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\begin{document}

\input{213-Lecture-Macros}		%% 	macros, settings, et.c


\title{Math 213 - Exam I Review}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{September 18, 2019}
\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%		Due Date Reminders
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Reminders}

\begin{itemize}

\item	Access your WebWork account \emph{only through Canvas}!

\medskip

\item	Homework A5 on section 13.1-13.2  is due tonight!
\item	Exam 1 takes place tonight at 5 PM. Section 17 will meet in 
		CB 118, and sections 18 and 19 will meet in CB 122.

\medskip
		
	
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
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\begin{frame}
\frametitle{Unit I: Geometry and Motion in Space}
\begin{itemize}

\item[{12.1}]		Lecture 1: Three-Dimensional Coordinate Systems
\item[{12.2}] 		Lecture 2: Vectors in the Plane and in Space
\item[{12.3}]		Lecture 3:The Dot Product
\item[{12.4}]		Lecture 4:The Cross Product
\item[{12.5}]		Lecture 5:	Equations of Lines and Planes, I
\item[{12.5}]		Lecture 6: Equations of  Lines and Planes, II
\item[{12.6}]		Lecture 7: Surfaces in Space
\item[{13.1}]		Lecture 8: Vector Functions and Space Curves
\item[{13.2}]		Lecture 9 Derivatives and Integrals of Vector Functions
\begingroup
\color{red}
\item[{}]				Lecture 10:  Exam I Review
\endgroup
\end{itemize}
\end{frame}

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%	Learning Goals
%
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\begin{frame}
\frametitle{Learning Goals}
\begin{itemize}
\item		Find out how to ace Exam I
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		New material begins here...
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\section{Cheat Sheet}

\begin{frame}
\frametitle{Dot Products and Cross Products}

\footnotesize

\begin{center}
{\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lllll}
	&		\textbf{Formula} 	&	\textbf{Type}	&	\textbf{Geometry}	&	\textbf{Zero if$\ldots$}	\\
\\
\textbf{Dot}	&	$\bfa \cdot \bfb$		&Scalar	&	Projections	&	$\bfa$, $\bfb$ orthogonal\\
\\
\textbf{Cross}	&	$\bfa \times \bfb$	&Vector	&	Area of a	&	$\bfa$, $\bfb$ parallel	\\
&&& Parallelogram	\\
\\
\textbf{Triple}	&	$\bfa \cdot (\bfb \times \bfc)$	& Scalar	& Volume of a &	$\bfa, \bfb, \bfc$ coplanar \\
&&&	Parallelepiped	\\
\end{tabular}
}
\end{center}
\end{frame}


\begin{frame}
\frametitle{Lines and Planes}
\footnotesize
\textbf{Equation of a Line}

\medskip

A line is specified by a point $P(x_0,y_0,z_0)$ that it contains and a vector $\bfv = \langle a, b, c \rangle$ that points along it

\medskip
Parametric:		$x(t) = x_0 + at, \quad y(t) = y_0 + bt, \quad z(t) = z_0 + ct$

\medskip
Symmetric:		$\dfrac{x-x_0}{a} = \dfrac{y-y_0}{b} = \dfrac{z-z_0}{c}$

\bigskip

\textbf{Equation of a Plane}

\medskip


A plane is specified by a point $P(x_0,y_0,z_0)$ in the plane and a vector $\bfn = \langle a,b,c \rangle$ that is normal to the plane
$$ ax+ by + cz = d$$
where $a$, $b$, $c$ are the components of $\bfn$ and $d$ is determined by substituting in $(x_0,y_0,z_0)$

\end{frame}

\begin{frame}
\frametitle{Quadric Surfaces}
\footnotesize

Ellipsoid:  $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $

Elliptic Paraboloid: $z = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} $

Cone:	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = z^2$

Hyperboloid of One Sheet: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1$

Saddle: $z = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}$

\medskip
You may also see similar equations but with $x$, $y$, $z$ permuted or with the origin shifted

\medskip
Know how to use the \emph{method of traces} to identify a surface
\end{frame}




\begin{frame}
\frametitle{Vector Functions and Space Curves}
\footnotesize

\medskip

\textbf{Vector Functions and Space Curves} 

\medskip
A \emph{vector function} takes the form 
$$ \bfr(t) = \langle f(t), g(t) \rangle \quad
\text{ or } \quad \bfr(t) = \langle f(t), g(t), h(t) \rangle $$

\medskip
A \emph{space curve} is the path in two- or three-dimensional space traced out by a vector function $\bfr(t)$ for a certain range of $t$

\medskip
The \emph{tangent vector} to a space curve is the vector
$$ \bfr'(t) = \langle f'(t), g'(t) \rangle \quad \text{ or } \quad  \bfr'(t) = \langle f'(t), g'(t), h'(t) \rangle$$

\medskip
The tangent vector $\bfr'(a)$  to a space curve gives its instantaneous rate of change at $t=a$

\smallskip
Its magnitude $| \bfr'(a) |$ is the instantaneous speed of the space curve at $t=a$

\bigskip
To find the angle of intersection of two curves described by $\bfr_1(t)$ and $\bfr_2(t)$
\begin{itemize}
\item	Find the time of intersection $t=a$
\item	Find the angle between the tangent vectors $\bfr_1'(a)$ and $\bfr_2'(a)$
\end{itemize}
\end{frame}

\section{Hot Spots}

\begin{frame}
\frametitle{Vector Algebra}
\footnotesize
If $|\bfa| = 3$ and $|\bfb| = 6$ what is $|\bfa + \bfb|$?

\bigskip

If a river $50$m wide  flows at $4$ m/sec and a swimmer swims straight from one bank toward the other at $5$ m/sec, how far downstream will the swimmer end up?  What distance will she/he swim?

\bigskip

\bigskip
Determine whether the triangle with vertices
$P(1,0,1)$, $Q(3,0,3)$, $R(-1,0,1)$ is isosceles.

\bigskip
Determine whether the points $P(1,1,0)$, $Q(3,1,2)$ and $R(4,2,2)$ are coplanar.


\end{frame}

\begin{frame}
\frametitle{Quadric Surfaces}
\footnotesize

Identify the surface
$$ z = x^2 + 2x - y^2 + 2y $$

\bigskip

Identify the surface
$$ z^2 - 2z	+ x^2 + y^2 -4y = 1 $$

\bigskip
Identify the surface
$$ y^2 = x^2 + z^2$$

\bigskip
Identify the surface
$$ x^2 = y $$

\bigskip
Identify the surface
$$ 4x^2 + 9y^2 + z^2 = 36$$

\end{frame}

\begin{frame}
\frametitle{Lines and Planes}
\footnotesize
Find the equation of a line segment from $P(1,-2,4)$ to $Q(3,2,3)$.



\bigskip
Determine whether the lines $\bfr_1(t) = \langle 1, 2 + 2t, 2 + 3t\rangle$ and $\bfr_2(t) = \langle 3, 2 + 5t, 7 -t \rangle$ are parallel, intersecting, or skew.

\bigskip
Find the equation of a plane containing the three points
$$ P(1,0,-1), \quad Q(2,3,4), \quad R(3,2,1). $$

\bigskip
Find the parametric equation of  line of intersection between the planes
$$ x - y +z  = 5$$
and
$$ 2x + y + z = 6$$
\end{frame}

\begin{frame}
\frametitle{Space Curves}
\footnotesize
Do the curves
$$ \bfr_1(s) = \langle s, 2s-1, s^2 \rangle $$
and 
$$ \bfr_2(t) = \langle t-1 , t^2 - 3, (t-1)^3 \rangle$$
intersect? Do they collide?

\bigskip
Find the point where the curves 
$$ \bfr_1(t) = \langle \cos t, \sin t, t \rangle $$
and 
$$ \bfr_2(t) = \langle t+1, t^2 , t^3 \rangle $$
intersect, and angle of intersection between the curves.

\end{frame}

\begin{frame}
\frametitle{Open Mike}

\begin{center}
Your questions?
\end{center}

\end{frame}

\begin{frame}
\frametitle{Reminders}

\footnotesize
\begin{itemize}
\item		Your exam is 5 PM to 7 PM. There will be 10 multiple choice questions and 4 free-response questions.

\item		If you are in section 17, you should go to CB 118. If you are in sections 18 or 19 you should go to CB 122. Please arrive at least 10 minutes before the start of the exam and please have your student ID with you.  Bring a one-page notebook-paper-sized sheet of formulas, notes, etc., if you wish.

\item		Raw scores should be posted in Canvas by late Thursday. We'll announce in Friday's class whether there will be a curve. 

\item		You will receive your exam papers back in Tuesday's recitation. 

\item		If you have any concerns about the grading on your exam, please turn in your exams with an explanatory note by the end of Tuesday's recitation. Regrading requests submitted after Tuesday recitation will not be considered.

\end{itemize}

\vfill
\begin{center}
\large
Good Luck!
\end{center}
\end{frame}



\end{document}