===== Math 445 HW5 ===== This homework is meant as review for the second-chance exam to be held Nov 26. It is due Thursday Nov 14th in lecture. Review topics * **fprintf:** formatted printing function * **plotting:** ''plot, contour, surf, mesh, meshgrid'' * **loops:** ''for'' and ''while'' * **anonymous functions:** defining and using * **solving equations:** with ''fsolve'' or ''newtonsearch'' * **monte carlo simulation** * **log-linear relationships**: ''semilogy, semilogx, loglog'' * **index operations** **Problem 1:** Find all real-valued solutions of the equation $x^4 - 3x^2 + 2x -4 = 0$ by plotting $y = x^4 - 3x^2 + 2x -4$ versus $x$, estimating the values of $x$ where $y(x) = 0$, and then solving the equation numerically using those estimates as initial guesses. Turn in your plot, your matlab code, and the numerical solutions to the equation. **Problem 2:** Define a ''rightshift'' function using Matlab's anonymous function facility that shifts the elements of a row vector one step to the right, wrapping the last element around to the first position. For example ''rightshift([1 2 3 4])'' should return ''[4 1 2 3]''. **Problem 3:** Write Matlab code to estimate the probability of drawing four-of-a-kind from a randomly shuffled 52 card deck. Turn in your code and your estimated probability. **Problem 4:** Make a 3d plot of $f$ versus $x,y$ for the function $f(x,y) = 3 e^{-(x^2/2 + xy + 2 y^2)}$ for $-3 \le x \le $ and $-3 \le y \le 3$, on a mesh with $Delta x = \Delta y = 0.1$. Use Matlab's ''meshgrid'' function. **Problem 5:** Print $e$ to five digits accuracy. (Hint: You get $e$ in Matlab from ''exp(1)''). **Problem 6:** Write an ''isPrime(n)'' function that returns 1 (true) if ''n'' is prime and 0 (false) for a composite. Don't worry about efficiency, just loop over 2 and the odd integers less than or equal to $\sqrt{n}$, and return 0 if any divide ''n'' evenly, and 1 if not. (You might have already done this as an optional problem for exam 1 prep.) **Problem 7:** Write Matlab code that will compute the first 20 prime numbers, using your ''isPrime'' function from problem 6. **Problem 8:** Deduce the functional relationship between $y$ and $x$ for the data shown in this plot {{:gibson:teaching:fall-2012:math445:fig1.png?direct&300}} **Bonus:** Write an ''primes(N)'' function that returns all primes less than or equal to ''N'' using the [[http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes | Sieve of Eratosthenes algorithm]].