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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 $\sin 2x=2\sin x\cos x$?
                      5: Symbolically proving it one way or another is impossible in general. Therefore,
                      6: LON-CAPA uses a sampling system. If your answer and the student's answer
                      7: agree at the sampling points within your given tolerance factor, the student's
1.3       vandui11    8: 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    9: 
                     10: To specify where to sample the formulas for determining whether the student's
                     11: answer is correct, you need to put a sampling specification in the \textbf{Sample
1.3       vandui11   12: Points}\index{Sample Points} field. The sampling specifications take the
1.1       bowersj2   13: following format:
                     14: 
                     15: \begin{enumerate}
1.3       vandui11   16: \item A comma-separated list of the variables you wish to interpret,
1.1       bowersj2   17: \item followed by {}``@'' (not in quotes),
                     18: \item followed by any number of the following two things, separated by semi-colons:
                     19: 
                     20: \begin{enumerate}
1.3       vandui11   21: \item a comma-separated list of as many numbers as there are variables, which specifies
1.1       bowersj2   22: one sampling point, OR
1.3       vandui11   23: \item a comma-separated list of as many numbers as there are variables, followed
1.1       bowersj2   24: by a colon, followed by another list of as many numbers as there are variables,
                     25: followed by a \#, followed by an integer.
                     26: \end{enumerate}
                     27: \end{enumerate}
                     28: The first form specifies one point to sample. The second form specifies a
                     29: range for each variable, and the system will take as many random samples
                     30: from that range as the number after the \#.
                     31: 
1.3       vandui11   32: For $2x^{2}+4$, with one variable {}``x'', one could specify:
1.1       bowersj2   33: 
                     34: \begin{itemize}
                     35: \item {}``x@2'', which will sample the answers only at 2. (This is generally
                     36: a bad idea, as the student could get lucky and match at that point)
                     37: \item {}``x@1:5\#4'' will takes 4 samples from somewhere between 1 and 5.
                     38: \item {}``x@1:5\#4;10'' will takes 4 samples from somewhere between 1 and 5,
                     39: and also sample at 10.
                     40: \end{itemize}
1.4     ! www        41: For $2x^{2}+3y^{3}+z$, which has three variables, one could specify:
1.1       bowersj2   42: 
                     43: \begin{itemize}
1.4     ! www        44: \item {}``x,y,z@4,5,3:10,12,8\#4;0,0,0'', which take four samples from the box determined
        !            45: by the points (4, 5, 3) and (10, 12, 8), and also sample the point (0, 0, 0).
1.1       bowersj2   46: \end{itemize}