%fa07 problem set 1


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\newtheorem{theorem}{Theorem}
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\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
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\lhead{\color{blue} \Large Lecture Notes}
\chead{\color{black} \LARGE Complete Analysis of a Function}
\rhead{\large page   \ \thepage}
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\lfoot{\small   \copyright $\;$ copyright  Hidegkuti,  2013}
\rfoot{\small   Last revised:  March 16, 2014}
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Equation: $f\left( x\right) =\rule{90pt}{1pt}$\bigskip \bigskip \bigskip

\begin{tabular}[t]{llll}
1. & domain: $\rule{90pt}{1pt}$ & 7. & one-to-one: $\rule{90pt}{1pt}$ \\ 
&  &  &  \\ 
& range: $\rule{90pt}{1pt}$ & 8. & concave up: $\rule{90pt}{1pt}$ \\ 
&  &  &  \\ 
2. & $y-$intercept: $\rule{90pt}{1pt}$ &  & concave down: $\rule{90pt}{1pt}$
\\ 
&  &  &  \\ 
& $x-$intercept(s): $\rule{90pt}{1pt}$ & 9. & point(s) of inflection: $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
3. & boundedness: $\rule{90pt}{1pt}$ & 10. & continuous: $\rule{90pt}{1pt}$
\\ 
&  &  &  \\ 
4. & horizontal asymptote(s): $\rule{90pt}{1pt}$ & 11. & even/odd: $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
& vertical asymptote(s): $\rule{90pt}{1pt}$ & 12. & end-behavior: \\ 
&  &  & $\lim\limits_{x\rightarrow -\infty }f\left( x\right) $\ \ \ \ and \
\ \ $\lim\limits_{x\rightarrow \infty }f\left( x\right) $ \\ 
5. & increasing: $\rule{90pt}{1pt}$ &  &  \\ 
&  & 13. & Graph \\ 
& decreasing: $\rule{90pt}{1pt}$ &  &  \\ 
&  &  &  \\ 
6. & absolute maximum(s): $\rule{90pt}{1pt}$ &  &  \\ 
&  &  &  \\ 
& relative maximum(s): $\rule{90pt}{1pt}$ &  &  \\ 
&  &  &  \\ 
& absolute minimum(s): $\rule{90pt}{1pt}$ &  &  \\ 
&  &  &  \\ 
& relative minimum(s): $\rule{90pt}{1pt}$ &  & 
\end{tabular}

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\pagebreak

\begin{center}
{\LARGE Definitions}\bigskip
\end{center}

\begin{enumerate}
\item The \textbf{domain} of a function is a non-empty set of elements to
which we assign things.

The \textbf{range} of a function is the set of elements that we assign to
elements of the domain.\bigskip

\item The $y-$\textbf{intercept} of a function is the point where the graph
of the function intersects the $y-$axis. \ \newline
A function can have at most one $y-$intercept.

The $x-$\textbf{intercept} of a function is the point where the graph of the
function intersects the $x-$axis. \ \newline
A function can have several $x-$intercepts.\bigskip

\item A set $S$ is \textbf{bounded from below} if there exists a real number 
$L$ (lower bound) \ such that for all $x$ in $S$, \ $x\geq L$.

A set $S$ is \textbf{bounded from above} if there exists a real number $U$
(upper bound) \ such that for all $x$ in $S$, \ $x\leq U$.

A set $S$ is \textbf{bounded} if it is bounded from above and from below.

A function $f$ is \textbf{bounded from below} if there exists a real number $%
L$ (lower bound) \ such that for all $x$ in the domain, \ $f\left( x\right)
\geq L$.

A function $f$ is \textbf{bounded from above} if there exists a real number $%
U$ (upper bound) \ such that for all $x$ in the domain, \ $f\left( x\right)
\leq U$.

A function $f$ is \textbf{bounded} if it is bounded from above and from
below.\bigskip

\item Asymptotes. \ At this point we will not rigorously define asymptotes
yet, just provide with an intuitive idea. \ There are two types of
asymptotes.

A function can have at most two \textbf{horizontal asymptote}s. \ A
horizontal asymptote is the graphical representation of the property that as
the numbers in the domain get larger and larger (positive or negative),
their assigned values get closer and closer to a fixed number.

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The three examples above show different ways a function can approach a
horizontal asymptote. \ The first picture shows a graph approaching a
horizontal asympotote from above. \ The second graph approaches a horizontal
asympotote from below. \ \ The third graph crosses it many times as it
oscillates around it.\bigskip

A \textbf{vertical asymptote} occurs when numbers in the domain \ close to a
fixed number take extremely large values. \ A function can have many
different vertical asymptotes. \ The function may behave in different ways
at the left-hand side and at the right-hand side of the asymptote. \
Sometimes the function is not even defined on both sides of a vertical
asymptote.

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\item Definition: \ A function $f$ is \textbf{increasing} on an interval $I$
if for all $a$ and $b$ in $I$, \ if $a<b$, then \ $f\left( a\right) \leq
f\left( b\right) $.

Definition: \ A function $f$ is \textbf{strictly increasing} on an interval $%
I$ if for all $a$ and $b$ in $I$,\ if $a<b$, then \ $f\left( a\right)
<f\left( b\right) $.

Definition: \ A function $f$ is \textbf{decreasing} on an interval $I$ if
for all $a$ and $b$ in $I$, if $a<b$, then \ $f\left( a\right) \geq f\left(
b\right) $.

Definition: \ A function $f$ is \textbf{strictly decreasing} on an interval $%
I$ if for all $a$ and $b$ in $I$,\ if $a<b$, then \ $f\left( a\right)
>f\left( b\right) $.\bigskip

\item Extremum is a common name for a maximum or a minimum.

A function $f$ has an \textbf{absolute maximum} at $x_{M}$ if for all $x$ in
the domain, $f\left( x_{M}\right) \geq f\left( x\right) $.

A function $f$ has an \textbf{absolute minimum} at $x_{m}$ if for all $x$ in
the domain, $f\left( x_{m}\right) \leq f\left( x\right) $.

A function $f$ has a \textbf{relative (or local) maximum} at $x_{M}$ if\
there exists an open interval $I=\left( a,b\right) $ that contains $x_{M}$
such that

\qquad i) \ the function is defined on $I$ and

\qquad ii) \ if we restrict the function to $I$ as its domain, then $\left(
x_{M},f\left( x_{M}\right) \right) $ is an absolute maximum.\medskip

A function $f$ has a \textbf{relative (or local) minimum} at $x_{m}$ if\
there exists an open interval $I=\left( a,b\right) $ that contains $x_{m}$
such that

\qquad i) \ the function is defined on $I$ and

\qquad ii) \ if we restrict the function to $I$ as its domain, then $\left(
x_{m},f\left( x_{m}\right) \right) $ is an absolute minimum.\bigskip

\item Definition: \ A function $f$ is \textbf{one-to-one (or injective)} if
for all $a$ and $b$ in its domain,\ if $a\not=b$, then \ $f\left( a\right)
\not=f\left( b\right) $.\medskip

Alternative definition: \ A function $f$ is \textbf{one-to-one (or injective)%
} if for all $a$ and $b$ in its domain,\ if $f\left( a\right) =f\left(
b\right) $, then \ $a=b$.\bigskip

\item A function $f$ is \textbf{concave up} on an interval $I$ if for all
pairs of numbers $a,b$ in $I$, the secant line connecting the points $\left(
a,f\left( a\right) \right) $ and $\left( b,f\left( b\right) \right) $ lies
above the graph of $f$ between $a$ and $b$.

A function $f$ is \textbf{concave down} on an interval $I$ if for all pairs
of numbers $a,b$ in $I$, the secant line connecting the points $\left(
a,f\left( a\right) \right) $ and $\left( b,f\left( b\right) \right) $ lies
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\item A function has \textbf{a point of inflection} at $x$ if there exists
real numbers $a$ and $b$ such that $a<x<b$ and $f$ has different concavity
behaviors on $\left( a,x\right) $ and on $\left( x,b\right) $.\bigskip

\item Continuity is an extremely important concept in calculus. \ At this
point, we will not yet define continuity but for now, we will have an
intuitive idea for it. \ Although this term will not be precisely defined,
the intuitive idea of a continuous function is that we can draw its graph
without lifting the pencil. \ For example, $f\left( x\right) =x^{2}$ is a
continuous function but $g\left( x\right) =\dfrac{1}{x}$ is not; it is not
continuous at $x=0$.\bigskip

\item A function $f$ is \textbf{even} if for all $x$ in its domain, if $%
f\left( x\right) $ is defined, then $f\left( -x\right) $ is defined and $%
f\left( -x\right) =f\left( x\right) $.\medskip

A function $f$ is \textbf{odd} if for all $x$ in its domain, if $f\left(
x\right) $ is defined, then $f\left( -x\right) $ is defined and $f\left(
-x\right) =-f\left( x\right) $.\bigskip

\item The end-behavior of a function $f$\ describes the behavior of $f$ for
very large negative and very large positive numbers.\bigskip \bigskip
\end{enumerate}

\begin{center}
{\LARGE Sample Problems}\bigskip
\end{center}

Sketch the graph and give a complete analysis for each of the following
functions.

\begin{enumerate}
\item $f\left( x\right) =\sqrt{x+1}-2$

\item $\ f\left( x\right) =10x-x^{2}+11$ \ where the domain is the closed
interval $\left[ 3,8\right] $

\item $\ f\left( x\right) =10x-x^{2}+11$ \ where the domain is the open
interval $\left( 3,8\right) $\pagebreak
\end{enumerate}

\begin{center}
{\LARGE Sample Problems - Answers}\bigskip
\end{center}

\begin{enumerate}
\item $f\left( x\right) =\sqrt{x+1}-2$

\begin{tabular}[t]{lll}
domain: \ $\left[ -1,\infty \right) $ & one-to-one & no point of inflection
\\ 
range: $\left[ -2,\infty \right) $ & no relative maximum & concave down on $%
\left( -1,\infty \right) $ \\ 
no asymptotes & no absolute maximum & continuous on $\left( -1,\infty
\right) $ \\ 
$y-$intercept: $\left( 0,-1\right) $ \ \ \ \ \ \ \  & no relative minimum & 
even/odd: neither \\ 
$x-$intercept: $\left( 3,0\right) $ & absolute minimum: $\left( -1,-2\right) 
$ \ \ \ \ \ \ \  & end-behavior: \\ 
bounded from below & increasing on $\left( -1,\infty \right) $ & $%
\lim\limits_{x\rightarrow -\infty }f\left( x\right) =$ undefined\ \ \ \ \ \ 
\\ 
&  & $\lim\limits_{x\rightarrow \infty }f\left( x\right) =\infty $%
\end{tabular}%
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\item $f\left( x\right) =10x-x^{2}+11$ \ where the domain is the closed
interval $\left[ 3,8\right] $.

\begin{tabular}[t]{lll}
domain: \ $\left[ 3,8\right] $ \ \ \ \ \ \ \  & not one-to-one & no point of
inflection \\ 
range: $\left[ 27,36\right] $ & relative maximum: $\left( 5,36\right) $ \ \
\ \ \ \  & concave down on $\left[ 3,8\right] $ \\ 
no asymptotes & absolute maximum: $\left( 5,36\right) $ & continuous on $%
\left( 3,8\right) $ \\ 
no intercepts & no relative minimum & even/odd: neither \\ 
bounded & absolute minimum: $\left( 8,27\right) $ & end-behavior: \\ 
& increasing: on $\left( 3,5\right) $ & $\lim\limits_{x\rightarrow -\infty
}f\left( x\right) =$ undefined\ \  \\ 
& decreasing: on $\left( 5,8\right) $ & $\lim\limits_{x\rightarrow \infty
}f\left( x\right) =$ undefined%
\end{tabular}%
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\item $f\left( x\right) =10x-x^{2}+11$ \ where the domain is the open
interval $\left( 3,8\right) $

\begin{tabular}[t]{lll}
domain: \ $\left( 3,8\right) $ \ \ \ \ \ \ \ \ \  & not one-to-one & no
point of inflection \\ 
range: $\left( 27,36\right] $ & relative maximum: $\left( 5,36\right) $ \ \
\ \ \ \ \  & concave down on $\left( 3,8\right) $ \\ 
no asymptotes & absolute maximum: $\left( 5,36\right) $ & continuous on $%
\left( 3,8\right) $ \\ 
no intercepts & no relative minimum & even/odd: neither \\ 
bounded & no absolute minimum & end-behavior: \\ 
& increasing: on $\left( 3,5\right) $ & $\lim\limits_{x\rightarrow -\infty
}f\left( x\right) =$ undefined\ \  \\ 
& decreasing: on $\left( 5,8\right) $ & $\lim\limits_{x\rightarrow \infty
}f\left( x\right) =$ undefined%
\end{tabular}%
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\vspace{0.7in}

\vspace{4in}

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\href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{%
For more documents like this, visit our page at\
https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail
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