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====== Math 445 final exam topics ======

The Math 445 final exam will be comprehensive, covering all material presented in lecture and lab (except for the derivation of differential equations from physics presented in lecture). Below is a broad overview but not exhaustive of topics that might be covered on the exam. 

----
===== Matlab syntax =====

====Vectors and matrices====

You should know how to create vectors and matrices, how to access and set their elements, and how to perform mathematical operations with them. For example, if asked "Write Matlab code that would assign the 3rd and 7th columns of the matrix ''A'' into a matrix variable ''B''," you should know that the answer is

<code matlab>
  B = A(:, [3 7])
</code>
Note that correctness and conciseness both count! The above answer is better and will get a better grade than ''B = [A(:,3), A(:,7)]'', because it is more readable, less error-prone, and generalizes more easily.

Relevant Matlab vocabulary:

  * ''zeros''
  * creating vectors and matrices with lists of elements in square brackets
  * creating vectors with colon syntax and ''linspace''
  * ''* \ + - =''

----
====Linear algebra versus elementwise operations===

You should know the difference between linear algebra and elementwise operations on matrices and vectors. For example, given two 2 x 2 matrices ''A'' and ''B'', you should be able to compute ''A * B'' and ''A .* B'' by hand. You should also know in what circumstances it's appropriate to use linear algebra operations or elementwise operations. 

----
==== Solving systems of equations ====

You should be able to translate a story problem into a system of linear equations and then write Matlab code to solve the system of equations. For example, suppose you have twenty coins worth %%$%%1.35 and they're all nickels and dimes. Translate this problem into a system of equations and write the Matlab code that would solve the equations numerically. 

\begin{eqnarray*}
n + d &= 20 \\
5 n + 10 d &= 135
\end{eqnarray*}

<code matlab>
A = [1 1 ; 5 10];
b = [20 ; 135];
x = A\b;
n = x(1)
d = x(2)
</code>

or even just

<code matlab>
[1 1 ; 5 10] \ [20 ; 135]
</code>

----
==== Plotting ====

You should know 
  * how to make $xy$-plots of given functions $y=f(x)$ using **linspace**, elementwise operations on vectors, **plot**, **semilogx**, **semilogy**, and **loglog**.
  * how to make 2D color plots with **pcolor**, using **linspace**, **meshgrid**, elementwise matrix operations as well
  * how to make 2D contour plots with **contourf** and **linspace**, **meshgrid** etc. 
  * how to make 2D quiver plots with **quiver** and **linspace**, **meshgrid** etc.
  * how to make 3D surface plots with **surf** and **linspace**, **meshgrid**, etc. 
  * how to load data from a file and plot it
  * how to label axes, title a plot, color the lines, show markers on data points, display a coordinate grid, show a colorbar, etc. 

----

==== Evaluating expressions ====

You should know how to write concise Matlab code to evaluate mathematical expressions. In particular, you should know how to evaluate expressions involving sums and products. 

For example, write two lines of Matlab code that would evaluate the sum

\begin{eqnarray*}
\sum_{n=1}^{10} \frac{n}{(n+1)^2}
\end{eqnarray*}

Answer:

<code matlab>
n = 1:10;
sum(n./(n+2).^2)
</code>

----
==== Functions ====

You should be able to write simple Matlab functions to perform specified computations. For example, if asked to write a Matlab function that, given a value of $N$, evaluates 

\begin{eqnarray*}
\sum_{n=1}^{N} \frac{n}{(n+1)^2}
\end{eqnarray*}

you should respond with 

<code matlab>
function s = f(N)
  n=1:N;
  s = sum(n./(n+2).^2)
end
</code>

----
====for loops====

You should know how to use ''for'' loops to perform repeated computations. For example, if asked to write a function that computes the above sum using a ''for'' loop (instead of using the ''sum'' function), you would write

<code matlab>
function s = f(N)
  s = 0;
  for n = 1:N
    s = s + n/(n+2)^2
  end
end
</code>

----
==== random numbers ====

You should know how to get random numbers of various kinds

  * **rand**: a random floating-point number between 0 and 1. 
  * **randi(max)**: a random integer between 1 and max.
  * **randn**: a random number in a Gaussian distribution with mean 0 and standard deviation 1.

Each of these random number generators has a matrix version, as well. E.g.
  * **rand(m,n)**: an m x n matrix of random floating-point numbers between 0 and 1.
  * **randi(m,n,max)**: an m x n matrix of random integers between 1 and max.
  * **randn(m,n)**: an m x n matrix of a random numbers in a Gaussian distribution with mean 0 and standard deviation 1.

==== if-else statements ====

You should know how to write simple ''if-else'' statements such as 

<code matlab>
x = randn();
if x < 0 
  fprintf('%d is negative\n', x)
elseif x == 0
  fprintf('%d is zero\n', x)
else
  fprintf('%d is positive\n', x)
end
</code>

----
===== Lab material ====

You should have a good grasp on the mathematics and Matlab programming of the lab material. For example,

----
==== log-linear relations ====

You should know how to infer a functional relation $y=f(x)$ given a logarithmic or linear plot
, and which of **plot**, **semilogx**, **semilogy**, and **loglog** is best for a given relation $y=f(x)$.

----
==== writing functions  =====

You should know how to write Matlab functions that do basic computations, like matrix-vector multiplication.

---- 
==== hamster dynamics / Google Page Rank ====

Given a graph of nodes and one-way links between them, you should be able to write a system of equations that governs random walks through the network of links, and then write Matlab code that would calculate the steady-state distribution.

----
==== nonlinear equations and Newton's method ==== 

You should know the mathematics behind Newton's method for solving nonlinear equations, how to code it in Matlab, and how to use Matlab's built-in solver **fsolve** to solve nonlinear equations. 

----
==== differential equations ====

 You should know now how to write an anonymous function for a system of first-order differential equations $d\vec{x}/dt = \vec{f}(t, \vec{x})$, and how to solve that system of equations numerically using Matlab's **ode45**. And given a quiver plot of a 2-d differential equation, you should be able to draw an approximate solution of the equation starting from a given initial condition, by tracing out a curve that is everywhere tangent to the arrows.
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