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made grammatical changes and simplified wording for clarity

    1: \label{Formula_Response_Sampling}
    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
    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.
    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
   12: Points}\index{Sample Points} field. The sampling specifications take the
   13: following format:
   14: 
   15: \begin{enumerate}
   16: \item A comma-separated list of the variables you wish to interpret,
   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}
   21: \item a comma-separated list of as many numbers as there are variables, which specifies
   22: one sampling point, OR
   23: \item a comma-separated list of as many numbers as there are variables, followed
   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: 
   32: For $2x^{2}+4$, with one variable {}``x'', one could specify:
   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}
   41: For $2x^{2}+3y^{3}$, which has two variables, one could specify:
   42: 
   43: \begin{itemize}
   44: \item {}``x,y@4,5:10,12\#4;0,0'', which take four samples from the box determined
   45: by the points (4, 5) and (10, 12), and also sample the point (0, 0).
   46: \end{itemize}