MATH 338       Quiz 1         Sample         Name______________

1. Find the dot product of the vectors [3 1 -1] and [2 0 5]
   
   
2. Reduce the following matrix to row echelon form and give the elementary matrices for any two of the row operations in the row reduction process.

                               2  3  3
                               1 -2  5
                               3  2  7
   

   
   
   
3. Find the rank of the matrix in problem 2. Is it singular or non-singular? Find a basis for the column space.
   
   
4. Determine whether or not the vectors [ 2 1 3], [3 -2 2], and [3 5 7] are linearly independent. Do they span R3?
   
   
5. Find a basis for the solution space of the following system:

             2x + 3y + 3z = 0
              x - 2y + 5z = 0
             3x + 2y + 7z = 0
   

   
   
   
6. Given the function f(x) = 2x exp(2x)
   1. Find the derivative of this function
      
      
   2. Find the critical/stationary points for this function
      
      
   3. Find the relative max and min for this function.
      
      
   4. Find the absolute max and min over the interval: [-1, 1]
   
   
   
7. Find the gradient of the function ax³ - 5y² + 7xy
   
   
8. Draw a graph of the feasible region satisfying the following inequalities:

    
                                x + y <= 5
                               2x + y <= 8
                                    x >= 0
                                    y >= 0
   

   
   
   
9. Plot the graph of the line perpendicular to the line x+2y=4 through the point (0,2)