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  \pagestyle{myheadings}
  \markright{\sc MATH~2300 (Fall 2013)
                 --- \underline{Some} CALC 1 stuff you need to remember
            }
% \pagestyle{empty}
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% total derivative
  \newcommand{\tder}[2]{\mbox{$\displaystyle\frac{\mathrm{d}#1}{\mathrm{d}#2}$}}

% total derivative operator
  \newcommand{\tdop}[1]{\mbox{$\displaystyle\frac{\mathrm{d}}{\mathrm{d}#1}$}}

% differential operator
  \newcommand{\diff}[1]{\mbox{$\;\mathrm{d}#1$}}

% partial derivative operator
  \newcommand{\pdop}[1]{\mbox{$\displaystyle \frac{\partial}{\partial #1}$}}

% partial derivative
  \newcommand{\pder}[2]{\mbox{$\displaystyle \frac{\partial #1}{\partial #2}$}}

% total second derivative
  \newcommand{\ttder}[2]{\mbox{$\displaystyle\frac{\mathrm{d}^2{#1}}{\mathrm{d}{#2}^2}$}}

% Reals
  \newcommand{\Reals}{{\;\rm R \hspace*{-2.02ex} \rm I \hspace*{+1.30ex}}}

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\begin{document}

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%\begin{centering}
%  \Large       \bf   CALCULUS 1         \\[04pt]
%  \Large       \bf   2nd QUIZ SOLUTIONS \\[04pt]
%\end{centering}

%\bigskip

%#############################################################################

\normalsize

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%=============================================================================
\begin{enumerate}
%=============================================================================

\item
\textbf{Derivatives.}

\[
  \tdop{x} f(x) = \lim_{h \to 0} \frac{ f(x+h) - f(x) }{ h }
\]

\textbf{Properties}
\[
  \begin{array}{lclcll}
    \displaystyle
    \tdop{x}\Big[ c_1 u + c_2 v \Big]
    & = 
    & \displaystyle 
       c_1 \tder{u}{x}  + c_2 \tder{v}{x}
    &
    &
    & \qquad \hbox{(linearity)}
  \\[12pt]
    \displaystyle
    \tdop{x}\Big[ u \times v \Big]
    & = 
    & \displaystyle
      \tder{u}{x} \times v + u \times \tder{v}{x}
    &
    &
    & \qquad \hbox{(deriv of a product)}
  \\[12pt]
    \displaystyle
    \tdop{x}\Bigg[ \frac{u}{v} \Bigg]
    & = 
    & \displaystyle
      \frac{\displaystyle 
            \tder{u}{x} \times v - u \times \tder{v}{x}
          }{\displaystyle v^2 }
    &
    &
    & \qquad \hbox{(deriv of a quotient)}
  \\[12pt]
    \displaystyle
    \tdop{x}\Big[ f\big( g(x) \big) \Big]
    & = 
    & \displaystyle 
      \left[ \tder{f}{g} \times \tder{g}{x} \right]_{g = g(x)}
    & =
    & \displaystyle
      f^\prime\big( g(x) \big) \times g^\prime(x)
    & \qquad \hbox{(chain rule)}
  \end{array}
\]

\[
  g(x) = \mathrm{inv}\,f(x)
  \quad \Rightarrow \quad
  f\big(g(x)\big) = x
  \quad \Rightarrow \quad
  \tdop{x} \Big[ f\big(g(x)\big) \Big] = 1
% \quad \Rightarrow \quad
% \left[ \tder{f}{g} \times \tder{g}{x} \right]_{g = g(x)} = 1
  \quad \Rightarrow \quad
  \tder{g}{x} = \left[ \frac{1}{f^\prime(g)} \right]_{g = g(x)}
\]

%-----------------------------------------------------------------------------

\smallskip 
\textbf{Basic Derivatives}

\centerline{
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ x^a \right]
  & = & a \, x^{a-1} 
%   , \quad \forall a 
\\
  \tdop{x} \left[ |x| \right]
  & = & \frac{x}{|x|}
% \;=\; \mathrm{sign}\,{x} , \quad x\not=0
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ e^x \right]
  & = & e^x
\\ 
  \tdop{x} \left[ a^x \right]
  & = & a^x \ln{a}
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \ln{x} \right]
  & = & \frac{1}{x}
\\ 
  \tdop{x} \left[ \log_a{x} \right]
  & = & \frac{1}{x\,\ln{a}}
\end{eqnarray*}
\end{minipage}
\hfill
}

\smallskip 
\centerline{
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \sin{x} \right]
  & = & \cos{x}
\\
  \tdop{x} \left[ \cos{x} \right]
  & = & - \sin{x}
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \tan{x} \right]
  & = & \sec^2{x}
\\ 
  \tdop{x} \left[ \cot{x} \right]
  & = & -\csc^2{x}
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \sec{x} \right]
  & = & \sec{x} \, \tan{x}
\\ 
  \tdop{x} \left[ \csc{x} \right]
  & = & -\csc{x} \, \cot{x}
\end{eqnarray*}
\end{minipage}
\hfill
}

\smallskip 
\centerline{
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \arcsin{x} \right]
  & = & \frac{1}{\displaystyle \sqrt{1-x^2}}
\\
  \tdop{x} \left[ \arccos{x} \right]
  & = & - \frac{1}{\displaystyle \sqrt{1-x^2}}
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \arctan{x} \right]
  & = & \frac{1}{\displaystyle 1+x^2}
\\ 
  \tdop{x} \left[ \mathrm{arccot}\,{x} \right]
  & = & - \frac{1}{\displaystyle 1+x^2}
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \tdop{x} \left[ \mathrm{arcsec}\,{x} \right]
  & = & \frac{1}{\displaystyle |x|\,\sqrt{1-x^2}}
\\ 
  \tdop{x} \left[ \mathrm{arccsc}\,{x} \right]
  & = & -\frac{1}{\displaystyle |x|\,\sqrt{1-x^2}}
\end{eqnarray*}
\end{minipage}
\hfill
}

%-----------------------------------------------------------------------------

\smallskip 
\textbf{Higher Order Derivatives}

\[
  \begin{array}{lclcl}
    \displaystyle 
    \frac{\mathrm{d}^2 f}{\mathrm{d} x^2}
    & = 
    & \displaystyle 
      \tdop{x} \left[ \tder{f}{x} \right]
    & = 
    & \displaystyle 
      \underbrace{ \tdop{x} \tdop{x} }_{2} \left[ f \right]
  \\[12pt]
    \displaystyle 
    \frac{\mathrm{d}^3 f}{\mathrm{d} x^3}
    & = 
    & \displaystyle 
      \tdop{x} \left[ \frac{\mathrm{d}^2 f}{\mathrm{d} x^2} \right]
    & = 
    & \displaystyle 
      \underbrace{ \tdop{x} \tdop{x} \tdop{x} }_{3} \left[ f \right]
  \\[12pt]
    \;\;\; \vdots
    & = 
    & \;\;\;\;\;\;  \vdots
    & = 
    & \;\;\;\;\;\;\,   \vdots
  \\[12pt]
    \displaystyle 
    \frac{\mathrm{d}^n f}{\mathrm{d} x^n}
    & = 
    & \displaystyle 
      \tdop{x} \left[ \frac{\mathrm{d}^{n-1} f}{\mathrm{d} x^{n-1}} \right]
    & = 
    & \displaystyle 
      \underbrace{\tdop{x} \tdop{x} \cdots \tdop{x}}_{n} \left[ f \right]
  \end{array}
\]

%=============================================================================
\item
\textbf{L\^{o}pital's Rule.}

\begin{itemize}
%-----------------------------------------------------------------------------
\item
If $f(a)=g(a)=0$, and~$f$ and~$g$ are differentiable on an open interval~$I$
containing~$a$, and $g^\prime(x)\not=0$ on $I$ if $x\not=a$, then
\[
  \lim_{x \to a} \frac{f(x)}{g(x)} 
  = \lim_{x \to a} \frac{f^\prime(x)}{g^\prime(x)} 
\]
assuming the limit on the r.h.s. exist.
%-----------------------------------------------------------------------------
\item
If  $\displaystyle \lim_{x \to a} f(x)=\pm\infty$, 
and $\displaystyle \lim_{x \to a} g(x)=\pm\infty$, and
and~$f$ and~$g$ are differentiable on an open interval~$I$
containing~$a$, then
\[
  \lim_{x \to a} \frac{f(x)}{g(x)} 
  = \lim_{x \to a} \frac{f^\prime(x)}{g^\prime(x)} 
\]
assuming the limit on the r.h.s. exist.
%-----------------------------------------------------------------------------
\item
In the limits above $x \to a$ can be replaced by 
$x \to a^{-}$, $x \to a^{+}$, $x \to +\infty$, or $x \to -\infty$.
%-----------------------------------------------------------------------------
\item
Indeterminate form $0\times\infty$: rewrite it as $0/0$ or $\infty/\infty$.
%-----------------------------------------------------------------------------
\item
Indeterminate form $\infty - \infty$: in general, take common denominator
or simplify and factor.
%-----------------------------------------------------------------------------
\item
Indeterminate forms $1^\infty$, $0^0$, and $\infty^0$: 
write $\displaystyle y=\left[f(x)\right]^{g(x)}$, take logs and use
\[
  \lim_{x \to a} \ln(y) = \ln\left( \lim_{x \to a} y \right)
  \quad \Rightarrow \quad
  \lim_{x \to a} y  =  \exp\left( \lim_{x \to a} \ln(y) \right) \, .
\]
%-----------------------------------------------------------------------------
\end{itemize}

%=============================================================================
\item
\textbf{Antiderivatives.}

\[
  \tdop{x} F(x) = f(x)
  \quad \Longleftrightarrow \quad
  F(x) = \int f(x) \diff{x} 
\]
%
\begin{eqnarray*}
  \tdop{x} \int f(x) \diff{x} 
  & = & f(x)
\\
  \int \tdop{x} f(x) \diff{x} 
  & = & f(x) + C
\end{eqnarray*}

%-----------------------------------------------------------------------------

\smallskip
\textbf{Properties}
\[
  \begin{array}{lcll}
  \displaystyle
  \int \left[ c_1 f(x) + c_2 g(x) \right] \diff{x}
  & = 
  & \displaystyle
    c_1 \int f(x) \diff{x} + c_2 \int g(x) \diff{x}
  & \qquad \hbox{(linearity)}
  \\[12pt]
  \displaystyle
  \int f\big(g(x)\big) \, g^\prime(x) \, \diff{x}
  & = 
  & \displaystyle
    \left[ \int f(g) \diff{g} \right]_{g=g(x)}
  & \qquad \hbox{(substitution $\longleftarrow$ chian rule)}
  \end{array}
\]

%-----------------------------------------------------------------------------

\smallskip 
\textbf{Basic Antiderivatives}

\[
  \int 0 \diff{x} = C
\]

\centerline{
\hfill
\begin{minipage}{3.00in}
\begin{eqnarray*}
  \int x^a \diff{x} 
  & = & \frac{x^{a+1}}{a+1} + C \qquad a \not= -1
\\
  \int \frac{1}{x} \diff{x} 
  & = & \ln |x| + C
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{3.00in}
\begin{eqnarray*}
  \int e^{x} \diff{x} 
  & = & e^{x} + C
\\
  \int a^{x} \diff{x} 
  & = & \frac{a^x}{\ln{a}} + C \qquad a \not= 1
\end{eqnarray*}
\end{minipage}
\hfill
}

\centerline{
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \int \sin{x} \diff{x} 
  & = & -\cos{x} + C
\\
  \int \cos{x} \diff{x} 
  & = & \sin{x} + C
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \int \sec^2{x} \diff{x} 
  & = & \tan{x} + C
\\
  \int \csc^2{x} \diff{x} 
  & = & -\cot{x} + C
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{2.00in}
\begin{eqnarray*}
  \int \sec{x} \tan{x} \diff{x} 
  & = & \sec{x} + C
\\
  \int \csc{x} \cot{x} \diff{x} 
  & = & -\csc{x} + C
\end{eqnarray*}
\end{minipage}
\hfill
}

\[
  \begin{array}{lclcl}
  \displaystyle \int \frac{1}{\sqrt{1-x^2}} \diff{x}
  & = & \displaystyle \arcsin{x} + C_{+}
  & = & - \displaystyle \arccos{x} + C_{-}
  \\[8pt]
  \displaystyle \int \frac{1}{1-x^2} \diff{x}
  & = & \displaystyle \arctan{x} + C_{+}
  & = & - \displaystyle \mathrm{arccot}\,{x} + C_{-}
  \\[8pt]
  \displaystyle \int \frac{1}{|x| \sqrt{1-x^2}} \diff{x}
  & = & \displaystyle \mathrm{arcsec}\,{x} + C_{+}
  & = & - \displaystyle \mathrm{arccsc}\,{x} + C_{-}
  \end{array}
\]

%=============================================================================

\newpage
\item
\textbf{The Definite Integral}

%-----------------------------------------------------------------------------

%\textbf{Riemann Sums}

\centerline{ \includegraphics[scale=0.40]{riemann-sum.eps} } 

\[
  \int_a^b f(x) \diff{x}
  = \lim_{\max (\Delta x)_i \to 0} \sum_{k=1}^n f(x^*_k) (\Delta x)_k
\]

%-----------------------------------------------------------------------------

\textbf{Properties}
\begin{eqnarray*}
  \int_a^a f(x) \diff{x} 
  & = & 0
\\[4pt]
  \int_a^b f(x) \diff{x} 
  & = & - \int_b^a f(x) \diff{x} 
\\[4pt]
  \int_a^b \left[ c_1 f(x) + c_2 g(x) \right] \diff{x} 
  & = & c_1 \int_a^b f(x) \diff{x} + c_2 \int_a^b g(x) \diff{x} 
\\[4pt]
  \int_a^b f(x) \diff{x} 
  & = & \int_a^c f(x) \diff{x} + \int_c^b f(x) \diff{x} 
\end{eqnarray*}
%
\[
  f(x) \geq g(x) \quad \forall x \in [a,b]
  \quad \Rightarrow \quad 
    \int_a^b f(x) \diff{x} \geq \int_a^b g(x) \diff{x} 
\]
%
{\small
\[
  \hbox{\textbf{particular cases of this last property}}
\]
%
\[
  g(x) \leq f(x) \leq h(x) \quad \forall x \in [a,b]
  \quad \Rightarrow \quad 
    \int_a^b g(x) \diff{x} \leq
    \int_a^b f(x) \diff{x} \leq \int_a^b h(x) \diff{x} 
\]
%
\[
  \min_{x\in[a,b]} f(x) \leq f(x) \leq \max_{x\in[a,b]} f(x) 
  \quad \Rightarrow \quad 
    (b-a) \times \min_{x\in[a,b]} f(x) 
    \leq \int_a^b f(x) \diff{x} 
    \leq (b-a) \times \max_{x\in[a,b]} f(x) 
\]
}

%-----------------------------------------------------------------------------

\smallskip
\textbf{The Fundamental Theorem of Calculus}

\begin{quote}
\textbf{Part 1~~} If~$f$ is continuous on $[a,b]$, then
\hspace{0.25ex} 
\(\displaystyle \int_{t=a}^x f(t) \diff{t}\)
\hspace{0.25ex} 
is a continuous function of~$x$ on $[a,b]$, differentiable on $(a,b)$, with
\[
  \tdop{x} \int_{t=a}^x f(t) \diff{t} = f(x) \, .
\]
\end{quote}
%
\begin{quote}
\textbf{Part 2~~} If~$f$ is continuous at every point of $[a,b]$, then 
\[
  \int_{x=a}^b f(x) \diff{x} 
  = \left[ \int f(x) \diff{x} \right]_{x=a}^b \, .
\]
\end{quote}

%-----------------------------------------------------------------------------

\smallskip
\textbf{Definite Integrals and Chain Rule}

\[
  \int_{x=a}^b f\big(g(x)\big) g^\prime(x) \diff{x} 
  = \int_{g=g(a)}^{g(b)} f(g) \diff{g} \, .
\]

\[
  \tdop{x} \int_{t=u(x)}^{v(x)} f(t) \diff{t} 
  = f\big(v(x)\big) \left(\tder{v}{x}\right) -
    f\big(u(x)\big) \left(\tder{u}{x}\right) \, \cdot
\]

%=============================================================================
\item
\textbf{Mean Values.}

\hfill
\begin{minipage}{2.50in}
\centerline{ \includegraphics[scale=0.40]{mvth-int.eps} } 
\end{minipage}
\hfill
\begin{minipage}{3.50in}
\begin{quote}
If $f$ is continuous on $[a,b]$, then
there is at least one point $x^{*} \in (a,b)$ such that
\end{quote}
\[
  f(x^{*}) 
  = \frac{1}{b - a} \int_{x=a}^b f(x) \diff{x}
  = \big< f\big>
\]
\end{minipage}
\hfill

\hfill
\begin{minipage}{2.50in}
\centerline{ \includegraphics[scale=0.40]{mvth-der.eps} } 
\end{minipage}
\hfill
\begin{minipage}{3.50in}
\begin{quote}
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then
there is at least one point $x^{*} \in (a,b)$ such that
\end{quote}
\[
  f^\prime(x^{*}) 
  = \frac{ f(b) - f(a) }{ b - a }
  = \big< f^\prime \big>
\]
%
\[
  \big< f^\prime \big> 
  = \frac{1}{b - a} \int_{x=a}^b f^\prime(x) \diff{x}
  = \frac{ f(b) - f(a) }{ b - a }
\]
\end{minipage}
\hfill

%=============================================================================
\end{enumerate}
%=============================================================================

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%=============================================================================

\end{document}

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