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\[ \ \]

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\begin{center}
    {\huge Calculus III Note-taking Guide Booklet}
\end{center}

\vfill


\vfill

\textbf{Created by:} Nicole Buczkowski, Worcester Polytechnic Institute

These open-source mathematics note taking guides are created to help students with note-taking and correspond to chapters in OpenStax Calculus Volume I$^1$, II$^2$, and III$^3$. They were created through the Worcester Polytechnic Institute Women's Impact Network EMPOwER grant program.

$^1$ Herman, Edwin, Gilbert Strang, Joseph Lakey, Elaine A. Terry, Alfred K. Mulzet, Sheri J. Boyd, Joyati Debnath et al. "Calculus Volume 1." (2016).

$^2$ Herman, Edwin, Gilbert Strang, William Radulovich, Erica A. Rutter, David Smith, Kirsten R. Messer, Alfred K. Mulzet et al. "Calculus Volume 2." (2016).

$^3$ Herman, Edwin, Gilbert Strang, Nicoleta Virginia Bila, Sheri J. Boyd, David Smith, Elaine A. Terry, David Torain et al. "Calculus Volume 3." (2016).


% Creative Commons License
% Creative Commons Attribution 4.0 International License
% This work is licensed under a Creative Commons Attribution 4.0 International License.

\pagebreak










\begin{center}
    \textbf{Table of Contents}
\end{center}
\begin{center}
        \textbf{From Volume 1}
    \end{center}
\begin{enumerate}
    \item 4.8 L’Hôpital’s Rule  \\ 
    \begin{center}
        \textbf{From Volume 2 $ \ \ \ \ \ $ }
    \end{center}
    \item 3.7 Improper Integrals
    \item 5.1 Sequences
    \item 5.2 Infinite Series
    \item 5.3 The Divergence and Integral Tests
    \item 5.4 Comparison Tests
    \item 5.5 Alternating Series
    \item 5.6 Ratio and Root Tests
    \item 6.1 Power Series and Functions
    \item 6.2 Properties of Power Series
    \item 6.3 Taylor and Maclaurin Series
    \item 6.4 Working with Taylor Series
    \item 7.1 Parametric Equations
    \item 7.2 Calculus of Parametric Curves
    \item 7.3 Polar Coordinates
    \item 7.4 Area and Arc Length in Polar Coordinates
\begin{center}
        \textbf{From Volume 3 $ \ \ \ \ \ $ }
    \end{center}
    \item 2.1 Vectors in the Plane
    \item 2.2  Vectors in Three Dimensions
    \item 2.3 The Dot Product
    \item 2.4 The Cross Product
    \item 2.5 Equations of Lines and Planes in Space
    \item 3.1 Vector-Valued Functions and Space Curves
    \item 3.2 Calculus of Vector-Valued Functions
    \item 3.3 Arc Length and Curvature
    \item 3.4 Motion in Space
\end{enumerate}


\pagebreak


\begin{center}
    \textbf{Calculus III}
\end{center}

\section{4.8 L'Hôpital's Rule}


\begin{prob}
Find the following limits
\vspace{0.1 in}

\begin{enumerate}
    \item \[\lim_{x \rightarrow 0} \frac{x^3}{x}= \hspace{6 in}\]
    \item \[\lim_{x \rightarrow 0} \frac{x}{x^3}= \hspace{6 in}\]
    \item \[\lim_{x \rightarrow 0} \frac{x}{x}= \hspace{6 in}\]
    \item \[\lim_{x \rightarrow 0} \frac{x+x^3}{2}= \hspace{6 in}\]
    \item \[\lim_{x \rightarrow 0} \frac{x}{x+4x^2}= \hspace{6 in}\]
\end{enumerate}
\end{prob}

\vspace{1 in}

\begin{theorem}
\textbf{L'Hôpital's Rule} \vspace{2 in}
\end{theorem}

\vspace{1 in}

\begin{prob}
    Evaluate the following limits. 
    \begin{enumerate}
        \item \[\lim_{x \rightarrow 0}\frac{\sin(x)}{x}= \hspace{6 in}\]\vspace{1 in}
         \item \[\lim_{x \rightarrow 0}\frac{e^{1/x}-1}{e^{1/x}}= \hspace{6 in}\]\vspace{1 in}
        \item \[\lim_{x \rightarrow 0}\frac{\sin{x}-x}{x^2}= \hspace{6 in}\]\vspace{1 in}
    \end{enumerate}
\end{prob}

\textbf{Other Indeterminant Forms}
\begin{itemize}
\item 
\item \vspace{0.5 in}
\item \vspace{0.5 in}
\end{itemize}
\vspace{0.5 in}

\pagebreak
\begin{ex}
\begin{align*}
    \lim_{x \rightarrow 0^+} \frac{1}{x^2}-\frac{1}{\tan(x)}\hspace{6 in}
\end{align*}

\vspace{1 in}
\end{ex}

\textbf{Indeterminant Powers} 

If $\lim_{x \rightarrow a} \ln(f(x))=L,$ then 
\vspace{1 in}

\begin{ex}
    Evaluate \[\lim_{x \rightarrow \infty} x^{\frac{1}{x}}\hspace{6 in}\]. 

    \vspace{1 in}
\end{ex}

\section{3.7 Improper Integrals}

\begin{defn}

\textbf{Integrating over an Infinite Interval} 

\begin{enumerate}
    \item If $f(x)$ is continuous on $[a, \infty)$, then \vspace{1 in}
    \item If $f(x)$ is continuous on $(-\infty, b]$, then\vspace{1 in}
    \item If $f(x)$ is continuous on $(-\infty, \infty)$, then \vspace{1 in}
\end{enumerate}
In each cases, if the limit exists, then the improper integral is said to \mbox{\parbox[t]{35ex}{\hrulefill}}. Otherwise, if the limit does not exist, then the improper integral is said to \mbox{\parbox[t]{35ex}{\hrulefill}}.
\end{defn}


\begin{ex} We evaluate
    \begin{align*}
        \int_1^\infty \frac{1}{x}dx&=\hspace{6 in}
    \end{align*}
    \vspace{2 in}
\end{ex}

\begin{prob}
    Evaluate
    \[\int_{-\infty}^0\frac{1}{x^2+4}dx\hspace{6 in}\]
    \vspace{2 in}
\end{prob}


\begin{defn}

\textbf{Integrating a Discontinuous Integrand} 

\begin{enumerate}
    \item If $f(x)$ is continuous on $[a, b)$, then \vspace{1 in}
    \item If $f(x)$ is continuous on $(a, b]$, then \vspace{1 in}
    \pagebreak
    \item If $f(x)$ is continuous on $[a,b]$ except at c in $(a,b)$, then \ \vspace{1 in}
\end{enumerate}
In each case, if the limit exists and is finite, then the improper integral  is said to \mbox{\parbox[t]{35ex}{\hrulefill}}. Otherwise, the improper integral is said to \mbox{\parbox[t]{35ex}{\hrulefill}}.
\end{defn}

\begin{ex} We evaluate
    \begin{align*}
        \int_{-1}^1 \frac{1}{x^3} dx&=\hspace{6 in}
    \end{align*}

    \vspace{1 in}
\end{ex}

\begin{theorem}
    \textbf{The Direct Comparison Test} Let $f,g$ be continuous on $[a, \infty)$ and assumme that  $0 \leq f(x) \leq g(x)$ for all $x \geq a$. Then
    \begin{enumerate}
        \item If $ \int_a^\infty g(x) dx$  \mbox{\parbox[t]{35ex}{\hrulefill}}, then $\int_a^\infty f(x) dx$ also  \mbox{\parbox[t]{35ex}{\hrulefill}}.
        \item If $ \int_a^\infty f(x) dx$  \mbox{\parbox[t]{35ex}{\hrulefill}}, then $\int_a^\infty g(x) dx$ also \mbox{\parbox[t]{35ex}{\hrulefill}}.
    \end{enumerate}
\end{theorem}
\begin{ex}
Consider for $p<1$
\begin{align*}
    \int_1^\infty (x+7)^pdx &\hspace{6 in}
\end{align*}
\vspace{1.5 in}

\end{ex}


\section{5.1 Sequences}

\begin{defn}
    An \mbox{\parbox[t]{35ex}{\hrulefill}} is an ordered list a of numbers of the form\\
    
    \vspace{0.75in}
    
    Each of the numbers is called a \mbox{\parbox[t]{35ex}{\hrulefill}}. The symbol $n$ is called the \mbox{\parbox[t]{35ex}{\hrulefill}} for the sequence. 

    We also use the notation 
    \vspace{0.5 in}
\end{defn}
\begin{ex}
    Examples of sequences:
    \vspace{1.75in}
\end{ex}

We sometimes would like to write sequences using its \mbox{\parbox[t]{35ex}{\hrulefill}}. 

\begin{prob} 
Write each sequences given using its explicity formula. We will do the second one together:
\begin{itemize}
        \item $1,2,3,4,...$ 
        
        \vspace{0.25 in}
        
        \item $2,4,6,8,10$ 
        
        \vspace{0.25 in}
        
        \item $1,-1,1,-1,...$
        
        \vspace{0.25 in}
        
        \item $1,1,2,3,5,8,...$
        
        \vspace{0.5 in}
        
    \end{itemize}
\end{prob}


\subsection{Limit of a Sequence}
\begin{defn}
   Given a sequence $\{a_n\}$, if the terms of $a_n$ become \mbox{\parbox[t]{35ex}{\hrulefill}} to a \mbox{\parbox[t]{15ex}{\hrulefill}} \mbox{\parbox[t]{25ex}{\hrulefill}} as \mbox{\parbox[t]{45ex}{\hrulefill}}, we say $\{a_n\}$ is a \mbox{\parbox[t]{35ex}{\hrulefill}}\textbf{} and $L$ is the \mbox{\parbox[t]{25ex}{\hrulefill}}. In this case, we write 
   \vspace{0.5 in}

   If a sequence is not convergent, we say it is \mbox{\parbox[t]{30ex}{\hrulefill}}. 
\end{defn}

More formally, we can instead use the definition: 
\begin{defn}
    A sequence $\{a_n\}$ \mbox{\parbox[t]{35ex}{\hrulefill}} to the number \mbox{\parbox[t]{25ex}{\hrulefill}} if for every $\varepsilon>0$ there corresponds an integer $N$ such that if $n\geq N$,\\
    
    \vspace{0.5 in}

    The number $L$ is the 
    \mbox{\parbox[t]{35ex}{\hrulefill}} and we write

    \vspace{0.5 in}
    
\end{defn}

\pagebreak 
\begin{ex}
    Let $\{a_n\}=\{\frac{1}{n}\}$ and $\{b_n\}=\{(-1)^n\}$. We investigate the convergence or divergence of each.

    \vspace{2.5 in} 
    
\end{ex}


\subsection{Calculating Limits of Sequences}
\begin{theorem}
    \textbf{Limit of a Sequence Defined by a function} Consider a sequence $\{a_n\}$ such that $a_n=f(n)$. If \mbox{\parbox[t]{55ex}{\hrulefill}} such that
    
    \vspace{0.5 in} 
    
    then $\{a_n\}$ converges and 
    
    \vspace{0.5 in}.
\end{theorem}

\begin{theorem} \textbf{Algebraic Limit Laws:}
    Given sequences $\{a_n\}$ and $\{b_n\}$ and a real number $C$, if there exist constants $A, B$ such that $\lim_{n\rightarrow \infty} a_n=A, \lim_{n\rightarrow \infty} b_n=B$. Then
    \begin{enumerate}
        \item 
        \item \vspace{0.5 in}
        \item \vspace{0.5 in}
        \item \vspace{0.5 in}
        \item \vspace{0.5 in}
    \end{enumerate}
    \vspace{0.5 in}
\end{theorem}






\begin{theorem}
    Consider a sequence $\{a_n\}$ and suppose there exists a real number  $L$ such that the sequence  $\{a_n\}$
  converges to $L$. Suppose $f$
  is a continuous function at $L$.
  Then there exists an integer  $N$
  such that  $f$
  is defined at all values  an
  for  $n\geq N$,
  and the sequence
  \vspace{0.5 in}
 
\end{theorem}

This allows us to use things like L'Hôpital's rule for sequences.

\begin{theorem}
    \textbf{The Squeeze Theorem for Sequences} 
    Consider sequences $\{a_n\}, \{b_n\},\{c_n\}$ and suppose that  \mbox{\parbox[t]{35ex}{\hrulefill}} for all $n \geq N$ for some $N$. If\\
    
    \vspace{0.5 in}
    
    then \vspace{0.5 in}
\end{theorem}

\begin{prob}
    If possible, find the limits of the following sequences.
        \begin{enumerate}
        \item $1,2,3,4,...$ \vspace{0.4 in}
        \item $5,19,5,19,5,19,...$ \vspace{0.4 in}
        \item $\{\frac{1}{n^2}\}$ \vspace{0.4 in}
    \end{enumerate}
\end{prob}

\subsection{Bounded and Monotonic Sequences}

\begin{defn}
    A sequence $\{a_n\}$ is \mbox{\parbox[t]{35ex}{\hrulefill}}  if there exists a number $M$ so that \mbox{\parbox[t]{35ex}{\hrulefill}}  for all $n$.\\
    A sequence $\{a_n\}$ is \mbox{\parbox[t]{35ex}{\hrulefill}}  if there exists a number $m$ so that \mbox{\parbox[t]{35ex}{\hrulefill}}  for all $n$. \\
    A sequence $\{a_n\}$
  is a \mbox{\parbox[t]{35ex}{\hrulefill}}  if it is bounded above and bounded below.

If a sequence is not bounded, it is an \mbox{\parbox[t]{35ex}{\hrulefill}} 
\end{defn}

\begin{defn}
    A sequence $\{a_n\}$ is\mbox{\parbox[t]{35ex}{\hrulefill}}  if \mbox{\parbox[t]{35ex}{\hrulefill}}  for all $n$. It is \mbox{\parbox[t]{35ex}{\hrulefill}}  if \mbox{\parbox[t]{35ex}{\hrulefill}}  for all $n$. A sequence is \mbox{\parbox[t]{35ex}{\hrulefill}}  if is either \mbox{\parbox[t]{35ex}{\hrulefill}}  or \mbox{\parbox[t]{35ex}{\hrulefill}} 
\end{defn}

\begin{theorem} \textbf{Montone Convergence Theorem}
    If  $\{a_n\}$
  is a \mbox{\parbox[t]{35ex}{\hrulefill}}  sequence and there exists a positive integer  $n_0$
  such that  $\{a_n\}$
  is \mbox{\parbox[t]{35ex}{\hrulefill}}  for all  $n\geq n_0$,
  then $\{a_n\}$
  \mbox{\parbox[t]{35ex}{\hrulefill}}  
\end{theorem}

\pagebreak 
\begin{prob}
    Classify each sequence as bounded or not and monotonic or not. Then using that information, decide if we know if the sequence converges. 
    \begin{enumerate}
        \item $1,2,3,4,...$
        \vspace{0.5 in}
        \item $5,19,5,19,5,19,...$
        \vspace{0.5 in}
        \item $\{(-1)^n\}$
        \vspace{0.5 in}
        \item $\{\frac{1}{n}\}$
        \vspace{0.5 in}
        \item $a_n=a_1$, where $a_1=7$
        \vspace{0.5 in}
    \end{enumerate}
\end{prob}


\section{5.2 Infinite Series}


\begin{defn}
An \mbox{\parbox[t]{35ex}{\hrulefill}} is a sum of infinitely many terms and is written in the form
\vspace{1 in}

For each $k$, $S_k$ is \mbox{\parbox[t]{35ex}{\hrulefill}}

\vspace{0.5 in}

If we can describe the convergence of a series to $S$, we call $S$ the \mbox{\parbox[t]{35ex}{\hrulefill}}, and we write
\vspace{0.5 in}

If the sequence of partial sums diverges, we have the \mbox{\parbox[t]{45ex}{\hrulefill}}
\end{defn}

\begin{ex}
Decide whether each sum converges or diverges.
    \begin{itemize}
        \item \[\sum_{n=1}^\infty 1 \hspace{6 in}\] 
        \item \[\sum_{n=1}^\infty 0 \hspace{6 in}\] 
    \end{itemize}
\end{ex}


\begin{ex}
    Find the sum of the \textbf{telescoping series}
    \[\sum_{n=1}^\infty \frac{1}{n(n+1)} \hspace{6 in}\]


    \vspace{2 in}
\end{ex}

\begin{theorem}
Let $\sum a_n$, $\sum b_n$ be convergent series. Then we have:

    \begin{enumerate}
        \item Sum/Difference Rule: \vspace{0.5 in}
        \item Constant Multiple Rule: \vspace{0.5 in}
    \end{enumerate}
\end{theorem}



\subsection{Geometric Series}

\begin{defn}
    A \mbox{\parbox[t]{35ex}{\hrulefill}} is any series that we can write in the form
    \vspace{0.75 in}
\end{defn}
where $a,r$ are fixed and $a\neq 0.$

\begin{prob}
    Identify if each is a geometric series. If it is, what are $a,r$?
    \begin{enumerate}
        \item $1+\frac{1}{2}+\frac{1}{4}+...+\left(\frac{1}{2}\right)^{n-1}+....$ \vspace{0.25 in}
        \item $2-\frac{2}{3}+\frac{2}{9}-\frac{2}{27}+...$ \vspace{0.25 in}
        \item $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$ \vspace{0.25 in}
        \item $5+5+5+5+...$ \vspace{0.25 in}
    \end{enumerate}
\end{prob}

\subsubsection{Convergence of the Geometric Series}

\textbf{Goal: Write $S_n$ in terms of $a, r$.} This way we know what the partial sum is of any geometric series. Consider 

\begin{align*}
    S_n&=a+ar+...+ar^{n-1}\\
\end{align*}
\vspace{2 in}

\begin{theorem}
    If \mbox{\parbox[t]{35ex}{\hrulefill}}in a geometric series, then
    \vspace{0.5 in}

    If \mbox{\parbox[t]{35ex}{\hrulefill}} in a geometric series, then it  \mbox{\parbox[t]{35ex}{\hrulefill}}. 
\end{theorem}

\begin{prob}
    Decide whether each geometric series converges or diverges. If it converges, what is its sum?
    \begin{enumerate}
        \item $1+\frac{1}{2}+\frac{1}{4}+...+\left(\frac{1}{2}\right)^{n-1}+....$ \vspace{0.5 in}
        \item $2-\frac{2}{3}+\frac{2}{9}-\frac{2}{27}+...$ \vspace{0.5 in}
        \item $5+5+5+5+...$ \vspace{0.5 in}
    \end{enumerate}
\end{prob}


 
 
\section{5.3 The Divergence and Integral Tests} 


\begin{theorem}
    \textbf{Divergence Test} If $\lim_{n\rightarrow\infty}a_n=c\neq 0$ or does not exist, then $\sum_{n=1}^\infty a_n$ \mbox{\parbox[t]{25ex}{\hrulefill}}. 
\end{theorem}



\pagebreak 
\begin{ex}
    Consider 
    \[\sum_{n=1}^\infty \frac{1}{n},\ \int_1^\infty \frac{1}{x}dx.\]

    Let $f(x)=\frac{1}{x}$. Then

    \vspace{1.5 in}

    
\end{ex}

\begin{theorem}\textbf{Integral Test} Suppose $\sum_{n=1}^\infty a_n$ is a series with positive terms. Suppose there exists a function $f$and a positive integer $N$ such that the following three conditions are satisfied:

\begin{enumerate}
    \item
    \item
    \vspace{0.25 in}
    \item
    \vspace{0.25 in}
\end{enumerate}

\vspace{0.5 in}

Then

\vspace{0.75 in}

both \mbox{\parbox[t]{35ex}{\hrulefill}} or both \mbox{\parbox[t]{35ex}{\hrulefill}}. 
    
\end{theorem}

\begin{prob}
    Use the integral test to decide whether each series converges or diverges.
    \begin{enumerate}
        \item $\sum_{n=1}^\infty \frac{1}{n^p}$, $p>1$ \vspace{1.5 in}
    \end{enumerate}
\end{prob}

\begin{defn}
    For any real number $p$, the series
    
    \vspace{0.5 in}
    
    is called a \mbox{\parbox[t]{35ex}{\hrulefill}}.
\end{defn}

\section{5.4 Comparison Tests}

\begin{theorem}
    \begin{enumerate}
        \item Suppose there exists an integer $N$   such that $0\leq a_n \leq b_n$ for all $n\geq N$. If \mbox{\parbox[t]{35ex}{\hrulefill}}, then  \mbox{\parbox[t]{35ex}{\hrulefill}}

         \item Suppose there exists an integer $N$   such that $a_n \geq b_n\geq 0$ for all $n\geq N$. If \mbox{\parbox[t]{35ex}{\hrulefill}}, then  \mbox{\parbox[t]{35ex}{\hrulefill}}
    \end{enumerate}
\end{theorem}

\begin{ex}
    We investigate the convergence or lack thereof of $\sum \frac{1}{n^3+3n+1}$.


\end{ex}
\vspace{2 in}

\begin{prob}
    Investigate the convergence or lack thereof of $\sum \frac{1}{2^n-1}$.
    \vspace{1 in}
\end{prob}

\begin{theorem}
    \textbf{Limit Comparison Test} Let $a_n, b_n>0$ for all $n\geq 1$. 
    \begin{enumerate}
        \item
        \item \vspace{0.3 in}
        \item \vspace{0.3 in} 
    \end{enumerate}
    \vspace{0.3 in}
\end{theorem}

\begin{ex}
    We use LCT to determine the convergence or divergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n^2}$. 

    \vspace{2 in}
\end{ex}

\pagebreak
\begin{prob}
    Using LCT, determine whether or not each series converges or diverges.
    \begin{enumerate}
        \item $\sum \frac{2n+1}{n^2+2n+1}$ \vspace{2 in}
        \item $\sum \frac{5^n}{3^n+2}$ \vspace{2 in}
    \end{enumerate}
\end{prob}



\section{5.5 Alternating Series}

\begin{defn}
    Any series whose terms alternate between positive and negative values is called an \mbox{\parbox[t]{15ex}{\hrulefill}} \mbox{\parbox[t]{15ex}{\hrulefill}}. An \mbox{\parbox[t]{35ex}{\hrulefill}} can be written in the form

    \vspace{1 in}
\end{defn}

\begin{theorem}
    \textbf{Alternating Series Test} An alternating series converges if 
    \begin{enumerate}
        \item 
        \item 
        \vspace{0.25 in}
    \end{enumerate}
    \vspace{0.25 in}
\end{theorem}

\begin{ex}
    Consider
     \[\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}.\]
     \vspace{0.75 in}
\end{ex}

\begin{theorem}
    \textbf{Remainders in Alternating Series} Consider an alternating series that satisfies the hypotheses of the alternating series test. Let $S$ denote the sum of the series and $S_N$ denote the $N$th partial sum. For any integer $N\geq1$,   the remainder \mbox{\parbox[t]{35ex}{\hrulefill}} satisfies
\vspace{0.5in}
\end{theorem}

\begin{defn}
    A series $\sum a_n$   exhibits \mbox{\parbox[t]{35ex}{\hrulefill}} if  \mbox{\parbox[t]{35ex}{\hrulefill}}. A series $\sum a_n$   exhibits \mbox{\parbox[t]{35ex}{\hrulefill}} if\mbox{\parbox[t]{35ex}{\hrulefill}}, but\mbox{\parbox[t]{35ex}{\hrulefill}}.
\end{defn}


\begin{prob}
    Decide whether or not $\sum \frac{\cos(n\pi)}{n^2}$ and $\sum \frac{\cos(n\pi)}{n}$  are alternating series and whether they converge or diverge. If they converge, does they converge absolutely or conditionally?
\end{prob}

\vspace{1 in}

\section{5.6 Ratio and Root Tests}

    
\begin{theorem}
    \textbf{Ratio Test} Let $\sum a_n$ be any series be a series with nonzero terms. Let
    
    \vspace{0.75 in}
    
    \begin{enumerate}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
    \end{enumerate}
\end{theorem}

\textbf{Note:} This extends the knowledge we already had for geometric series. 

\begin{theorem}
    \textbf{Root Test} Consider the series $\sum a_n$. Let
    
    \vspace{0.75 in}
    
    \begin{enumerate}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
        \item If \mbox{\parbox[t]{15ex}{\hrulefill}}, then \mbox{\parbox[t]{45ex}{\hrulefill}}
    \end{enumerate}
\end{theorem}

\begin{prob}
    Determine if the following series converge absolutely.
    \begin{enumerate}
        \item $\sum \frac{2^n}{n!}$ \vspace{1.5 in}
        \item $\sum \frac{(-1)^n(n!)^2}{(2n)!}$ \vspace{1.5 in}
        \item $\sum \left( \frac{1}{n}\right)^n$ \vspace{1.5 in}
    \end{enumerate}
\end{prob}



\section{6.1 Power Series and Functions}

\begin{defn}
    A series of the form\\
 
\vspace{0.5 in}

is a \mbox{\parbox[t]{55ex}{\hrulefill}}. A series of the form\\

\vspace{0.5 in}

is a \mbox{\parbox[t]{55ex}{\hrulefill}}. 
\end{defn}

\begin{ex}
    \[\sum_{n=0}^\infty x^n \hspace{4 in}\]

    \vspace{0.25 in}
    
\end{ex}

\begin{theorem}
    \textbf{Convergence of a Power Series}
    Consider the power series $\sum_{n=0}^\infty c_n(x-a)^n$. The series satisfies exactly one of the following properties:

\begin{enumerate}
    \item
    \item \vspace{0.25 in}
    \item \vspace{0.25 in}
\end{enumerate}
\vspace{0.25 in}
\end{theorem}

\begin{defn}
    Consider the power series  $\sum_{n=0}^\infty c_n(x-a)^n$. The set of real numbers x where the series converges is the \mbox{\parbox[t]{35ex}{\hrulefill}}. If there exists a real number $R>0$ such that the series \mbox{\parbox[t]{25ex}{\hrulefill}} \mbox{\parbox[t]{15ex}{\hrulefill}} and \mbox{\parbox[t]{45ex}{\hrulefill}}, then $R$ is the \mbox{\parbox[t]{40ex}{\hrulefill}}. If the series converges only at  $x=a$, we say the radius of convergence is \mbox{\parbox[t]{20ex}{\hrulefill}}. If the series converges for all real numbers x, we say the radius of convergence is \mbox{\parbox[t]{20ex}{\hrulefill}}. 
\end{defn}

\textbf{How to Test a Power Series for Convergence}
\begin{enumerate}
    \item Use the \mbox{\parbox[t]{45ex}{\hrulefill}} to find the largest open interval where the series converges 
    \vspace{0.25 in}
    \item 
    \item 
\end{enumerate}

\begin{ex}
    Determine where the Power Series $\sum (-1)^{n-1} \frac{x^n}{n}$ converges or diverges. 

    \vspace{2.75 in}
\end{ex}

\begin{prob}
    Determine where the power series below converge or diverge.
    \begin{enumerate}
        \item $\sum \frac{x^n}{n!}$ \vspace{1.5 in}
        \item $\sum n!x^n$ \vspace{1.25in}
    \end{enumerate}
\end{prob}

\pagebreak
\section{6.2 Properties of Power Series}

\begin{theorem}
    \textbf{Combining Power Series}
    Suppose that the two power series $\sum_{n=0}^\infty c_nx^n$ and  $\sum_{n=0}^\infty d_nx^n$ converge to the functions $f$ and $g$, respectively, on a common interval $I$.
    \begin{enumerate}
    \item 
    \item \vspace{0.25 in}
    \item \vspace{0.25 in}
  \end{enumerate}
  \vspace{0.25 in}
\end{theorem}


\begin{theorem}
   Suppose that the power series $\sum_{n=0}^\infty c_nx^n$ and $\sum_{n=0}^\infty d_nx^n$ converge to $f$ and $g$, respectively, on a common interval $I$. Let

   \vspace{0.5 in}
   
   Then
   
   \vspace{0.5 in}
   
   and

   \vspace{0.5 in}
   
\end{theorem}


\begin{theorem}
     \textbf{Term-by-Term Differentiation and Integration of Power Series} Suppose that the power series $\sum_{n=0}^\infty c_n x^n$ converges on the interval $(a-R,a+R)$ for some $R>0$. Let $f$ be the function defined by the series

   \vspace{0.5 in}
  
   Then $f$ is \mbox{\parbox[t]{20ex}{\hrulefill}} on the interval $(a-R,a+R)$
  and we can find $f'$
  by differentiating the series term-by-term:

  \vspace{0.5 in}
  
  for $|x-a|<R.$
  Also, to find \mbox{\parbox[t]{20ex}{\hrulefill}},
  we can integrate the series term-by-term. The resulting series converges on $(a-R,a+R)$,
  and we have

   \vspace{0.5 in}
  
  for $|x-a|<R.$
\end{theorem}


\textit{Warning! This may not work for series that are not Power Series.} 

\pagebreak
\begin{ex}
Let 
\[f(x)=\sum_0^\infty \frac{(-1)^nx^{2n+1}}{2n+1},\ -1\leq x \leq 1\]

We identify this as a function we more commonly know.

\vspace{2 in}

\end{ex}


\section{6.3 Taylor and Maclaurin Series} 
 
\begin{defn}
    If $f$ has derivatives of all orders at $x=a$, then the \mbox{\parbox[t]{45ex}{\hrulefill}} is
    \vspace{1 in}

    The Taylor series for $f$ at \mbox{\parbox[t]{10ex}{\hrulefill}} is known as the \mbox{\parbox[t]{45ex}{\hrulefill}}
\end{defn}



\begin{defn}
    If $f$ has $n$ derivatives at $x=a$, then the $n$th \mbox{\parbox[t]{35ex}{\hrulefill}} for $f$ at a is
    \vspace{1 in}
\end{defn}

\begin{ex}
    We find the Taylor series generated by $1/x^2$ at $a=1$.
\vspace{2.5 in}
\end{ex}

\pagebreak
\begin{prob}
    Find the Taylor Series generated by $f(x)=x^3$ at $x=3$.
\end{prob}
\vspace{2.5 in}

\begin{ex}
    We find the Taylor Polynomial of degree $n$ of $e^x$.
\end{ex}
\vspace{2.2 in}


\begin{theorem}
    \textbf{Taylor's Theorem with Remainder} Let $f$  be a function that can be differentiated $n+1$  times on an interval $I$ containing the real number $a$. Let $p_n$ be the $n$th Taylor polynomial of $f$ at $a$ and let

    \vspace{0.5 in}
    
    be the \mbox{\parbox[t]{35ex}{\hrulefill}}. Then for each $x$ in the interval $I$, there exists a real number $c$ between $a$ and $x$ such that

    \vspace{0.5 in}
    
    If there exists a real number $M$ such that  \mbox{\parbox[t]{35ex}{\hrulefill}} for all $x\in I$,
    then 
  
    \vspace{0.25 in}
    
    for all $x$ in $I$.
\end{theorem}



    \begin{ex}
        We find the Taylor Series of $\sin(x)$. 

        \vspace{2 in}
    \end{ex}
       


    
   


    \section{6.4 Working with Taylor Series} 
    \textbf{Note:} We can use the following from here on without proof
    \begin{align*}
        e^x&= \hspace{4 in}\\
        \cos(x)&=\\
        \sin(x)&=\\
    \end{align*}
    
    \begin{ex}
      We express $\int e^{-x^2} dx$ as an infinite series.

      \vspace{4.5 in}
    \end{ex}

    \begin{prob}
        Find the Taylor Series for $\sinh(x)$. \textit{Hint:} Recall $\sinh(x) = \frac{e^x-e^{-x}}{2}$.

        \vspace{4.5 in}
    \end{prob}


    \section{7.1 Parametric Equations}

\begin{defn}
   If $x$ and $y$ are continuous functions of $t$ on an interval $I$, then the equations

   \vspace{0.5 in}
   
   are called \mbox{\parbox[t]{35ex}{\hrulefill}} and  \mbox{\parbox[t]{15ex}{\hrulefill}} is called the \mbox{\parbox[t]{35ex}{\hrulefill}}. The set of points $(x,y)$ obtained as $t$ varies over the interval $I$ is called the  \mbox{\parbox[t]{45ex}{\hrulefill}}. The graph of parametric equations is called a \mbox{\parbox[t]{35ex}{\hrulefill}} or plane curve, and is denoted by $C$.
\end{defn}

\begin{ex}
    Consider $x=\sin \frac{\pi t}{2}, y=2t+4, 0\leq t \leq 4$.

    \vspace{2 in}
\end{ex}

\begin{prob}
    Sketch the curve
    \[x=3t+2,\  y=t^2+1,\  -\infty <t< \infty. \]

    \vspace{2 in}
\end{prob}

\begin{ex}
    We find two different parametric equations to represent the graph of $y=2x^2+3$.

    \vspace{1 in}
\end{ex}

\pagebreak
\subsection{Cycloids}

\begin{ex}
    A wheel of radius $a$ rolls along a horizontal straight line. We find parametric equations for the path traced by a point on the wheel. The path is called a \mbox{\parbox[t]{35ex}{\hrulefill}}.

    \vspace{2 in}
\end{ex}


\section{7.2 Calculus of Parametric Curves}

\subsection{Derivatives of Parametric Equations}
\textbf{Parametric Formula for $\frac{dy}{dx}$}

\vspace{1 in}


\textbf{Parametric Formula for $\frac{d^2y}{dx^2}$}

\vspace{1 in}


\begin{ex}
    Find the tangent line to the plane curve defined by the parametric equations  
  \[x(t)=t^2-3,\ y(t)=2t-1,\ t\geq 0\]
  at $t=0$.
\end{ex}

\vspace{2 in}

\pagebreak
\begin{prob}
Find $\frac{d^2y}{dx^2}$ as a function of $t$ if $x=1-t^2$, $y=t-t^2$.

\vspace{2.5 in}


\end{prob}


\begin{ex}
   We set up, but do not evaluate, an integral that gives the area under the curve of the cycloid defined by the equations
    \[x=t-\sin(t),\ y=1-\cos(t),\ 0\leq t \leq 2\pi.\]

    \vspace{2.5 in}
\end{ex}


\begin{defn}
    Consider the plane curve defined by the differentiable parametric equations $x=x(t)$, $y=y(t)$, $t_1\leq t\leq t_2$. Then the \mbox{\parbox[t]{30ex}{\hrulefill}} is given by
   \[L= \hspace{3 in}\]
\end{defn}


\begin{prob}
    Find the length of the circle of radius $r$ defined parametrically by 
    \[x=r \cos t,\ y=\sin t,\ 0 \leq t \leq 2 \pi.\]

    \vspace{2 in}
\end{prob}


\textbf{Areas of Surface of Revolution for Parametrized Curves}

\begin{enumerate}
    \item \textbf{Revolution about the $x$-axis:}
    \vspace{1 in}
    \item \textbf{Revolution about the $y$-axis:}
    \vspace{1 in}
\end{enumerate}



\section{7.3 Polar Coordinates} 


\begin{defn}
    The point $P$ has Cartesian coordinates $(x,y).$
  The line segment connecting the origin to the point $P$
  measures the distance from the origin to $P$
  and has \mbox{\parbox[t]{30ex}{\hrulefill}}.
  The \mbox{\parbox[t]{30ex}{\hrulefill}} \mbox{\parbox[t]{60ex}{\hrulefill}}.
  
  This is the basis of the \mbox{\parbox[t]{35ex}{\hrulefill}}. In the \mbox{\parbox[t]{35ex}{\hrulefill}}, each point also has two values associated with it: \mbox{\parbox[t]{30ex}{\hrulefill}}.
\end{defn}

\begin{ex}
    We find all of the polar coordinates for the point $P(2,\pi/6)$.

   \vspace{2 in}
\end{ex}



\begin{theorem}
    \textbf{Converting Points between Coordinate Systems:} Given a point $P$ in the plane with \mbox{\parbox[t]{15ex}{\hrulefill}} \mbox{\parbox[t]{25ex}{\hrulefill}} and \mbox{\parbox[t]{40ex}{\hrulefill}} the following conversion formulas hold true:

  \vspace{1 in}
\end{theorem}

\pagebreak
Some curves are easier to express in polar and some are easier to express in Cartesian.


\begin{ex}
    We find the polar equation for the circle $x^2+(y-3)^2=3^2$ (circle centered at (0,3) with radius 3).

    \vspace{2 in}
\end{ex}

\subsection{Polar Curves}

\begin{ex}
    We graph $r=4\sin \theta$.

\vspace{2 in}
\end{ex}


\begin{prob}
    Graph the curve $r= 1-\cos(\theta).$

    \vspace{2 in}

\end{prob}




\begin{ex}
Polar objects can have multiple representations

    \begin{itemize}
        \item 
        \item \vspace{0.5 in}
    \end{itemize}
    \vspace{0.5 in}
\end{ex}


\pagebreak
\begin{prob}
    Graph the sets of points using the conditions:

    \begin{itemize}
        \item $1\leq r\leq 3$ and $0\leq \theta \leq \pi/2$

        \vspace{0.75 in}
        
        \item $-3\leq r\leq 2$ and $\theta = \pi/4$

         \vspace{0.75 in}
         
        \item $2\pi/3\leq \theta \leq 5\pi/6$

         \vspace{0.75 in}
         
    \end{itemize}
    
\end{prob}


\subsection{Transforming Polar Equations to Rectangular Coordinates}


\begin{prob}
    Write the polar equation as a Cartesian equation.
    \begin{itemize}
        \item $r \cos(\theta)=2 \hspace{5 in}$

        \vspace{0.5 in}

        \item $r=6\cos\theta-8 \sin \theta \hspace{5 in}$

\vspace{0.5 in}

    \item $r =1-\cos(\theta) \hspace{5 in}$
    %x^4+y^4+2x^2y^2+2x^3+2xy^2-y^2=0
    \end{itemize}
    \vspace{0.5 in}
    
\end{prob}





\section{7.4 Area and Arc Length in Polar Coordinates} 

\subsection{Slope of Polar Curves}

\textbf{Recall:} Slope of a curve in Cartesian is $\frac{dy}{dx}$. This is \textbf{not} true in polar. When $r=f(\theta)$:

\vspace{2 in}


\pagebreak 

\textbf{Areas of Regions Bounded by Polar Curves}

\vspace{1 in}

\begin{theorem}
    Suppose $f$ is continuous and nonnegative on the interval $\alpha\leq \theta \leq \beta$ with  \mbox{\parbox[t]{30ex}{\hrulefill}}. The area of the region bounded by the graph of  \mbox{\parbox[t]{30ex}{\hrulefill}}
  between the radial lines $\theta=\alpha$ and $\theta = \beta$ is
  \vspace{0.5 in}
\end{theorem}


\begin{prob}
    Find the area of the region enclosed by $r=1-\cos\theta$.

    \vspace{2 in}
\end{prob}


\begin{ex}
    We find the area of the region that lies outside the cardioid $r=2+2\sin \theta$ and inside the circle $r=6\sin\theta$.
   \vspace{2.5 in}
\end{ex}




\subsection{Arc Length in Polar Coordinates}


\begin{theorem}
    Let $f$ be a function whose derivative is continuous on an interval $\alpha\leq \theta \leq \beta$. The length of the graph of $r=f(\theta)$ from $\theta = \beta$ to $\theta=\beta$ is
  \vspace{0.5 in}
\end{theorem}

\pagebreak
\begin{prob}
    Find the arc length of the $r=2+2 \cos \theta$.

    \vspace{2 in}
\end{prob}



\section{2.1 Vectors in the Plane}

\begin{defn}
   A \mbox{\parbox[t]{30ex}{\hrulefill}} is a quantity that has both \mbox{\parbox[t]{30ex}{\hrulefill}} and \mbox{\parbox[t]{30ex}{\hrulefill}}.
\end{defn}

\begin{defn}
   Vectors are said to be \mbox{\parbox[t]{30ex}{\hrulefill}} vectors if they have the \mbox{\parbox[t]{20ex}{\hrulefill}} \mbox{\parbox[t]{45ex}{\hrulefill}}.
\end{defn}

\begin{defn}
    The vector with initial point $(0,0)$ and terminal point $(x,y)$
  can be written in component form as
  
  \vspace{0.5 in}
  
  The scalars $x$ and $y$
  are called the \mbox{\parbox[t]{30ex}{\hrulefill}} of $\textbf{v}$.
\end{defn}


\begin{ex}
    Consider $\textbf{v}=\overrightarrow{\rm PQ}$ with $P(-3,4)$ and $Q(-5,2)$. The vector $\textbf{v}$ has components 
    \begin{itemize}
        \item 
        \item 
    \end{itemize}

    So the component form is \mbox{\parbox[t]{30ex}{\hrulefill}}.  The length is

    \vspace{0.5 in}
\end{ex}

\subsection{Combining Vectors}

\begin{defn}
    Let $k$ be a \textbf{scalar} (a real number). Then if $\textbf{u,v}$ are vectors then we have 
    \begin{itemize}
        \item Addition/Subtraction: 
        \item Scalar Multiplication: 
    \end{itemize}
\end{defn}


\begin{prob}
    If $\textbf{u}= \langle -1,3\rangle $ and $\textbf{u}= \langle 4,7\rangle$ find:
    \begin{itemize}
        \item $2\textbf{u}-3\textbf{v}$

        \vspace{0.5 in}
        \item $\|\frac{1}{2}\textbf{u}\|$
        
         \vspace{0.5 in}
        \item $\frac{1}{2} \|\textbf{u}\|$

         \vspace{0.5 in}
    \end{itemize}
\end{prob}

\textbf{Properties of Vector Operations}
\begin{enumerate}
    \item $\textbf{u}+\textbf{v}=$
    \item $\textbf{u}+\textbf{0}=$
     \item $0\textbf{u}=$
     \item $a(b\textbf{u})=$
     \item $(a+b)\textbf{u}=$
     \item $(\textbf{u} + \textbf{v)} + \textbf{w} =$
     \item $\textbf{u}-\textbf{u}=$
     \item $1\textbf{u}=$
     \item $a(\textbf{u}+\textbf{v})=$
\end{enumerate}

\subsection{Unit Vectors}

\begin{defn}
    A \mbox{\parbox[t]{40ex}{\hrulefill}} is a vector with \mbox{\parbox[t]{40ex}{\hrulefill}}. For any nonzero vector $\textbf{v}$, we can use scalar multiplication to find a unit vector $\textbf{u}$ that has the same direction as v.

    \vspace{0.5 in}
\end{defn}


\begin{prob}
    Find a unit vector \textbf{u} in the direction of the vector from $P_1(1,0)$ and $P_2(3,2)$.
\end{prob}

\vspace{2 in}

\section{2.2 Vectors in Three Dimensions}

\begin{defn}
    The \mbox{\parbox[t]{40ex}{\hrulefill}} rectangular coordinate system consists of three perpendicular axes: the x-axis, the y-axis, \mbox{\parbox[t]{40ex}{\hrulefill}}, and an origin at the point of intersection of the axes. 
\end{defn}

\begin{theorem}
The \mbox{\parbox[t]{30ex}{\hrulefill}} between points \mbox{\parbox[t]{20ex}{\hrulefill}} and \mbox{\parbox[t]{20ex}{\hrulefill}} is given by the formula

  \vspace{0.5 in}
  
\end{theorem}

\begin{defn}
    A \mbox{\parbox[t]{25ex}{\hrulefill}} is the set of all points in space \mbox{\parbox[t]{40ex}{\hrulefill}} from a fixed point, the \mbox{\parbox[t]{25ex}{\hrulefill}} of the sphere. In a sphere, the distance from the center to a point on the sphere is called the \mbox{\parbox[t]{25ex}{\hrulefill}}.

    The sphere with center $(a,b,c)$ and radius $r$ can be represented by the equation
    \vspace{0.5 in}
\end{defn}

\subsection{Graphing Other Equations in Three Dimensions}

\begin{ex}
    We describe the set of points in three-dimensional space that satisfies  $(x-2)^2+(y-1)^2=4$, and graph the set.

    \vspace{0.5 in}
\end{ex}

\subsection{Working with Vectors in 3D}

\begin{ex}
    Let $\overrightarrow{\rm PQ}$
  be the vector with initial point $P=(3,12,6)$ and terminal point  $Q=(-4,-3,2)$. We express $\overrightarrow{\rm PQ}$ in both component form and using standard unit vectors.

  \vspace{2 in}
\end{ex}

\begin{prob}
    If $\textbf{u}= \langle -1,3,0\rangle $ and $\textbf{v}= \langle 4,7,11\rangle$ find:
    \begin{itemize}
        \item $2\textbf{u}-3\textbf{v}$

        \vspace{0.5 in}
        \item $\|\frac{1}{2}\textbf{u}\|$
        
         \vspace{0.5 in}
        \item $\frac{1}{2} \|\textbf{u}\|$

         \vspace{0.5 in}
    \end{itemize}
\end{prob}

\pagebreak
\section{2.3 The Dot Product}

\begin{defn}
    The \mbox{\parbox[t]{30ex}{\hrulefill}} of two vectors is $\textbf{u}=\langle u_1, u_2, u_3 \rangle$ and $\textbf{v}=\langle v_1, v_2, v_3 \rangle$
    \vspace{1 in}
\end{defn}

\begin{theorem}
    The \mbox{\parbox[t]{30ex}{\hrulefill}} of two vectors is the product of the \mbox{\parbox[t]{30ex}{\hrulefill}} of each vector and the \mbox{\parbox[t]{30ex}{\hrulefill}} of the angle between them:
    
    \vspace{1 in}
\end{theorem}



\begin{prob}
    \begin{enumerate}
        \item Find the dot product of $\textbf{u}=\langle 1,-2,-2 \rangle$ and $\textbf{v}=\langle -6,2,-3 \rangle$.

        \vspace{1 in}

        \item Find the angle between $\textbf{u} =  \textbf{i} -2\textbf{j}-2\textbf{k}$ and $\textbf{v} =  6\textbf{i} +3\textbf{j}+0\textbf{k}$.

        \vspace{1 in}
    \end{enumerate}
\end{prob}


\begin{theorem}
    The nonzero vectors $\textbf{u}$ and $\textbf{v}$ are \mbox{\parbox[t]{30ex}{\hrulefill}} if and only if \mbox{\parbox[t]{25ex}{\hrulefill}}.
\end{theorem}

\textbf{Properties of the Dot Product}

\begin{enumerate}
    \item $\textbf{u} \cdot \textbf{v}=$
     \item $c\textbf{u} \cdot \textbf{v}= $
    \item $\textbf{u} \cdot (\textbf{v}+\textbf{w})=$
    \item $\textbf{u} \cdot \textbf{u}= $
    \item $\textbf{0} \cdot \textbf{u}= $
\end{enumerate}

\pagebreak

\subsection{Vector Projections}

The \mbox{\parbox[t]{65ex}{\hrulefill}} is the vector labeled \mbox{\parbox[t]{25ex}{\hrulefill}}. It has the same \mbox{\parbox[t]{35ex}{\hrulefill}} as $\textbf{u}$ and $\textbf{v}$ and the same \mbox{\parbox[t]{35ex}{\hrulefill}}, and represents the \mbox{\parbox[t]{35ex}{\hrulefill}} that acts in the \mbox{\parbox[t]{35ex}{\hrulefill}}. If $\theta$ represents the angle between $\textbf{u}$ and $\textbf{v}$, then the length of $proj_\textbf{u}\textbf{v}$ is \mbox{\parbox[t]{15ex}{\hrulefill}}. When expressing $\cos\theta$
  in terms of the dot product, this becomes
    \vspace{2 in}
    
    We now multiply by a unit vector in the direction of $\textbf{u}$ to get $proj_u v$

    \vspace{2 in}
    The length of this vector is also known as the \mbox{\parbox[t]{65ex}{\hrulefill}} and is denoted by
    \vspace{2 in}

\begin{prob}
    Find the vector projection of $\textbf{u}=6\textbf{i} +3\textbf{j}+2\textbf{k}$ onto $\textbf{v}=\textbf{i} -0\textbf{j}-0\textbf{k}$.
\end{prob}

\vspace{2 in}



\section{2.4 The Cross Product} 


\begin{defn}
    The \textbf{cross product} of two vectors is
    
    \vspace{0.5 in}
    
    where $\textbf{n}$ is the \mbox{\parbox[t]{75ex}{\hrulefill}}
\end{defn}

\textbf{Parallel Vectors}
Nonzero vectors $\textbf{u}, \textbf{v}$ are \mbox{\parbox[t]{15ex}{\hrulefill}} if and only if \mbox{\parbox[t]{25ex}{\hrulefill}}.

\textbf{Properties of the Cross Product} If $\textbf{u}, \textbf{v}, \textbf{w}$ are vectors and $r,s$ are scalars, then 

\begin{enumerate}
    \item $r(\textbf{u}) \times (s\textbf{v})=$
    \item $\textbf{u} \times (\textbf{v}+\textbf{w}) =$
    \item $\textbf{v} \times \textbf{u}=$
    \item $(\textbf{v}+\textbf{w}) \times \textbf{u} = $
    \item $\textbf{0} \times \textbf{u}=$
    \item $\textbf{u} \times (\textbf{v} \times \textbf{w}) =$
\end{enumerate}

\begin{ex} \textbf{Area of a Parallelogram}
\vspace{1 in}

\end{ex}

\begin{ex}
    We find the cross product of the three dimensional vectors $\textbf{u}= 2 \textbf{i}+\textbf{j} + \textbf{k}$ and $\textbf{v} = -4\textbf{i} +3 \textbf{j}+ \textbf{k} $.
    \vspace{2.5 in}
\end{ex}

\begin{prob}
    Find the cross product of the three dimensional vectors $\textbf{u}=  \textbf{i}$ and $\textbf{v} = \textbf{j} $.
    \vspace{2.5 in}
\end{prob}

\section{2.5 Equations of Lines and Planes in Space} 

\textbf{Recall:}
\begin{itemize}
    \item Slope-intercept form of a line: $y=mx+b$
    \item A parametric form of a line: $x(t)=m_1t+x_0,\ y(t)=m_2t+b, -\infty <t<\infty$
\end{itemize}

\begin{defn}
    A \mbox{\parbox[t]{20ex}{\hrulefill}} parallel to vector \mbox{\parbox[t]{20ex}{\hrulefill}} and passing through point \mbox{\parbox[t]{20ex}{\hrulefill}} can be described by the following parametric equations:
    
    \vspace{1 in}
    
    If the constants $a,b,$ and $c$ are all nonzero, then $L$ can be described by the symmetric equation of the line:
  
    \vspace{1 in}
\end{defn}

\begin{prob}
    Find parametric and symmetric equations of the line passing through points $(1,0,-2)$ and $(-3,5,0)$.
    
    \vspace{1 in}
\end{prob}

\begin{ex}
A mouse travels from its home (the origin) to a piece of  cheese in the direction of the point (1,1,1) at a speed of 60 cm per second. What is its position after 10 seconds?

\vspace{2.75 in}
\end{ex}


\subsection{Distance between a Point and a Line}

\vspace{1.5 in}



\subsection{Equations for a Plane}

\begin{defn}
    Given a point $P$ and vector $\textbf{n}$, the set of all points $Q$ satisfying the equation \mbox{\parbox[t]{20ex}{\hrulefill}} forms a \mbox{\parbox[t]{15ex}{\hrulefill}}. The equation
    
    \vspace{0.5 in}
    
    is known as the \mbox{\parbox[t]{45ex}{\hrulefill}}. 
    
    The \mbox{\parbox[t]{45ex}{\hrulefill}} containing point $P=(x_0,y_0,z_0)$ with normal vector $\textbf{n}=\langle a,b,c\rangle$ is

    \vspace{0.5 in}
    
This equation can be expressed as \mbox{\parbox[t]{45ex}{\hrulefill}} where \mbox{\parbox[t]{30ex}{\hrulefill}}. This form of the equation is sometimes called the \mbox{\parbox[t]{55ex}{\hrulefill}}
\end{defn}


\begin{prob}
    Find an equation for the plane through $P_0(-3,0,7)$ perpendicular to \[\textbf{n}=2\textbf{j} - \textbf{k}. \]
    \vspace{1 in}
\end{prob}


\begin{ex}
    We find an equation for the plane through $A(0,0,1)$, $B(2,0,0)$, and $C(0,3,0)$.


    \vspace{2 in}
\end{ex}

\pagebreak

\begin{prob}
    Find a vector parallel to the line of intersection of the planes $3x-6y-2z=15$ and $2x+y-2z=5$. \textit{Hint: This line of intersection is perpendicular to both planes normal vectors.}
    \vspace{3 in}
\end{prob}


\begin{ex}
    We find the point where the line 
    \[x(t)= \frac{8}{3}+2t,\ y(t) = -2t,\ z(t) = 1+t\]
    intersects the plane $3x+2y+6z=6$.

    
\vspace{2.5 in}
\end{ex}


\section{3.1 Vector-Valued Functions and Space Curves}

\begin{defn}
    A \mbox{\parbox[t]{45ex}{\hrulefill}} is a function of the form

    \vspace{1 in}
 
where the \mbox{\parbox[t]{45ex}{\hrulefill}} $f,g,h$, are real-valued functions of the parameter $t$. Vector-valued functions are also written in the form

    \vspace{1 in}
\end{defn}

\begin{ex}
    We sketch $\textbf{r}(t) = \cos(t) \textbf{i} + \sin(t) \textbf{j} + t \textbf{k}.$ 

    \vspace{2 in}
\end{ex}

\begin{prob}
    Describe how the following compare to  $\textbf{r}(t) = \cos(t) \textbf{i} + \sin(t) \textbf{j} + t \textbf{k}.$ 

    \begin{itemize}
        \item  $\textbf{r}(t) = \cos(2t) \textbf{i} + \sin(2t) \textbf{j} + t \textbf{k}.$ 
        \item  $\textbf{r}(t) = \cos(t) \textbf{i} + \sin(t) \textbf{j} + 2t \textbf{k}.$ 
    \end{itemize}
\end{prob}

\begin{defn}
    A vector-valued function r approaches the \mbox{\parbox[t]{25ex}{\hrulefill}} as t approaches a, written
    
    \vspace{0.75 in}
    
    provided
    
    \vspace{0.75 in}
\end{defn}

\begin{theorem}
    Let f, g, and h be functions of t. Then the limit of the vector-valued function  $\textbf{r}(t)=f(t)\textbf{i}+g(t)\textbf{j} +h(t)\textbf{k}$  as $t$ approaches $a$ is given by

\vspace{1 in}
 
provided the limits exist.

\end{theorem}

\begin{prob}
    Let $\textbf{r}(t) = \frac{2t-4}{t+1} \textbf{i}+ \frac{t}{t^2+1} \textbf{j}+ (4t-3) \textbf{k}$. Find $\lim_{t \rightarrow 3} \textbf{r}(t)$. 

    \vspace{1.5 in}
\end{prob}

\pagebreak

\begin{defn}
    Let $f, g, h$ functions of $t$. Then, the vector-valued function $\textbf{r}(t)=f(t)\textbf{i}+g(t)\textbf{j} +h(t)\textbf{k}$ is \mbox{\parbox[t]{45ex}{\hrulefill}} if the following three conditions hold:

    \begin{itemize}
        \item
        \item \vspace{0.25 in}
        \item \vspace{0.25 in}
    \end{itemize}
    \vspace{0.25 in}
\end{defn}

\section{3.2 Calculus of Vector-Valued Functions}

\begin{theorem}
    Let $f, g, h$ be differentiable functions of $t$ and let $\textbf{r}(t)=f(t)\textbf{i}+g(t)\textbf{j} +h(t)\textbf{k}$. Then 

    \vspace{1 in}
\end{theorem}

\begin{prob}
    Let  $\textbf{r}(t)=t \ln(t) \textbf{i}+5e^t\textbf{j} +\cos(t)\textbf{k}$. Find $\textbf{r}'(t)$ and $\textbf{r}''(t)$. 

    \vspace{1 in}
\end{prob}

\subsection{Tangent Vectors and Unit Tangent Vectors}


\begin{defn}
    Let $C$ be a curve defined by a vector-valued function $\textbf{r}$, and assume that $\textbf{r}'(t)$ exists when $t=t_0$. A \mbox{\parbox[t]{45ex}{\hrulefill}} $\textbf{v}$ at $t=t_0$ is any vector such that, when the tail of the vector is placed at point $\textbf{r}(t_0)$ on the graph, vector $\textbf{v}$ is \mbox{\parbox[t]{20ex}{\hrulefill}} to curve $C$. Vector \mbox{\parbox[t]{15ex}{\hrulefill}} is an example of a tangent vector at point  $t=t_0.$ The \mbox{\parbox[t]{45ex}{\hrulefill}} at $t$ is defined to be

    \vspace{0.5 in}
\end{defn}

\begin{prob}
    Find the a tangent vector and the unit tangent vector for each of $\textbf{r}(t) = \cos(t) \textbf{i} + \sin(t) \textbf{j}$.

    \vspace{1 in}
\end{prob}


\subsection{Integrals of Vector-Valued Functions}

\begin{defn}
    The \mbox{\parbox[t]{45ex}{\hrulefill}}of a vector-valued function $\textbf{r}(t)=f(t)\textbf{i}+g(t)\textbf{j}+h(t)\textbf{k}$ is

\vspace{0.5 in}
 

The \mbox{\parbox[t]{45ex}{\hrulefill}} of the vector-valued function is

\vspace{0.5 in}


\end{defn}

\begin{prob}
    Calculate $\int_1^3\left((2t+4)\textbf{i}-t^2 \textbf{j} \right) dt$.

    \vspace{1 in}
\end{prob}

\section{3.3 Arc Length and Curvature}

Recall: Arc Length of a Parametric Curve

\vspace{1 in}

\begin{theorem}
    Given a smooth curve $C$ defined by the function $\textbf{r}(t) = f(t)\textbf{i}+g(t)\textbf{j}+h(t)\textbf{k}$, where $t$ lies within the interval $[a,b],$ the \mbox{\parbox[t]{25ex}{\hrulefill}} of $C$ over the interval is

  \vspace{1 in}
\end{theorem}

\begin{prob}
    Calculate the arc length for $\textbf{r}(t)=\sin(t)\textbf{i}+\cos(t)\textbf{j}+(10-t)\textbf{k}$, from $t=0$ to $t=2\pi$.

    \vspace{2 in}
\end{prob}



\begin{theorem}
    Let  $\textbf{r}(t)$ describe a smooth curve for $t\geq a$. Then the arc-length function is given by

    \vspace{1 in}
    
    Furthermore, \mbox{\parbox[t]{45ex}{\hrulefill}}.
\end{theorem}

\pagebreak 
\subsection{Curvature}

\begin{defn}
    Let $C$ be a smooth curve in the plane or in space given by $\textbf{r}(s)$, where $s$ is the arc-length parameter. The  \mbox{\parbox[t]{45ex}{\hrulefill}} is

    \vspace{1 in}
\end{defn}


\begin{theorem}
    If $C$ is a smooth curve given by $\textbf{r}(t)$,  then the curvature $\kappa$ of $C$ at $t$ is given by

    \vspace{1 in}
    
    or

    \vspace{1 in}
 
    If C is the graph of a function $y=f(x)$ and both $y'$ and $y''$ exist, then the curvature at point $(x,y)$ is given by

    \vspace{1 in}

\end{theorem}

We show the first formula:

\vspace{2 in}

\begin{prob}
    Find the curvature for each of the following curves at the given point:
    \[\textbf{r}(t)=4\cos t\textbf{i}+4\sin t\textbf{j}+3t\textbf{k},t=4\pi/3\]

    \vspace{2 in}
\end{prob}

\subsection{The Normal and Binormal Vectors}

\begin{defn}
    Let $C$ be a three-dimensional smooth curve represented by $\textbf{r}$ over an open interval $I$. If $\textbf{T}(t)\neq0$, then the \mbox{\parbox[t]{45ex}{\hrulefill}} is 
    
    \vspace{0.5 in}
 
    The \mbox{\parbox[t]{45ex}{\hrulefill}} is 

    \vspace{0.5 in}

    where $\textbf{T}(t)$ is the unit tangent vector.
\end{defn}

\textbf{Note:}

\vspace{1 in}


\begin{prob}
  Find the principal unit normal vector and the binormal vector for $\textbf{r}(t) = 4\cos t\textbf{i}+4\sin t\textbf{j}+3t\textbf{k}$.

    \vspace{3 in}
\end{prob}

\begin{defn}
    Suppose we form a circle in the osculating plane of $C$ at point $P$ on the curve. Assume that the circle has the \mbox{\parbox[t]{45ex}{\hrulefill}} as the curve does at point $P$ and let the circle have \mbox{\parbox[t]{25ex}{\hrulefill}}. Then, the \mbox{\parbox[t]{45ex}{\hrulefill}} is given by \mbox{\parbox[t]{20ex}{\hrulefill}} \mbox{\parbox[t]{25ex}{\hrulefill}}.
    
    We call \mbox{\parbox[t]{35ex}{\hrulefill}} of the curve, and it is equal to the reciprocal of the curvature. If this circle lies on the \mbox{\parbox[t]{35ex}{\hrulefill}} side of the curve and is \mbox{\parbox[t]{45ex}{\hrulefill}}  at point $P$, then this circle is called the \mbox{\parbox[t]{45ex}{\hrulefill}}.
\end{defn}

\pagebreak

\begin{ex}
    Find the equation of the osculating circle of the curve defined by the vector-valued function $y=x^2$ at the origin.

    \vspace{2 in}
\end{ex}

\section{3.4 Motion in Space}


\begin{defn}
    Let $\textbf{r}(t)$ be a twice-differentiable vector-valued function of the parameter $t$ that represents the \mbox{\parbox[t]{25ex}{\hrulefill}} of an object as a function of time. The \mbox{\parbox[t]{45ex}{\hrulefill}} is

    \vspace{0.5 in}

    The \mbox{\parbox[t]{45ex}{\hrulefill}} is

    \vspace{0.5 in}

    The \mbox{\parbox[t]{45ex}{\hrulefill}} is

    \vspace{0.5 in}
\end{defn}

\begin{prob}
    Find the velocity, speed, and acceleration of a particle whose path is

    \[\textbf{r}(t) = t^2 \textbf{i} +(t+2) \textbf{j} +3t\textbf{k}.\]

    \vspace{2 in}
\end{prob}

\subsection{Components of the Acceleration Vector}

\vspace{2 in}

\begin{theorem}
    The acceleration vector $\textbf{a}(t)$ of an object moving along a curve traced out by a twice-differentiable function $\textbf{r}(t)$ lies in the plane formed by the \mbox{\parbox[t]{35ex}{\hrulefill}} and the \mbox{\parbox[t]{35ex}{\hrulefill}} to $C$. Furthermore,

    \vspace{1.5 in}
 
    The coefficients of $\textbf{T}(t)$ and $\textbf{N}(t)$ are referred to as the \mbox{\parbox[t]{45ex}{\hrulefill}} and the \mbox{\parbox[t]{45ex}{\hrulefill}}, respectively.
\end{theorem}

\begin{prob}
    A particle moves in a path defined by the vector-valued function $\textbf{r}(t)=t^2\textbf{i}+(2t-3)\textbf{j}+(3t^2-3t)\textbf{k},$  where $t$ measures time in seconds and distance is measured in feet. Find $a_T$ and $a_N$. 

    \vspace{2 in}
\end{prob}

 \end{document}
