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====== Math 445 HW 2: array operations and basic plotting ======

Helpful Matlab functions

  sum, linspace, colon syntax, dot syntax
  * (mat-vec mult), \, norm, 
  plot, semilogy, semilogx, loglog, xlabel, ylabel, legend, axis, grid, axis
  hist, randn
  exp

**Problem 1:** Write a Matlab expression that sums the first N of each series and evaluate it for N=100.

(a) 1 + 1/2 + 1/3 + 1/4 + ... \\
(b) 1 + 1/2 + 1/4 + 1/8 + ... \\
%%(c)%% 1 + 1/3 + 1/9 + 1/27 + ...\\
(d) 1/2 + 2/3 + 3/4 + 4/5 + ...\\
(e) 1/2 - 2/3 + 3/4 - 4/5 + ...\\
  
**Problem 2:** A theater has a seating capacity of 900 and charges %%$%%2.50 for children, %%$%%4 for students, and %%$%%5.50 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The total money brought in was %%$%%3825. How many children, students, and adults attended the show? Write out the equations in algebra, translate to a matrix-vector $Ax=b$ problem, and solve that numerically in Matlab. Show that your answer is correct to floating-point accuracy by computing $\| Ax-b\|$ (the norm of difference between the left and right-hand sides of the equation $Ax=b$). 

**Problems 3-9:** Turn in the Matlab code and the figures. Always label the figures appropriately, using ''title'', ''xlabel'' and either ''ylabel'' or ''legend''. The figures should be titled "Problem 5", etc. Read ''help'' for the following functions and then experiment with them to see how they affect the plot.

  figure, clf, hold on, hold off
  grid, xticks, yticks 
  axis([xmin xmax ymin ymax), axis equal, axis square
  randn, hist

**Problem 3:** Plot sin(x) versus x for 100 evenly space points in x from 0 to 2pi, using a solid blue line.

**Problem 4:** Plot $y = 5x^2 - 4x - 3$ for 40 evenly space points in x from -2 to 2, using a red line and a circle on each data point. Superimpose an x,y grid on the plot.

**Problem 5:** Make a plot of the unit circle. Make sure it's closed --no gap! Hint: use ''linspace'' to specify a range of angles, then ''cos'' and ''sin'' to produce vectors of the x,y coordinates of points on the unit circle.

**Problem 6:**  Plot $10^{3x}$ for 100 points in x evenly space between -2 and 5. Choose the most appropriate plotting function. Hint: it ain't ''plot''!
  
**Problem 7:** Plot $3 x^5$ for 100 points in x evenly space between 1 and 4. Choose the most appropriate plotting function. Again, it ain't ''plot''!

**Problem 8:** Make a histogram of 1000 random numbers from a normal (Gaussian) distribution. Use Matlab's ''hist'' and ''randn'' functions.

**Problem 9:** Chemicals A and B combine in reaction to form chemical C. The rate of production of C is proportional to the amount of A and the amount of C present at any given time: $dC/dt = \alpha A(t) B(t)$. 2 gm of A combine with 1 gm of B to form 3 gm of C. At the start of the reaction there are 40 gm A, 50 gm B, and 0 gm C. After 5 minutes there are 10 gm of C. 

This is a problem in differential equations. If you take MATH 527 you will learn how to solve this problem and determine the amounts of A, B, and C (in grams) as a function of time t (in minutes). These functions are

\begin{eqnarray*}
C(t) &= 300 (e^{0.0227 t}-1)/(5e^{0.0227 t} - 2)
\end{eqnarray*} 
\begin{eqnarray*}
A(t) &= 40 - 2/3 \; C(t)
\end{eqnarray*}
\begin{eqnarray*}
B(t) &= 50 - 1/3 \; C(t)
\end{eqnarray*}

Make a plot of A, B, and C versus t for t between 0 and two hours. Label the horizontal axis and indicate which curve is which with a legend.

**Problem 10:** Use the graph from problem 9 to answer the following questions about the reaction. You might need to refine your plot and zoom in answer some of these. 

(a) How much A,B, and C are present after one hour?\\
(b) How much C is eventually produced?\\
%%(c)%% Are A and B fully depleted in the long-term limit? If not, how much of each are left?\\
(d) Does your answer to %%(c)%% make sense, given the problem description? Explain.\\
(e) How long does it take for the reaction to reach 95% completion?
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