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1. Produce a plot of //sin(x)// versus //x// in blue and //cos(x)// versus //x// in 
red for 100 evenly spaced points between 0 and 2pi. Label the axes.

2. Given matrix A, set //v// to the //j//th column of //A//.

3. Given matrix A, set //v// to the //i//th row of //A//.

4. Suppose matrix //A// has //M// rows and //N// cols. Set //B// to //A// with its columns in reversed order.

5. Solve the system of equations

<code>
  2y - x  = 1 
  3y + 2z = 5 
  -3x + z = -2
</code>

6. Produce a vector of 100 random floating-point numbers between 0 and 10.

7. Produce a vector of 100 random integers between 0 and 10.

8. Produce a random permutation of the integers between 1 and 10.

9. Produce a random permutation of the vector v = [3 3 4 5 7];

10. Produce all permutations of the vector v = [3 3 4 5 7];

11. Produce a quiver plot of the vector field $v = [v_x, v_y]$ where  $v_x = \sin(x) \cos(y)$,  
$v_y = x y$, and //x// and //y// range from -pi to pi. Label the axes.

12. Produce a contour plot of $f(x,y) = \sin(x^2+y^2)/\sqrt{x^2 + y^2}$ 
where //x// and //y// range from -10 to 10. Label the axes.


13. Produce a 3d surface plot of the function from problem 12 over the same range, and with a color bar.
Label the axes.


14. Write down the connectivity matrix //C// for the links in this small 
network of websites.

{{:gibson:teaching:fall-2012:math445:network2.png?direct&300}}

15. Write Matlab code that converts the connectivity matrix //C// to a 
transition matrix //T// that governs the transition of probabilities 
under random surfing. 


16. Given //T//, write Matlab code that computes the vector $x$ of 
probabilities $x_j$ that you'll end up at website $j$ after a 
long night of random websurfing. 


17. Write an equation for //y// as a function of //x// for
the following data plot. Bonus: express exponential functions 
as powers of //e// rather than powers of 10. Use $e^{2.3}\approx 10$ 
to convert between the two.   

 {{:gibson:teaching:fall-2012:math445:fig1.png?direct&300}}



18. How would you graph the function $y(x) = x^c$, in a way that highlights
this functional relationship? I.e. given vectors $x$ and $y$ satisfying
$y_i = x_i^c$, what Matlab command should you use to plot $y$ versus $x$?


19. How would you graph the function $y(x) = c^x$, in a way that highlights
this functional relationship? 


20. Write an anonymous function that returns the square of its input.

21. Write an anonymous function that, for an input vector $x = [x_1, ~x_2]$
returns the output vector $f(x) = [4 x_1 x_2, ~\sin(x_1) \cos(x_2)]$

22.  Convert the following 2nd order ODE to a 1st order system of ODE in 
two variables.

$dx^2/dt^2 + 3\; dx/dt + \sin(x) = 0$


22. Show how to integrate the system of ODEs from problem 22 from 
t = 0 to 10 using the initial condition x(0) = 0, dx/dt(0) = 1.
Write the ODE system in Matlab using an anonymous function. 


23. Compute the terminal velocity of a ping-pong ball dropped from 
a great height, using this system of equations

<latex> dy/dt = v_y </latex>

<latex> d v_y/dt = -g - \mu v_y |v_y| </latex>

where $g = 9.81, ~\mu =0.35$, $y$ represents the vertical position, and
$v_y$ represents the vertical velocity. Represent the two free variables
with the vector $x = [y, ~v_y]$ and reexpress the two equations above as
an ODE system of the form

$dx/dt = f(x)$ 

Note that both sides of this equation are vectors: $dx/dt = [dx_1/dt, ~dx_2/dt]$ and 
$f(x) = [f_1(x_1, x_2), ~f_2(x_1, x_2)]$. Your job is to find the functions $f_1$ and $f_2$.

Write an anonymous function in Matlab that computes $dx/dt = f(x)$ for an input vector $x$, 
and then use //ode45// to integrate this system from $t=0$ to $t=100$
from the initial conditions $x(0) = [y(0), ~v_y(0)] = [0, 0]$.

24. Plot the position of the ping-pong ball as a function of time.
Label the axes. 

25. Write a function that performs matrix-vector multiplication for
a **sparse** matrix //A//, that accesses only nonzero elements of //A//. 

For an // M x N // matrix //A// and an //N//-dimensional vector //x//, 
the matrix-vector product $y = Ax$ is defined by

<latex>
y_i = \sum_{j=1}^N A_{ij} x_j
</latex>

for each component $y_i$ of the //M// dimensional vector $y$. But don't code that 
formula directly! Instead start your function with

<code>
  K = nnz(A);
  [i,j,a] = find(A);
</code>

and write the matrix-vector multiplication as a loop over the K nonzero elements
of A.
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