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Problem 1: Write Matlab code for plotting $y = 1.2 e^{-0.8x}$ as a red line, for 100 evenly spaced points in $x$ between $-10$ and $10$ . Choose a plotting function that highlights the functional relationship between $y$ and $x$ . Label the axes and title the graph.

  x = linspace(-10,10,100);
  y = 1.2*exp(-0.8*x);
  semilogy(x,y,'r-');
  xlabel('x')
  ylabel('y')
  title('graph of y = 1.2 exp(-0.8x)'); % one choice of title

Problem 2: Write an equation for $y$ as a function of $x$ for the following data plot.

We see a straight line on a plot where both axes are logarithmic. That means a power-law relationship, i.e. the form $y = c x^b$ . Increasing $x$ by a factor of $10^2$ causes an increase in $y$ by a factor of $10^6$ . Therefore $b=3$ . At $x=1$ , $y=10^{-2}$ , so $c=10^{-2}$ . Therefore the functional relationship is $y=10^{-2} x^3$ .

Problem 3: Write Matlab code that produces a contour plot of $(1-y^2) \sin x$ over the rectangle $-1 \leq y \leq 1$ and $-\pi \leq x \leq \pi$ , with 30 evenly spaced points in each of $x$ and $y$ .

  x = linspace(-pi,pi,30);
  y = linspace(-1,1,30);
  [X,Y] = meshgrid(x,y);
  Z = (1-Y.^2).*sin(X);
  contour(x,y,Z);
  xlabel('x')
  ylabel('y')
  title('contours of (1-y^2) sin(x)')

Problem 4: Write Matlab code that produces the solution of the following system of equations

$\begin{align*} w - x + z &= 2 \\ 2x + 3y - 4z &= -5 \\ 4w - 3x + 2y + z &= 3 \\ -w + 3x + 6y &= 11 \end{align*}$

 
A = [1 -1 0 1; 0 2 3 4; 4 -3 2 1; -1 3 6 0];
b = [2; -5; 3; 11];
x = A\b

Problem 5: Write one line of Matlab code that would verify that the solution computed in the previous problem satisfies the given equations. Hint: you can answer this in as little as five characters, or perhaps even three.

  Ax-b

Problem 6: Write a Matlab function that returns a numerical approximation to $e$ using the following series, where $N$ is an input argument to the function.

$e \approx \sum_{n=0}^{N} \frac{1}{n!}$

function e = eulernumber(N)
% return approximation of e from sum_{n=0}^N 1/n! 

  % it's easiest to take care of n=0 term before loop
  nfactorial = 1;
  e = 1;          

  for n=1:N
     nfactorial = n*nfactorial; 
     e = e + 1/nfactorial;
  end

end

Problem 7: The Lorenz system is a ordinary differential equation in three variables $[x, y, z]$ defined by

$\begin{align*} \frac{dx}{dt} = s (y - x), \quad \frac{dy}{dt} = x (r -z) - y, \quad \frac{dz}{dt} = xy - bz \end{align*}$

Write Matlab code for an anonymous function that returns the vector $dv/dt = [dx/dt, ~dy/dy, ~dz/dt]$ given an input vector $v = [x, y, z]$ , using parameter values $s=10$ , $b=8/3$ , and $r=28$ .

  f = @(v) [10*(v(2)-v(1)); v(1)*(28-v(3)) - v(2); v(1)*v(2) - 8*v(3)/3];

Problem 8: Revise your answer to Problem 7 so that the anonymous function is in the right form to be passed into Matlab's {\tt ode45} function, and write Matlab code that would integrate the Lorenz system from $t=0$ to $t=100$ , from the initial condition $x=0$ , $y=1$ , $z=0$ .

  f = @(t, v) [10*(v(2)-v(1)); v(1)*(28-v(3)) - v(2); v(1)*v(2) - 8*v(3)/3];
  [t,v] = ode45(f, [0 100], [0; 1; 0]);

Problem 9: Suppose you have a biased coin that has a 53.8\% chance of landing ``heads'' on any given toss, and you want to figure out how likely it is that, out of ten coin tosses, you'll get exactly $n$ heads, for each value of $n$ between 0 and 10. Write a Matlab function that would estimate the probability getting $n$ heads in ten tosses.

  function p = biasedcoin(n);
  % estimate prob of getting n heads in 10 tosses of a 53.8\% head biased coin

  Ntrials = 1e04;
  count = 0;

  for n=1:Ntrials
    flips = rand(10,1);        % simulate ten coin flips
    if sum(flips < 0.538) == n % test if there are n heads in ten flips
      count = count + 1;
    end
  end
 
  p = count/Ntrials;

Problem 10: Write Matlab code that would set $C$ to a matrix representing the connectivity of the following network of links.

 
  C = zeros(4,4);
  C(3,1) = 1;
  C(4,1) = 1;
  C(1,2) = 1;
  C(4,2) = 1;
  C(1,3) = 1;
  C(4,3) = 1;
  C(2,4) = 1;

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