For each of the following, provide an answer in Matlab-digestable syntax

1. Assign the value 0.00014 to the variable //x//, using compact scientific notation.

2. Set the variable //x// to a vector whose components are the even numbers between 14 and 36.

3. Set the variable //x// to 200 evenly spaced points between 0 and 2.

4. Produce a plot of //y = x^2 - 2x + 3// for the //x// of the previous problem. Label the axes.

5. Plot //tan(x)// versus //x// for 200 evenly spaced points between 0 and 1.57 ≈ pi/2, using
a logarithmic scale on the //y// axis. 

6. Plot //sin(x)// and //cos(x)// for 200 evenly spaced points between -π and π, on the same plot,
using red for //sin(x)// and blue for //cos(x)//. Add a legend that indicates which function is which color.

7. Produce a vector of length 13 whose components are random integers between 0 and 10, inclusive.

8. Produce a vector of length 13 whose components are random real numbers between 0 and 10, inclusive.

9. Show how you would solve the following system of equations with Matlab

<code>
  x + 2y -  z =   5
 3x +  y + 6z =  37
-3x +  y + 2z = -11
</code>

10. Write a conditional expression that is true if scalar variables //x// and //y// are both nonzero 
and false otherwise.

11. Set variable //A// to a 3 x 5 matrix of zeros.

12. Set variable //A// to a 4 x 7 matrix of random real numbers, using a guassian (normal) distribution.

13. Write a conditional expression that is true if a matrix //A// is square and false otherwise.

14. Write a conditional expression that is true if either //x// or //y// is an integer.



Write a Matlab function that

15. returns 1 (true) if its argument is divisible by 3 and 0 (false) if it's not. 

16. takes a vector //x// as input and returns 1 if the components of //x// are sorted
in ascending order, 0 if not.

17. computes the product //y = Ax// of an //m x n// matrix //A// and an //n x 1// vector //x//, according the formula
<latex>
y_i = \sum_{j=1}^n A_{ij} x_j
</latex>

**A few more problems, added 1/14/2013**

18. Set vector v to the 3rd column of matrix A.

19. Set vector v to the 4th row of matrix A.

20. Set vector v to the first 3 entries in the 7th row of A.

21. Set the 4th row of the 8 x 5 matrix A to (1, 2, 9, 5, 0).

22. Set the 2nd column of the 4 x 3 matrix A to a column vector with entries (3, 0, -1, 5).

23. Zero all the entries in the last column of a matrix A.

24. 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.

----

**A few more problems, added late 10/14/2013**. You don't have to turn these in, but they are fair game for the exam. I will add more example problems over the next few days.

25. Write an expression that returns the positive elements of a vector v.

26. Write an expression that returns the sum of the positive elements of a vector v.

27. Write an expression that returns the number of positive elements in a vector v.

** Even more practice problems! Hooray! added 10/22/3013 **

28. Write a function that computes the factorial of an integer n using a ''while'' loop.

29. Write an ''isPrime(n)'' function that returns 1 (true) is n is prime and 0 (false) if n is not prime. Don't worry about making the test efficient or doing it with integer arithmetic.

30. Write a function that returns a vector of all the integer divisors of an integer n. Again, don't worry about efficiency or integer arithmetic.