formula $\ec s(x)=\sin(x)$
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), the instantaneous rate of change at 
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 is 
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.
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\noindent
(If you have no idea how I computed 0.54, don't worry.
 You're not supposed to know that yet.)
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{
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Tangent = Limit of Secants
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It might not be obvious at first, but as 
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 and 
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 usually approach the tangent line at 
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.
 The next few slides illustrate this.
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<2>
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The secant line between 
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 and 
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<3>
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The secant line between 
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 and 
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<4>
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The secant line between 
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 and 
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<5>
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The secant line between 
\begin_inset Formula $\ec x=0.9$
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 and 
\begin_inset Formula $\ec x=1$
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: 
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<6>
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The tangent line at 
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{
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Okay, so which one is the derivative?
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\begin_layout Definition
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[
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(The derivative at a point)
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The 
\emph on

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 derivative
\emph default
 of 
\begin_inset Formula $\ec f$
\end_inset

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

 is the 
\emph on
instantaneous
\emph default
 rate of change of 
\begin_inset Formula $\ec f$
\end_inset

.
\end_layout

\begin_layout Standard
As we pointed out earlier, this is the slope of the tangent line.
 Thus the derivative is the 
\emph on
slope
\emph default
 of the line tangent to 
\begin_inset Formula $\ec f$
\end_inset

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

.
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\begin_layout Pause

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\begin_layout Standard
So the derivative is the 
\emph on
limit
\emph default
 of the slopes of the secant lines between 
\begin_inset Formula $\ec x=a$
\end_inset

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

 as 
\begin_inset Formula $\ec x=a$
\end_inset

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

.
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\end_inset

 
\end_layout

\end_inset


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Definition of the derivative
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\begin_layout Standard
We write this formally as
\end_layout

\begin_layout Definition
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[
\end_layout

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(The derivative at a point)
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]
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\end_inset


\begin_inset Formula 
\[
\ec f'(b)=\lim_{x\rightarrow b}\frac{f(x)-f(b)}{x-b}
\]

\end_inset

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

 is the symbol used for the derivative of 
\begin_inset Formula $\ec f$
\end_inset

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

.
\end_layout

\begin_layout Standard
(This assumes that the above limit exists.)
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Definition
 If the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 exists and is finite at 
\begin_inset Formula $\ec x=b$
\end_inset

, we say that 
\begin_inset Formula $\ec f$
\end_inset

 is 
\emph on

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\backslash
hilite
\end_layout

\end_inset

 differentiable at
\emph default
 
\begin_inset Formula $\ec x=b$
\end_inset

.
 If the derivative does not exist, then 
\begin_inset Formula $\ec f$
\end_inset

 is not differentiable at 
\begin_inset Formula $\ec x=b$
\end_inset

.
 
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\noindent
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Differentiability
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\begin_layout Standard
\noindent
An important point to remember is that
\end_layout

\begin_layout Theorem
\noindent
 If a function 
\begin_inset Formula $\ec f$
\end_inset

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

, then 
\begin_inset Formula $\ec f$
\end_inset

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

.
 
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\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Why? The definition of continuity requires that 
\begin_inset Formula $\ec f(b)$
\end_inset

 exists, and 
\begin_inset Formula $\ec\lim_{x\rightarrow b}f(x)=f(b)$
\end_inset

.
 
\end_layout

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

 does not exist, then we cannot compute the derivative , whose definition
 requires 
\begin_inset Formula $\ec f(b)$
\end_inset

.
 (
\emph on
Go back and check where!
\emph default
) 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec\lim_{x\rightarrow b}f(x)\neq f(b)$
\end_inset

, then the difference quotient
\begin_inset Formula 
\[
\ec\frac{f(x)-f(b)}{x-b}
\]

\end_inset

 approaches a non-zero constant in the numerator, and zero in the denominator.
 In a previous module, we stated that this was 
\emph on
not a finite quantity
\emph default
.
 Differentiability requires a 
\emph on
finite
\emph default
 limit for the difference quotient.
 (
\emph on
Go back and check where!
\emph default
) 
\end_layout

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Numerical approach
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\begin_deeper
\begin_layout Standard
We turn our attention to the question of computing derivatives.
 Because it can be hard to estimate slopes from looking at a graph, it is
 often easy to estimate the derivative using the 
\emph on
numerical approach
\emph default
 to computing a limit.
\end_layout

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<2>
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\begin_deeper
\begin_layout Standard
Build a table of 
\begin_inset Formula $a$
\end_inset

-
\begin_inset Formula $m$
\end_inset

 values, where 
\begin_inset Formula $\ec m$
\end_inset

 is the slope of the secant line of 
\begin_inset Formula $\ec s(x)=\sin(x)$
\end_inset

 between 
\begin_inset Formula $\ec x=a$
\end_inset

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

, where 
\begin_inset Formula $\ec a=0,0.9,0.99,0.9999,1.0001,1.01,1.1,2$
\end_inset

.
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0.9 
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0.99 
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0.9999 
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 1
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1.01 
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1.1 
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2 
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? 
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? 
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??? 
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? 
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? 
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? 
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<cell alignment="center" valignment="top" topline="true" usebox="none">
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? 
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status open

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<3>
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\begin_deeper
\begin_layout Standard
Build a table of 
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-
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 values, where 
\begin_inset Formula $m$
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 is the slope of the secant line of 
\begin_inset Formula $s(x)=\sin(x)$
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 between 
\begin_inset Formula $x=a$
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 and 
\begin_inset Formula $x=1$
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, where 
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.
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\begin_layout Standard
\begin_inset ERT
status collapsed

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{
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\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
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\align center
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
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0 
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0.9 
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
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0.99 
\end_layout

\end_inset
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.9999 
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{
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\backslash
hilite
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 1
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status collapsed

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}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0001 
\end_layout

\end_inset
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\begin_inset Text

\begin_layout Plain Layout
1.01 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.1 
\end_layout

\end_inset
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\begin_layout Plain Layout
2 
\end_layout

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<row>
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
??? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
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\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Remember that you can make SAGE do this computation using a 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 for
\family default
 loop if you like.
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\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Build a table of 
\begin_inset Formula $a$
\end_inset

-
\begin_inset Formula $m$
\end_inset

 values, where 
\begin_inset Formula $m$
\end_inset

 is the slope of the secant line of 
\begin_inset Formula $s(x)=\sin(x)$
\end_inset

 between 
\begin_inset Formula $x=a$
\end_inset

 and 
\begin_inset Formula $x=1$
\end_inset

, where 
\begin_inset Formula $a=0,0.9,0.99,0.9999$
\end_inset

.
\end_layout

\begin_layout Standard
\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

 
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="10">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

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0 
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.9 
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\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.99 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.9999 
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</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\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

 1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.01 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.1 
\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2 
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<row>
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $m$
\end_inset

 
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
0.84 
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\begin_inset Text

\begin_layout Plain Layout
0.58 
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0.54 
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0.54 
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
??? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.54 
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.54 
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.5 
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.07 
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</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Notice that the slopes do indeed approach 0.54, which is what we had predicted
 before.
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example Problem
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now that we've touched on the concepts, let's look at the example problem
 again.
 It was:
\end_layout

\begin_layout Standard
\noindent
An electronic instrument makes a sound by vibrating a speaker's voice coil
 according to the function 
\begin_inset Formula $\ec s(x)=v\sin(x)$
\end_inset

, where
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec v$
\end_inset

 is a constant; 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x$
\end_inset

 is the time in seconds; and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec s(x)$
\end_inset

 is the distance in mm that the coil moves.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

What is the rate of change of the position of the coil at time 
\begin_inset Formula $\ec x=1$
\end_inset


\begin_inset space ~
\end_inset

second when 
\begin_inset Formula $\ec v=1$
\end_inset

 and when 
\begin_inset Formula $\ec v=4$
\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
If you look back, you will see that we have in fact solved the problem for
 
\begin_inset Formula $\ec v=1$
\end_inset

.
 We found that the derivative was 
\begin_inset Formula $\ec s'(1)\approx0.54$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example Problem: Units
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

What are the 
\emph on
units
\emph default
 of the derivative? mm? sec? something else?
\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 units are mm/sec.
 
\emph on
Why?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Remember that the derivative is the limit of the slopes of the secant lines.
 These slopes represent the ratio 
\emph on
change in 
\begin_inset Formula $\ec s$
\end_inset

 to change in 
\begin_inset Formula $\ec x$
\end_inset


\emph default
.
 The problem told us that 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec s$
\end_inset

 represents distance in mm, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x$
\end_inset

 represents time in secondes.
 
\end_layout

\begin_layout Standard
Thus the units should be mm/sec.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example Problem: 
\begin_inset Formula $v=4$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now that we've touched on the concepts, let's look at the example problem
 again.
 It was:
\end_layout

\begin_layout Standard
An electronic instrument makes a sound by vibrating a speaker's voice coil
 according to the function 
\begin_inset Formula $s(x)=v\sin(x)$
\end_inset

, where
\end_layout

\begin_layout Itemize
\begin_inset Formula $v$
\end_inset

 is a constant; 
\end_layout

\begin_layout Itemize
\begin_inset Formula $x$
\end_inset

 is the time in seconds; and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $s(x)$
\end_inset

 is the distance in mm that the coil moves.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

What is the rate of change of the position of the coil at time 
\begin_inset Formula $x=1$
\end_inset


\begin_inset space ~
\end_inset

second when 
\begin_inset Formula $v=1$
\end_inset

 and when 
\begin_inset Formula $v=4$
\end_inset

?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Since we solved the problem for 
\begin_inset Formula $\ec v=1$
\end_inset

, take a few moments on your own to solve the problem for 
\begin_inset Formula $\ec v=4$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should find that 
\begin_inset Formula $\ec s'(4)\approx2.16$
\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 Itemize
The 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 derivative
\emph default
 of 
\begin_inset Formula $\ec f(x)$
\end_inset

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

 is the 
\emph on
slope of the line tangent to 
\begin_inset Formula $\ec f(x)$
\end_inset


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

.
 This is the limit of the slopes of the secant lines.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec f$
\end_inset

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

 is formally defined to be
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{4in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(The derivative at a point)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset Formula 
\[
\ec f'(b)=\lim_{x\rightarrow b}\frac{f(x)-f(b)}{x-b}
\]

\end_inset

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

 is the symbol used for the derivative of 
\begin_inset Formula $\ec f$
\end_inset

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

.
\end_layout

\end_deeper
\begin_layout Itemize
If the derivative of a function exists at a point, we say that the function
 is 
\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

 differentiable
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 at that point.
 
\end_layout

\begin_layout Itemize
We can estimate this value graphically or numerically.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
