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\begin_body

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%MODULE 1: Curve sketching
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%USER DEFINED COMMANDS
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%INPUT THE INSTRUCTIONS PAGE 
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%BUILD  TITLE PAGE 
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\begin_layout Title
Curve Sketching
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\begin_layout Subtitle
We use the principles of Calculus to help sketch the graphs of functions.
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{
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Introduction
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\begin_layout Standard
The graph of a function can help us visualize how a function behaves.
 Graphing calculators and computer algebra systems allow us to see many
 aspects of a graph easily, but before using a graphing calculator or a
 computer algebra system you need to know 
\emph on
where
\emph default
 to look for interesting features.
 For this we need a tool called 
\emph on

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\backslash
hilite
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\end_inset

 curve sketching
\emph default
.
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\noindent
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{
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Sample problems
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}
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\noindent
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\backslash
begin{question}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Make a sketch of the curve 
\begin_inset Formula $\ec x^{3}+2x-1$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Sketch the curve of 
\begin_inset Formula $\ec-10x^{3}+\frac{x}{10}+\frac{1}{10}$
\end_inset

, labeling every point where 
\end_layout

\begin_deeper
\begin_layout Itemize
the curve starts and stops increasing to decreasing; and 
\end_layout

\begin_layout Itemize
the curve's slope starts and stops increasing and decreasing.
 
\end_layout

\end_deeper
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\backslash
end{question}
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{
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SAGE worksheets
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}
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\begin_layout Standard
You will not need a SAGE worksheet for this module.
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\begin_layout BeginFrame
\begin_inset ERT
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{
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\end_inset

Four components of curves
\begin_inset ERT
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}
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\end_inset


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\begin_layout Standard
\noindent
We are interested in four general shapes that a curve can take:
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name "pg:four shapes"

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	filename figures/inc_concavedown.png
	lyxscale 20
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\end_layout

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	filename figures/dec_concavedown.png
	lyxscale 20
	scale 20

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	filename figures/dec_concaveup.png
	lyxscale 20
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	lyxscale 20
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\begin_layout Standard
\noindent
We can assemble these four shapes, along with lines and asymptotes, to provide
 a general idea of the shape of any curve.
\end_layout

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{
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Four components of curves
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textwidth}{6in}
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\begin_deeper
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What is interesting about those four shapes? We need to discuss two basic
 ideas.
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[
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(Increasing and decreasing)
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]
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\begin_deeper
\begin_layout Itemize
We say that a function is 
\emph on

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\backslash
hilite
\end_layout

\end_inset

 increasing
\emph default
 on the interval 
\begin_inset Formula $\ec I=(a,b)$
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 when for any 
\begin_inset Formula $\ec x_{1},x_{2}\in I$
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, if 
\begin_inset Formula $\ec x_{2}>x_{1}$
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, then 
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\end_inset

.
 
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\begin_layout Itemize
We say that a function is 
\emph on

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status collapsed

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\backslash
hilite
\end_layout

\end_inset

 decreasing
\emph default
 on the interval 
\begin_inset Formula $\ec I=(a,b)$
\end_inset

 when for any 
\begin_inset Formula $\ec x_{1},x_{2}\in I$
\end_inset

, if 
\begin_inset Formula $\ec x_{2}>x_{1}$
\end_inset

, then 
\begin_inset Formula $\ec f(x_{2})<f(x_{1})$
\end_inset

.
 
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<3-4>
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\end_layout

\begin_deeper
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Which of the four basic shapes correspond to 
\emph on
increasing
\emph default
 functions?
\end_layout

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\begin_layout Pause

\end_layout

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<4>
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\begin_inset Graphics
	filename figures/inc_concavedown.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

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</cell>
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	filename figures/inc_concaveup.png
	lyxscale 20
	scale 30

\end_inset

 
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<5-6>
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\begin_deeper
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Which of the four basic shapes correspond to 
\emph on
decreasing
\emph default
 functions?
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

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<6>
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\begin_inset Graphics
	filename figures/dec_concaveup.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

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	filename figures/dec_concavedown.png
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{
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Derivatives and the direction of a function
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}
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\begin_layout Standard
Finding extrema allows us to determine when a function is increasing or
 decreasing.
 To review, 
\begin_inset ERT
status collapsed

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\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Find all places where 
\begin_inset Formula $\ec f'(x)=0$
\end_inset

 or 
\begin_inset Formula $\ec f'(x)$
\end_inset

 does not exist.
 These 
\begin_inset Formula $x$
\end_inset

 values are called 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset


\begin_inset ERT
status collapsed

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\backslash
hilite
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\end_inset

 critical points
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Itemize
Find the 
\begin_inset Formula $\ec y$
\end_inset

 values at each critical points 
\emph on
and
\emph default
 at the endpoints.
 
\end_layout

\begin_layout Itemize
The smallest 
\begin_inset Formula $\ec y$
\end_inset

 value found is the minimum, and the largest 
\begin_inset Formula $\ec y$
\end_inset

 value found is the maximum.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

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\backslash
end{method}
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\end_inset

 Once we locate the extrema, continuity tells us that we can connect them
 with one of the curves give on 
\begin_inset ERT
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\backslash
RefPage
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\end_inset


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{
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pg:four shapes
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}
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\end_inset

.
 
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\begin_layout Itemize
An 
\emph on
increasing
\emph default
 curve has a minimum on the left and a maximum on the right; and 
\end_layout

\begin_layout Itemize
a 
\emph on
decreasing
\emph default
 curve has a maximum on the left and a minimum on the right.
 
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard

\emph on
Which
\emph default
 shape do we use to connect a minimum to a maximum, or vice versa? To answer
 that, we need a new concept.
\end_layout

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\noindent
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{
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The rate of change of the rate of change
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}
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\end_layout

\begin_layout Standard
\noindent
Reexamine the four basic shapes that interest us.
 In each case, the derivative--the slope of the tangent line--is either
 increasing or decreasing.
 
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sideways false
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	filename figures/concdn_decrderiv2.png
	lyxscale 10
	scale 10

\end_inset

 
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</cell>
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	filename figures/concdn_decrderiv.png
	lyxscale 10
	scale 10

\end_inset


\end_layout

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

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\begin_inset Graphics
	filename figures/concup_incrderiv2.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
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	filename figures/concup_incrderiv.png
	lyxscale 10
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</cell>
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\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
What is the 
\emph on
derivative
\emph default
 (
\series bold
not
\series default
 the 
\emph on
function
\emph default
) doing in each plot?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\align center
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\begin_inset Text

\begin_layout Plain Layout
decreasing 
\end_layout

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

\begin_layout Plain Layout
decreasing 
\end_layout

\end_inset
</cell>
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<row>
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\begin_inset Text

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increasing 
\end_layout

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increasing 
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\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
We can already narrow the shape of a curve to increasing or decreasing curves.
 If we can decide whether the 
\emph on
derivative
\emph default
 is increasing or decreasing, then we can decide which of the two remaining
 shapes to choose.
\end_layout

\begin_layout BeginFrame
\noindent
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status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Concavity
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
How do we decide whether the derivative is increasing or decreasing? We
 use the derivative of the derivative, or the 
\emph on
second
\emph default
 derivative.
 If 
\begin_inset Formula $\ec f'$
\end_inset

 is continuous, then the maximum and minimum values of 
\begin_inset Formula $\ec f'$
\end_inset

 tell you whether 
\begin_inset Formula $\ec f'$
\end_inset

 is increasing or decreasing.
 How do you find the minimum and maximum values of 
\begin_inset Formula $\ec f'$
\end_inset

? The same way that you find the minimum and maximum values of 
\begin_inset Formula $\ec f$
\end_inset

.
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Concavity)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f'$
\end_inset

 is increasing on the interval 
\begin_inset Formula $\ec I$
\end_inset

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

 is 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 concave up
\emph default
 on 
\begin_inset Formula $\ec I$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f'$
\end_inset

 is decreasing on the interval 
\begin_inset Formula $\ec I$
\end_inset

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

 is 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 concave up
\emph default
 on 
\begin_inset Formula $\ec I$
\end_inset

.
 
\end_layout

\begin_layout Itemize
if 
\begin_inset Formula $\ec f$
\end_inset

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

, then the point 
\begin_inset Formula $\ec(a,f(a))$
\end_inset

 is called an 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 inflection point
\emph default
.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Concavity and curve sketching
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We can classify each of the four basic shapes according to the curve's (a)
 rate of change and (b) concavity.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
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\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/concdn_decrderiv2.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
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	filename figures/concdn_decrderiv.png
	lyxscale 10
	scale 10

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\end_layout

\end_inset
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<row>
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\begin_layout Plain Layout
increasing, concave down 
\end_layout

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

\begin_layout Plain Layout
decreasing, concave down 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/concup_incrderiv2.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/concup_incrderiv.png
	lyxscale 10
	scale 10

\end_inset


\end_layout

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

\begin_layout Plain Layout
decreasing, concave up 
\end_layout

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

\begin_layout Plain Layout
increasing, concave up 
\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
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status collapsed

\begin_layout Plain Layout

{
\end_layout

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Curve Sketching
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We can now summarize a method for curve sketching by extrema.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Determine asymptotes.
 
\end_layout

\begin_layout Itemize
Determine intervals where the curve is 
\emph on
increasing
\emph default
 or 
\emph on
decreasing
\emph default
: 
\end_layout

\begin_deeper
\begin_layout Itemize
Find all critical points: that is, places where 
\begin_inset Formula $\ec f'(x)=0$
\end_inset

 or 
\begin_inset Formula $\ec f'(x)$
\end_inset

 does not exist.
 These 
\begin_inset Formula $x$
\end_inset

 values are called 
\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

 critical points
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Itemize
Classify the critical points as local minima or maxima.
 This indicates intervals where the curve is increasing or decreasing.
 
\end_layout

\end_deeper
\begin_layout Itemize
Determine intervals where the curve is concave 
\emph on
up
\emph default
 or 
\emph on
down
\emph default
: 
\end_layout

\begin_deeper
\begin_layout Itemize
Find all places where 
\begin_inset Formula $\ec f''(x)=0$
\end_inset

 or 
\begin_inset Formula $\ec f''(x)$
\end_inset

 does not exist.
 These 
\begin_inset Formula $x$
\end_inset

 values are called 
\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

 possible inflection points
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Itemize
Find the values of 
\begin_inset Formula $\ec f'$
\end_inset

 at each possible inflection point 
\emph on
and
\emph default
 at the endpoints.
 
\end_layout

\begin_layout Itemize
Classify the possible inflection points as minima or maxima of 
\begin_inset Formula $\ec f'$
\end_inset

.
 This indicates intervals the curve is concave up or down.
 
\end_layout

\end_deeper
\begin_layout Itemize
Combine the information from the first and second derivatives to determine
 the shape of the curve on each interval.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

First example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We apply this method to the first question posed at the beginning of this
 module: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

Make a sketch of the curve 
\begin_inset Formula $\ec x^{3}+2x-1$
\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
There are no asymptotes (
\emph on
why not?
\emph default
), so first we find the intervals where the curve is increasing or decreasing.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Setting the derivative to zero, we have
\begin_inset Formula 
\[
{\ec3x^{2}+2=0}.
\]

\end_inset

 The derivative has 
\emph on
no real roots
\emph default
, so it is either 
\emph on
always increasing
\emph default
 or 
\emph on
always decreasing
\emph default
.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Which is it? There are several ways to decide, but one easy way is to check
 that 
\begin_inset Formula 
\begin{align*}
\mathrm{at\ }{\ec x=0},\  & {\ec x^{3}+2x-1=-1}\mathrm{\ and}\\
\mathrm{at\ }{\ec x=1},\  & {\ec x^{3}+2x-1=2}.
\end{align*}

\end_inset

 Since the curve must increase to rise from 0 to 1, the curve is 
\emph on
always increasing
\emph default
.
\end_layout

\begin_layout BeginFrame
First example
\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
We apply this method to the first question posed at the beginning of this
 module: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

Make a sketch of the curve 
\begin_inset Formula $x^{3}+2x-1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 There are no asymptotes (
\emph on
why not?
\emph default
).
 The curve 
\emph on
always decreases
\emph default
.
\end_layout

\begin_layout Standard
\noindent
Next we find the intervals where the curve is concave up or down.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2-3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Setting the second derivative to zero, we have 
\begin_inset Formula ${\ec6x=0}$
\end_inset

.
 There is a 
\emph on
possible
\emph default
 inflection point at 
\begin_inset Formula $\ec x=0$
\end_inset

.
\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
Since no endpoints have been specified, we check the first derivative around
 
\begin_inset Formula $\ec x=0$
\end_inset

 to see whether it's a maximum or a minimum of the first derivative.
 
\begin_inset Formula 
\begin{align*}
\mathrm{at\ } & {\ec x=-1},\ {\ec3x^{2}+2=5},\\
\mathrm{at\ } & {\ec x=0},\ {\ec3x^{2}+2=2},\mathrm{\ and}\\
\mathrm{at\ } & {\ec x=1},\ {\ec3x^{2}+2=5}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

\end_layout

\end_inset


\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
Thus 
\begin_inset Formula $\ec x=0$
\end_inset

 is a 
\emph on
minimum
\emph default
 of the first derivative.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
This tells us that
\end_layout

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

 is 
\emph on
decreasing
\emph default
 on 
\begin_inset Formula $\ec(-\infty,0)$
\end_inset

 
\begin_inset Formula $\Longrightarrow$
\end_inset

 
\begin_inset Formula $\ec f(x)$
\end_inset

 is concave 
\emph on
down
\emph default
 on 
\begin_inset Formula $\ec(-\infty,0)$
\end_inset

; and 
\end_layout

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

 is 
\emph on
increasing
\emph default
 on 
\begin_inset Formula $\ec(0,\infty)$
\end_inset

 
\begin_inset Formula $\Longrightarrow$
\end_inset

 
\begin_inset Formula $\ec f(x)$
\end_inset

 is concave 
\emph on
up
\emph default
 on 
\begin_inset Formula $\ec(0,\infty)$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout BeginFrame
First example
\end_layout

\begin_layout Standard
\noindent
We apply this method to the first question posed at the beginning of this
 module: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

Make a sketch of the curve 
\begin_inset Formula $x^{3}+2x-1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 There are no asymptotes (
\emph on
why not?
\emph default
).
 The curve is 
\emph on
always decreasing
\emph default
.
 It is 
\end_layout

\begin_layout Itemize
concave 
\emph on
down
\emph default
 on 
\begin_inset Formula $(-\infty,0)$
\end_inset

; and 
\end_layout

\begin_layout Itemize
concave 
\emph on
up
\emph default
 on 
\begin_inset Formula $(0,\infty)$
\end_inset

.
 
\end_layout

\begin_layout Standard
 When 
\begin_inset Formula $x=0$
\end_inset

, 
\begin_inset Formula $y=-1$
\end_inset

.
 That suggests the following shape of the curve: 
\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 Tabular
<lyxtabular version="3" rows="2" columns="5">
<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">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
left: inc, conc dn 
\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
right: incr, conc up 
\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
together 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/ex1left.png
	lyxscale 20
	scale 20

\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 Graphics
	filename figures/ex1right.png
	lyxscale 20
	scale 20

\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 Graphics
	filename figures/ex1together.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second example
\begin_inset ERT
status collapsed

\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
Now turn to the second question: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Sketch the curve of 
\begin_inset Formula $\ec-10x^{3}+\frac{x}{10}+\frac{1}{10}$
\end_inset

, labeling every point where 
\end_layout

\begin_layout Itemize
the curve starts and stops increasing to decreasing; and 
\end_layout

\begin_layout Itemize
the curve's slope starts and stops increasing and decreasing.
 
\end_layout

\begin_layout Standard
\noindent
\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
Again there are no asymptotes 
\emph on
(why not?)
\emph default
 so we look for critical points.
 Take a moment to find the extrema of the function.
\end_layout

\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
Checking 
\begin_inset Formula $y$
\end_inset

 values, we find these extrema: 
\begin_inset Formula 
\begin{align*}
\mathrm{local\ max\ at\ } & {\ec x=\frac{1}{10\sqrt{3}}\approx0.058},\ {\ec y\approx0.104}\mathrm{\ and}\\
\mathrm{local\ min\ at\ } & {\ec x=-\frac{1}{10\sqrt{3}}\approx-0.058},\ {\ec y=\approx0.096}.
\end{align*}

\end_inset

 These are very close to each other---you probably would not notice them
 if you merely glanced at a graph.
\end_layout

\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
\noindent
That implies that the function
\end_layout

\begin_layout Itemize

\emph on
increases
\emph default
 on 
\begin_inset Formula $\ec(-0.058,0.058)$
\end_inset

, while it 
\end_layout

\begin_layout Itemize

\emph on
decreases
\emph default
 on 
\begin_inset Formula $\ec(-\infty,-0.058)$
\end_inset

 and 
\begin_inset Formula $\ec(0.058,\infty)$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
Second example
\end_layout

\begin_layout Standard
\noindent
Now turn to the second question: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Sketch the curve of 
\begin_inset Formula $-10x^{3}+\frac{x}{10}+\frac{1}{10}$
\end_inset

, labeling every point where 
\end_layout

\begin_layout Itemize
the curve starts and stops increasing to decreasing; and 
\end_layout

\begin_layout Itemize
the curve's slope starts and stops increasing and decreasing.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 Again there are no asymptotes.
 The curve increases on 
\begin_inset Formula $(-0.058,0.058)$
\end_inset

, while it decreases on 
\begin_inset Formula $(-\infty,-0.058)$
\end_inset

 and 
\begin_inset Formula $(0.058,\infty)$
\end_inset

.
 Now find the extrema of 
\begin_inset Formula $\ec f'$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

There is only one extremum of 
\begin_inset Formula $\ec f'$
\end_inset

: at 
\begin_inset Formula $\ec x=0$
\end_inset

, there is a local maximum of 
\begin_inset Formula $\ec f'$
\end_inset

.
 So 
\begin_inset Formula $\ec f'$
\end_inset

 is increasing on 
\begin_inset Formula $\ec(-\infty,0)$
\end_inset

 and decreasing on 
\begin_inset Formula $\ec(0,\infty)$
\end_inset

, which means that the 
\begin_inset Formula $\ec f$
\end_inset

 is concave up on 
\begin_inset Formula $\ec(-\infty,0)$
\end_inset

 and concave down on 
\begin_inset Formula $\ec(0,\infty)$
\end_inset

.
\end_layout

\begin_layout BeginFrame
Second example
\end_layout

\begin_layout Standard
\noindent
Now turn to the second question: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Sketch the curve of 
\begin_inset Formula $-10x^{3}+\frac{x}{10}+\frac{1}{10}$
\end_inset

\SpecialChar \ldots{}

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 Again there are no asymptotes.
 The curve increases on 
\begin_inset Formula $(-0.058,0.058)$
\end_inset

, and decreases on 
\begin_inset Formula $(-\infty,-0.058)$
\end_inset

 and 
\begin_inset Formula $(0.058,\infty)$
\end_inset

.
 It is concave up on 
\begin_inset Formula $\left(-\infty,0\right)$
\end_inset

, concave down on 
\begin_inset Formula $\left(0,\infty\right)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

We divide the intervals as follows: 
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<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">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

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

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $(-\infty,-0.058)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $(-0.058,0)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $(0,0.058)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $(0.058,\infty)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
incr/decr 
\end_layout

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

\begin_layout Plain Layout
decr 
\end_layout

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

\begin_layout Plain Layout
incr 
\end_layout

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

\begin_layout Plain Layout
incr 
\end_layout

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

\begin_layout Plain Layout
decr 
\end_layout

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

\begin_layout Plain Layout
concavity 
\end_layout

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

\begin_layout Plain Layout
up 
\end_layout

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

\begin_layout Plain Layout
up 
\end_layout

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

\begin_layout Plain Layout
down 
\end_layout

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

\begin_layout Plain Layout
down 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
shape 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/dec_concaveup.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/inc_concaveup.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/inc_concavedown.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/dec_concavedown.png
	lyxscale 10
	scale 10

\end_inset

 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\end_inset


\end_layout

\begin_layout BeginFrame
Second example
\end_layout

\begin_layout Standard
\noindent
Now turn to the second question: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 Sketch the curve of 
\begin_inset Formula $-10x^{3}+\frac{x}{10}+\frac{1}{10}$
\end_inset

\SpecialChar \ldots{}

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset

 Again there are no asymptotes.
 The curve increases on 
\begin_inset Formula $(-0.058,0.058)$
\end_inset

, and decreases on 
\begin_inset Formula $(-\infty,-0.058)$
\end_inset

 and 
\begin_inset Formula $(0.058,\infty)$
\end_inset

.
 It is concave up on 
\begin_inset Formula $\left(-\infty,0\right)$
\end_inset

, concave down on 
\begin_inset Formula $\left(0,\infty\right)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

Evaluating 
\begin_inset Formula $y$
\end_inset

 values and assembling the pieces, we obtain: 
\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 Tabular
<lyxtabular version="3" rows="2" columns="1">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
dec, conc up + inc, conc up + incr, conc dn + dec, conc dn = sketch 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/ex2together.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Alternate Method
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
An alternate method to determining the intervals where 
\begin_inset Formula $\ec f$
\end_inset

 is increasing, decreasing, concave up, and concave down is the following.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
Determine the critical points.
 
\end_layout

\begin_layout Itemize
Determine the possible points of inflection.
 
\end_layout

\begin_layout Itemize
Divide the real line into intervals at the critical points and possible
 points of inflection.
 
\end_layout

\begin_layout Itemize
From each interval choose a sample point.
 Evaluate 
\begin_inset Formula $\ec f'$
\end_inset

 and 
\begin_inset Formula $\ec f''$
\end_inset

 at that sample point.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f'>0$
\end_inset

 at the sample point, then 
\begin_inset Formula $\ec f$
\end_inset

 is increasing on the entire interval; otherwise 
\begin_inset Formula $\ec f$
\end_inset

 is decreasing on the entire interval.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f''>0$
\end_inset

 at the sample point, then 
\begin_inset Formula $\ec f$
\end_inset

 is concave up on the entire interval; otherwise 
\begin_inset Formula $\ec f$
\end_inset

 is concave down on the entire interval.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second example: alternate method
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We illustrate the alternate method using the second example.
 We already found the critical points and possible points of inflection,
 so we can divide the real line into intervals: 
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="6">
<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">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset space \quad{}
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec-\infty$
\end_inset


\begin_inset space \quad{}
\end_inset


\begin_inset Formula $\ec-0.058$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec-0.058$
\end_inset


\begin_inset space \quad{}
\end_inset


\begin_inset Formula $\ec0$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec0$
\end_inset


\begin_inset space \quad{}
\end_inset


\begin_inset Formula $\ec0.058$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec0.058$
\end_inset


\begin_inset space \quad{}
\end_inset


\begin_inset Formula $\ec\infty$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset space \quad{}
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec f'$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
neg 
\end_layout

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

\begin_layout Plain Layout
pos 
\end_layout

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

\begin_layout Plain Layout
pos 
\end_layout

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

\begin_layout Plain Layout
neg 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $\ec f''$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
pos 
\end_layout

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

\begin_layout Plain Layout
pos 
\end_layout

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

\begin_layout Plain Layout
neg 
\end_layout

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

\begin_layout Plain Layout
neg 
\end_layout

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

\begin_layout Plain Layout

\end_layout

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

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

 We evaluate 
\begin_inset Formula $f'$
\end_inset

 and 
\begin_inset Formula $f''$
\end_inset

 at sample points 
\begin_inset Formula $-1$
\end_inset

, 
\begin_inset Formula $-0.01$
\end_inset

, 
\begin_inset Formula $0.01$
\end_inset

, and 
\begin_inset Formula $1$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

We now assemble the pieces of the graph the same way as before: 
\end_layout

\begin_layout Standard
\align center
dec, con up + inc, con up + incr, con dn + dec, con dn = sketch 
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/ex2together.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Proof of the alternate method
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
This alternate method is based on the 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 Mean Value Theorem
\emph default
.
 Recall that it states
\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Mean Value Theorem)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

Suppose that a function 
\begin_inset Formula $\ec f$
\end_inset

 is continuous on 
\begin_inset Formula $\ec[a,b]$
\end_inset

 and differentiable on 
\begin_inset Formula $\ec(a,b)$
\end_inset

.
 Then there exists 
\begin_inset Formula $\ec c\in(a,b)$
\end_inset

 such that
\begin_inset Formula 
\[
{\ec f'(c)=\frac{f(b)-f(a)}{b-a}}.
\]

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
Essentially, 
\begin_inset Formula $f$
\end_inset

 is increasing if, and only if, 
\begin_inset Formula $f(b)-f(a)$
\end_inset

 is positive.
 If 
\begin_inset Formula $\ec f'(c)$
\end_inset

 is 
\emph on
negative
\emph default
 for all 
\begin_inset Formula $\ec c\in(a,b)$
\end_inset

, then the fact that 
\begin_inset Formula $\ec b-a$
\end_inset

 is positive means that, by the Mean Value Theorem, 
\begin_inset Formula $\ec f(b)-f(a)$
\end_inset

 must is negative.
\end_layout

\begin_layout BeginFrame
\noindent
\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
We showed how to sketch the graphs of curves of a function.
 
\end_layout

\begin_layout Itemize
There are two different methods to use, one based on analysis of extrema
 and the other based on the Mean Value Theorem.
 Both methods give identical results.
 
\end_layout

\begin_layout Itemize
Curve sketching requires an identification of 
\emph on
concavity
\emph default
:
\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

(Concavity)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
If 
\begin_inset Formula $\ec f'$
\end_inset

 is increasing on the interval 
\begin_inset Formula $\ec I$
\end_inset

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

 is 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 concave up
\emph default
 on 
\begin_inset Formula $\ec I$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $\ec f'$
\end_inset

 is decreasing on the interval 
\begin_inset Formula $\ec I$
\end_inset

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

 is 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 concave up
\emph default
 on 
\begin_inset Formula $\ec I$
\end_inset

.
 
\end_layout

\begin_layout Itemize
if 
\begin_inset Formula $\ec f$
\end_inset

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

, then the point 
\begin_inset Formula $\ec(a,f(a))$
\end_inset

 is called an 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 inflection point
\emph default
.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout EndFrame

\end_layout

\end_body
\end_document
