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%MODULE 1: Squeeze Theorem
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%BUILD  TITLE PAGE 
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\begin_layout Title
Derivatives by experimentation
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\begin_layout Subtitle
Geometrically, the instantaneous rate of change is the slope of a line tangent
 to 
\begin_inset Formula $\ec f$
\end_inset

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

.
 In this lab, you will explore how we can use the tangent lines to deduce
 a formula for the derivative from its values at various points.
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{
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A Sample problem
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}
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\backslash
begin{question}
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If 
\begin_inset Formula $\ec f\left(x\right)=\sin x$
\end_inset

, what is 
\begin_inset Formula $\ec f'(x)$
\end_inset

 
\emph on
as a function?
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\begin_layout Standard
\noindent
That is, what formula gives us 
\begin_inset Formula $\ec f(x)$
\end_inset

 for any value of 
\begin_inset Formula $\ec x$
\end_inset

?
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\backslash
end{question}
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\noindent
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{
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SAGE worksheets
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}
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\noindent
In this lab you will need the following SAGE worksheets:
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\begin_layout Itemize
Function Factory 
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\begin_layout Itemize
Tangent Line Guesser 
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\noindent
Open each one in separate windows or tabs of your browser.
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{
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Setup Function Factory
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}
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First we will set up the Function Factory SAGElet.
 When you first start it, you should see something like this:
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\noindent
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\end_layout

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\align center
\begin_inset Graphics
	filename figures/function_factory_small.png
	scale 40

\end_inset

 
\end_layout

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Setup Function Factory
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\noindent
Change the values of the input boxes to match the following:
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\begin_layout Itemize
delete everything in 
\begin_inset Formula $\ec\mathit{points}$
\end_inset

, leaving it empty; 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x_{\min}\rightarrow-\mathit{pi}/2$
\end_inset

; 
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\begin_layout Itemize
\begin_inset Formula $\ec x_{\max}\rightarrow5*pi/2$
\end_inset

.
 
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{
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Setup Tangent Line Guesser
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}
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\begin_layout Standard
Now turn to the Tangent Line Guesser SAGElet.
 You should see something like this:
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\noindent
\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/tanline_guesser_small.png
	scale 20

\end_inset

 
\end_layout

\end_inset


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\noindent
Things are already set up for this exercise.
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{
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Estimating the derivative
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\begin_layout Standard
We can estimate the values of the derivative of 
\begin_inset Formula $\ec f$
\end_inset

 using the Tangent Line Guesser.
 For any 
\begin_inset Formula $x$
\end_inset

 value 
\begin_inset Formula $\ec a$
\end_inset

, you can change the value of 
\begin_inset Formula $\ec m$
\end_inset

, the slope of the red line.
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\begin_layout Pause

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\begin_layout Standard
Experiment with different values of 
\begin_inset Formula $\ec m$
\end_inset

 until the red line seems to be tangent to the curve.
 For 
\begin_inset Formula $\ec a=0$
\end_inset

, try 
\begin_inset Formula $\ec m=2$
\end_inset

, 
\begin_inset Formula $\ec m=1$
\end_inset

, 
\begin_inset Formula $\ec m=0.5$
\end_inset

, 
\begin_inset Formula $\ec m=0$
\end_inset

, and 
\begin_inset Formula $\ec m=-1$
\end_inset

.
 Which value of 
\begin_inset Formula $\ec m$
\end_inset

 gives the correct line?
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\begin_layout Pause

\end_layout

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\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/sin_a0_m1.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The value 
\begin_inset Formula $\ec m=1$
\end_inset

 appears to create a tangent line.
 Now turn to the Function Factory.
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{
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Points in derivative
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}
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\begin_layout Standard
In the Function Factory SAGElet, we enter points that belong to the derivative,
 and see if they match the function specified by 
\begin_inset Formula $\ec f$
\end_inset

.
 We decided in the Tangent Line Guesser SAGElet that 
\begin_inset Formula $\ec f'\left(0\right)\approx1$
\end_inset

.
 Type 
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{
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\backslash
ec
\end_layout

\end_inset

(0,1)
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 in the input box next to 
\begin_inset Formula $\ec\mathit{points}$
\end_inset

.
 Press the tab or enter key to see an update.
 You should see something like this:
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\noindent
\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/sin_derivative_ff_first_point.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Right away you can see that 
\begin_inset Formula $\ec x^{2}$
\end_inset

 is 
\emph on
not
\emph default
 the correct derivative for 
\begin_inset Formula $\ec\sin x$
\end_inset

, because the point 
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{
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\backslash
ec
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\end_inset

(0,1)
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}
\end_layout

\end_inset

 does not lie on the curve.
 However, we won't change 
\begin_inset Formula $\ec f$
\end_inset

 yet.
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More points!
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}
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{
\backslash
textwidth}{6in}
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\end_layout

\begin_deeper
\begin_layout Standard
Return to the Tangent Line Guesser SAGElet.
 This time, try different values for 
\begin_inset Formula $\ec a$
\end_inset

.
 At each value of 
\begin_inset Formula $\ec a$
\end_inset

, try to find a value of 
\begin_inset Formula $\ec m$
\end_inset

 that gives the correct tangent line.
 For example:
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<2>
\end_layout

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\begin_deeper
\begin_layout Standard
When we try 
\begin_inset Formula $\ec a=\pi/2$
\end_inset

, we find 
\begin_inset Formula $\ec m\approx0$
\end_inset

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/sin_api2_m0.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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<3>
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\begin_deeper
\begin_layout Standard
When we try 
\begin_inset Formula $\ec a=3\pi/4$
\end_inset

, we find 
\begin_inset Formula $\ec m\approx-0.75$
\end_inset

.
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\begin_inset Float figure
placement h!
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\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/sin_a3pi4_m0_75.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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<4>
\end_layout

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\end_layout

\begin_deeper
\begin_layout Standard
When we try 
\begin_inset Formula $\ec a=-\pi/6$
\end_inset

, we find 
\begin_inset Formula $\ec m\approx0.85$
\end_inset

.
\end_layout

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\begin_inset Float figure
placement h!
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sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/sin_a-pi6_m0_85.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
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{
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Back to Function Factory
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}
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\begin_layout Standard
Add the points
\begin_inset Formula 
\[
\ec\left(-\frac{\pi}{6},0.85\right),\left(\pi/2,0\right),\left(\frac{3\pi}{4},-0.75\right)
\]

\end_inset

 into Function Factory and you get the following image.
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\noindent
\begin_inset Float figure
placement h!
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\end_layout

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

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
At this point we can see that 
\begin_inset Formula $\ec x^{2}$
\end_inset

 is 
\emph on
definitely
\emph default
 wrong.
 But we still don't know what the correct function is!
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{
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Take some more points
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}
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\begin_layout Standard
The only thing we can do now is try some more values of 
\begin_inset Formula $\ec a$
\end_inset

.
 Choose values that lie between and outside the values we have used so far.
 You might want to try 
\begin_inset Formula $\ec a=\pi/4$
\end_inset

, for example, or 
\begin_inset Formula $\ec a=5\pi/4$
\end_inset

, or 
\begin_inset Formula $\ec a=3\pi/2$
\end_inset

, or more! For each value of 
\begin_inset Formula $\ec a$
\end_inset

, find the corresponding value of 
\begin_inset Formula $\ec m$
\end_inset

.
 When you think you have enough points, add them to Function Factory.
\end_layout

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\noindent
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{
\end_layout

\end_inset

Things fall into place
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\begin_layout Standard
Here is an image reflecting those choices, as well as some others:
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\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/sin_derivative_ff_many_points.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
You should be able to guess the correct value of 
\begin_inset Formula $\ec f$
\end_inset

 at this point...
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
{\ec f=\cos x.}
\]

\end_inset

 Change the value of 
\begin_inset Formula $\ec f$
\end_inset

 in the Function Factory to 
\begin_inset Formula $\ec\cos x$
\end_inset

.
\end_layout

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{
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Things fall into place
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}
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\begin_layout Standard
It looks as if 
\begin_inset Formula $\ec\cos x$
\end_inset

 matches the points perfectly.
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\noindent
\begin_inset Float figure
placement h!
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sideways false
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\begin_layout Plain Layout
 
\end_layout

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

\end_inset

 
\end_layout

\end_inset


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{
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Try more!
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}
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\end_layout

\begin_layout Standard
Repeat this exercise with several different functions.
 (Your professor may assign a particular function to you.) Enter each function
 into the Tangent Line Guesser SAGElet, then determine approximate values
 of the derivative.
 Enter the appropriate points into the Function Factory SAGElet, repeating
 until you have found enough points to guess a formula for the derivative.
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Tabular
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<features tabularvalignment="middle">
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<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ec e^{x}$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\ln\left(x\right)$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec x^{2}$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\sqrt{x}$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\cos\left(x\right)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\sin\left(2x\right)$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\tan\left(x\right)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec\sin\left(x+\pi/2\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


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{
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Conclusion
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}
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\begin_layout Itemize
The derivative of a function 
\begin_inset Formula $y$
\end_inset

 can correspond to another function; let's call it 
\begin_inset Formula $y'$
\end_inset

.
\end_layout

\begin_layout Itemize
We can often guess at this 
\begin_inset Formula $y'$
\end_inset

 by considering our geometric intuition of the derivative.
\end_layout

\begin_layout Itemize
In class you will study and learn algebraic methods that should explain
 why you obtain the result you obtained here.
 Feel free to experiment with other functions.
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
