#LyX file created by tex2lyx 2.0.0
\lyxformat 345
\begin_document
\begin_header
\textclass beamer
\begin_preamble

\usepackage{appearance}\newcommand{\reals}{\mathbb R}

\end_preamble
\use_default_options false
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\paperfontsize default
\spacing single
\use_hyperref 0
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\end_header

\begin_body

\begin_layout Standard


\begin_inset ERT
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\begin_layout Standard

%MODULE: Continuity
\end_layout

\begin_layout Standard


\end_layout

\end_inset


\end_layout

\begin_layout Pause

First Example
\end_layout

\begin_layout Standard
\noindent


\begin_inset ERT
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\begin_layout Standard

{
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Standard


\backslash
dims
\end_layout

\end_inset

 We demonstrate this using the first example.
\begin_inset ERT
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\begin_layout Standard

}
\end_layout

\end_inset


\begin_inset Formula \begin{align*}
	\dims y'&\dims=-\frac{x}{y}.
\end{align*}
\end_inset

 We illustrate this by computing the slopes of lines tangent to the circle at various points.
\end_layout

\begin_layout Standard

The point 
\begin_inset Formula $\hilite (1,0)$
\end_inset

 is on the curve, since 
\begin_inset Formula $\ec 1^2+0^2=1$
\end_inset

. At this point, 
\begin_inset Formula \[{\ec { y'}=-\frac{1}{0}},\mathrm{\ corresponding\ to\ a\ vertical\ line}.\]
\end_inset

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


\begin_layout Standard

 
\end_layout

\begin_layout Standard
\align center


\begin_inset Graphics 
	filename figures/tanline_at_right.png
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard


\begin_inset ERT
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\begin_layout Standard


\backslash
lyxframe
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Standard

{
\end_layout

\end_inset

Implicit Plots
\begin_inset ERT
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\begin_layout Standard

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent

It isn't always easy to view the graph of a relation. If you can't separate 
\begin_inset Formula $\ec y$
\end_inset

 and 
\begin_inset Formula $\ec x$
\end_inset

, you can still plot it in SAGE using the 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 implicit_plot()
\family default
 command. For example, the graphs of the circles were generated using these commands:
\end_layout

\begin_layout Standard


\begin_inset Box Boxed
position "c"
hor_pos "c"
has_inner_box 1
inner_pos "c"
use_parbox 1
width "8.75in"
special "none"
height "1in"
height_special "totalheight"
status open


\begin_layout Standard


\size larger

\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 circleplot = implicit_plot(x‸2+y‸2-1,(x,-2,2),(y,-2,2),cmap='jet',plot_points=500)
\family default

\begin_inset Newline newline
\end_inset

 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 circleplot.show(aspect_ratio=1)
\family default

\begin_inset Newline newline
\end_inset

 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 upperplot = implicit_plot(x‸2+y‸2-1,(x,-2,2),(y,0,2),cmap='jet',plot_points=500)
\family default

\begin_inset Newline newline
\end_inset

 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 lowerplot = implicit_plot(x‸2+y‸2-1,(x,-2,2),(y,-2,0),cmap='hsv',plot_points=500)
\family default

\begin_inset Newline newline
\end_inset

 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 (upperplot+lowerplot).show(aspect_ratio=1)
\family default

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent


\emph on
(Try it!)
\emph default

\end_layout

\begin_layout Standard

The command takes as its first argument an expression 
\family typewriter

\begin_inset ERT
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\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 exp
\family default
 which corresponds to the equation 
\begin_inset Formula $\ec\mathtt{\ttcolor exp}=0$
\end_inset

. We want to plot 
\begin_inset Formula $\ec x^2+y^2=1$
\end_inset

, which corresponds to 
\begin_inset Formula $\ec x^2+y^2-1=0$
\end_inset

, so 
\begin_inset Formula $\ec\mathtt{\ttcolor exp}=x^2+y^2-1$
\end_inset

. The second and third arguments are a range of 
\begin_inset Formula $x$
\end_inset

- and 
\begin_inset Formula $y$
\end_inset

-values over which the expression should be graphed. The third argument corresponds to a color (the usual colors don't apply in the currect version of SAGE; we advise you to stick with 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 'jet'
\family default
 (blue), 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 'hsv'
\family default
 (red), and 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
ttcolor
\end_layout

\end_inset

 summer
\family default
 (green)). The final argument should be included in order to obtain an accurate graph from SAGE; if the number is too small, the graph will not be accurate.
\end_layout

\begin_layout Standard


\begin_inset ERT
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\begin_layout Standard


\backslash
lyxframe
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Standard

{
\end_layout

\end_inset

Second Example
\begin_inset ERT
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\begin_layout Standard

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent

We can also differentiate the second example, where it is not possible to isolate 
\begin_inset Formula $\ec y$
\end_inset

. 
\begin_inset Formula \begin{align*}
	\ec x^2e^{xy}&{\ec=x+y}.
\end{align*}
\end_inset


\end_layout

\begin_layout Pause

Second Example
\end_layout

\begin_layout Standard
\noindent

We can also differentiate the second example, where it is not possible to isolate 
\begin_inset Formula $\ec y$
\end_inset

. 
\begin_inset Formula \begin{align*}
	\dims x^2e^{xy}&{\dims=x+y}\\
	\ec 2xe^{xy}+x^2e^{xy}\cdot\left(y+x{\hilite y'}\right)&{\ec=1+{\hilite y'}}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Use the Product Rule and the Chain Rule. We only write 
\begin_inset Formula $ y'$
\end_inset

 when we take the derivative of 
\begin_inset Formula $ y$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Second Example
\end_layout

\begin_layout Standard
\noindent

We can also differentiate the second example, where it is not possible to isolate 
\begin_inset Formula $\ec y$
\end_inset

. 
\begin_inset Formula \begin{align*}
	\dims x^2e^{xy}&{\dims=x+y}\\
	\dims 2xe^{xy}+x^2e^{xy}\cdot\left(y+xy'\right)&\dims=1+y'\\
	\ec 2xe^{xy}+x^2ye^{xy}+x^3e^{xy}\cdot {\hilite y'}&{\ec=1+{\hilite y'}}.
\end{align*}
\end_inset

 
\begin_inset ERT
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\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Distribution.
\begin_inset ERT
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\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Second Example
\end_layout

\begin_layout Standard
\noindent

We can also differentiate the second example, where it is not possible to isolate 
\begin_inset Formula $\ec y$
\end_inset

. 
\begin_inset Formula \begin{align*}
	\dims x^2e^{xy}&{\dims=x+y}\\
	\dims 2xe^{xy}+x^2e^{xy}\cdot\left(y+xy'\right)&\dims=1+y'\\
	 2xe^{xy}+x^2ye^{xy}+x^3e^{xy}\cdot { y'}&=1+{ y'}\\
	\ec \left(x^3e^{xy}-1\right){\hilite y'}&{\ec=1-e^{xy}\left(x^2+2x\right)}.
\end{align*}
\end_inset

 
\begin_inset ERT
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\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Everything with 
\begin_inset Formula $ y'$
\end_inset

 on one side;
\begin_inset Newline newline
\end_inset

everything without it to the other.
\begin_inset ERT
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\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Second Example
\end_layout

\begin_layout Standard
\noindent

We can also differentiate the second example, where it is not possible to isolate 
\begin_inset Formula $\ec y$
\end_inset

. 
\begin_inset Formula \begin{align*}
	\dims x^2e^{xy}&{\dims=x+y}\\
	\dims 2xe^{xy}+x^2e^{xy}\cdot\left(y+xy'\right)&\dims=1+y'\\
	 2xe^{xy}+x^2ye^{xy}+x^3e^{xy}\cdot { y'}&=1+{ y'}\\
	 \left(x^3e^{xy}-1\right){ y'}&{=1-e^{xy}\left(x^2+2x\right)}\\
	\ec { y'}&{\ec=\frac{1-e^{xy}\left(x^2+2x\right)}{x^3e^{xy}-1}}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Solve for 
\begin_inset Formula $ y'$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Standard


\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
lyxframe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Standard

{
\end_layout

\end_inset

Third Example
\begin_inset ERT
status collapsed

\begin_layout Standard

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\ec y&{\ec=\arcsin x}.
\end{align*}
\end_inset


\end_layout

\begin_layout Pause

Third Example
\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\dims y&{\dims=\arcsin x}\\
	\ec\sin y&{\ec=x}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Properties of inverse functions.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Third Example
\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\dims\sin y&{\dims=x}\\
	\ec\cos y \cdot y'&{\ec=1}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Differentiated both sides with respect to 
\begin_inset Formula $x$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Third Example
\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\dims\cos y\cdot y'&{\dims=1}\\
	\ec y'&{\ec=\frac{1}{\cos y}}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Solve for 
\begin_inset Formula $ y'$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Third Example
\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\dims y'&{\dims=\frac{1}{\cos y}}\\
	\ec y'&{\ec=\frac{1}{\cos\left(\arcsin x\right)}}.
\end{align*}
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

Substitute 
\begin_inset Formula $y=\arcsin x$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

Third Example
\end_layout

\begin_layout Standard
\noindent

The third example asks us to find the derivatives of the inverse functions of 
\begin_inset Formula $\ec\sin x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, and 
\begin_inset Formula $\ec\sec x$
\end_inset

. The derivatives of these functions have an interesting form. Implicit differentation allows us to find these derivatives. Start with 
\begin_inset Formula $\ec\arcsin x$
\end_inset

:. 
\begin_inset Formula \begin{align*}
	\dims y'&{\dims=\frac{1}{\cos\left(\arcsin x\right)}}\\
	\ec y'&{\ec=\frac{1}{\sqrt{1-x^2}}}.
\end{align*}
\end_inset

 
\begin_inset ERT
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\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

This is a property learned in precalculus; we explain it briefly. We want to know 
\begin_inset Formula $\cos\alpha$
\end_inset

 where 
\begin_inset Formula $\alpha=\arcsin x$
\end_inset

. Thus 
\begin_inset Formula $x=\sin\alpha$
\end_inset

, so we can draw a right triangle with hypotenuse of length 
\begin_inset Formula $1$
\end_inset

 that has an angle of measure 
\begin_inset Formula $\alpha$
\end_inset

 and a leg opposite 
\begin_inset Formula $\alpha$
\end_inset

 of length 
\begin_inset Formula $x$
\end_inset

. The length of the side adjacent to 
\begin_inset Formula $\alpha$
\end_inset

 will then be 
\begin_inset Formula $\sqrt{1-x^2}$
\end_inset

. 
\begin_inset ERT
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\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Standard


\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
lyxframe
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Standard

{
\end_layout

\end_inset

Third Example
\begin_inset ERT
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\begin_layout Standard

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent

We have the following. 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{theorem}
\end_layout

\end_inset

[(Derivatives of inverse trig functions)] 
\end_layout

\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arcsin x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{\sqrt{1-x^2}}$
\end_inset

. 
\end_layout

\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arctan x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{{1+x^2}}$
\end_inset

. 
\end_layout

\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arcsec x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{x\sqrt{x^2-1}}$
\end_inset

. 
\end_layout

\begin_layout Standard


\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{theorem}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{explanation}
\end_layout

\end_inset

You should check on your own that the other two are correct.
\begin_inset ERT
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\begin_layout Standard


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Standard


\begin_inset ERT
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\begin_layout Standard


\backslash
lyxframe
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Standard

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
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\begin_layout Standard

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize


\emph on

\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
hilite
\end_layout

\end_inset

 Implicit differentiation
\emph default
 allows us to compute the derivative in cases where 
\begin_inset Formula $\ec y$
\end_inset

 may not be a function of 
\begin_inset Formula $\ec x$
\end_inset

. 
\end_layout

\begin_layout Itemize

Implicit differentation uses the chain rule: whenever you have to take the derivative of 
\begin_inset Formula $\ec y$
\end_inset

, you multiply by 
\begin_inset Formula $\hilite y'$
\end_inset

. 
\end_layout

\begin_layout Itemize

Implicit differentiation allowed us to determine the derivatives of three new functions: 
\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
begin{theorem}
\end_layout

\end_inset

[(Derivatives of inverse trig functions)] 
\end_layout

\begin_deeper
\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arcsin x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{\sqrt{1-x^2}}$
\end_inset

. 
\end_layout

\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arctan x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{{1+x^2}}$
\end_inset

. 
\end_layout

\begin_layout Itemize

If 
\begin_inset Formula $\ec f(x)=\arcsec x$
\end_inset

, then 
\begin_inset Formula $\ec f'(x) = \frac{1}{x\sqrt{x^2-1}}$
\end_inset

. 
\end_layout

\end_deeper
\begin_deeper
\begin_layout Standard


\begin_inset ERT
status collapsed

\begin_layout Standard


\backslash
end{theorem}
\end_layout

\end_inset

 You should remember these derivatives. 
\end_layout

\end_deeper
\begin_layout EndFrame


\end_layout

\end_body
\end_document
