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1.2       bowersj2    1: \label{Formula_Response_Sampling}
1.1       bowersj2    2: 
                      3: As you may know, it is extremely difficult to determine whether a given expression
                      4: is exactly equal to another expression in general. For example, is $\sin 2x=2\sin x\cos x$?
1.5     ! www         5: LON-CAPA has two ways of finding out if it is:
        !             6: \begin{itemize}
        !             7: \item algebraically, using a symbolic algebra system
        !             8: \item numerically, using sampling points
        !             9: \end{itemize}
        !            10: 
        !            11: You need to determine which way is the safest in a given situation.
        !            12: 
        !            13: If you don't specify sampling points, the symbolic algebra system is used.
        !            14: 
        !            15: If you do specify sampling points, LON-CAPA uses them.
        !            16: If your answer and the student's answer
1.1       bowersj2   17: agree at the sampling points within your given tolerance factor, the student's
1.3       vandui11   18: answer will be accepted. If the student's answer does not agree at the sampling points within your given tolerance factor, it will be rejected.
1.1       bowersj2   19: 
                     20: To specify where to sample the formulas for determining whether the student's
                     21: answer is correct, you need to put a sampling specification in the \textbf{Sample
1.3       vandui11   22: Points}\index{Sample Points} field. The sampling specifications take the
1.1       bowersj2   23: following format:
                     24: 
                     25: \begin{enumerate}
1.3       vandui11   26: \item A comma-separated list of the variables you wish to interpret,
1.1       bowersj2   27: \item followed by {}``@'' (not in quotes),
                     28: \item followed by any number of the following two things, separated by semi-colons:
                     29: 
                     30: \begin{enumerate}
1.3       vandui11   31: \item a comma-separated list of as many numbers as there are variables, which specifies
1.1       bowersj2   32: one sampling point, OR
1.3       vandui11   33: \item a comma-separated list of as many numbers as there are variables, followed
1.1       bowersj2   34: by a colon, followed by another list of as many numbers as there are variables,
                     35: followed by a \#, followed by an integer.
                     36: \end{enumerate}
                     37: \end{enumerate}
                     38: The first form specifies one point to sample. The second form specifies a
                     39: range for each variable, and the system will take as many random samples
                     40: from that range as the number after the \#.
                     41: 
1.3       vandui11   42: For $2x^{2}+4$, with one variable {}``x'', one could specify:
1.1       bowersj2   43: 
                     44: \begin{itemize}
                     45: \item {}``x@2'', which will sample the answers only at 2. (This is generally
                     46: a bad idea, as the student could get lucky and match at that point)
                     47: \item {}``x@1:5\#4'' will takes 4 samples from somewhere between 1 and 5.
                     48: \item {}``x@1:5\#4;10'' will takes 4 samples from somewhere between 1 and 5,
                     49: and also sample at 10.
                     50: \end{itemize}
1.4       www        51: For $2x^{2}+3y^{3}+z$, which has three variables, one could specify:
1.1       bowersj2   52: 
                     53: \begin{itemize}
1.4       www        54: \item {}``x,y,z@4,5,3:10,12,8\#4;0,0,0'', which take four samples from the box determined
                     55: by the points (4, 5, 3) and (10, 12, 8), and also sample the point (0, 0, 0).
1.1       bowersj2   56: \end{itemize}