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1.1       www         1: \label{Function_Plot_Response_Evaluation_Rule}
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1.5     ! www         3: Overview - This box is used to create a rule that determines whether or not a submitted graph is correct.  In general, it takes the form of testing the function, its integral, or its first or second derivative over a given set of x-values.  The test can be to see if it equals, is greater than, or less than a specified value.  Anywhere a number is needed, a variable can also be used.  (Skip to the bottom for examples of rules.)
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1.5     ! www         5: \begin{itemize}
        !             6: \item ``Index/Name" - This is an internal label for the rule.  Something must be entered here, and it must be different for each rule.  This same value will be used to add a conditional hint.
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1.5     ! www         8: \item ``Function" - This determines what the server will be testing.  For instance, choose 'First derivative' causes the server to evaluate the derivative of the entered answer over the given domain.
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1.5     ! www        10: \item ``Initial x-value'' and ``Initial x-value label'' - A value must be entered for one of these.  Either choose a numerical value for x (the first option), or choose the beginning of the submitted answer, the end, or a previously chosen named point (see below).
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1.5     ! www        12: \item ``Final x-value'' and ``Final x-value label'' - This determines the end of the domain over which the rule examines.  To test only a single point (the initial value), leave these blank.  If a label is entered, such as `positive', the point at which the rule fails will be given this special label.  This label can then be used in subsequent rules as an 'Initial x-value label'.
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1.5     ! www        14: \item ``Minimum length for range'' - This tests that the difference between the initial and final x-values are at least a certain length apart.  This is only useful if there is at least one label.
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1.5     ! www        16: \item ``Maximum length for range'' - This tests that the difference between the initial and final x-values are at most a certain length apart.  This is only useful if there is at least one label.
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1.5     ! www        18: \item ``Relationship''  - The heart of the rule.  This choice determines whether the chosen `Function' is greater than, less than, equal to, etc. a certain `Value'.
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1.5     ! www        20: \item ``Value'' - Enter the number you wish to compare to.  It is also possible to choose 'not defined', in the event the answer should not have a value for the given domain. Within the value argument, the function itself can be evaluated using \&fpr\_f(), its derivative using \&fpr\_dfdx(), and its second derivative using \&fpr\_d2fdx2(). This allows for a comparison of two points on the graph.  The value of a previously defined label can be retrieved using the function \&fpr\_val(), e.g., \&fpr\_val(`positive'). Previous defined values from script blocks can also be retrieved as normal variables, e.g., \$x.
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1.5     ! www        22: \item ``Percent error'' - This allows for a margin of error in the y-direction.  For instance, if the rule requires that the derivative be equal to 5, the server will accept values close enough to 5 that are within the percent error defined here. Note: Choosing 10\% would not mean that the answer is correct as long as it is within the range 4.5-5.5.  Instead, the percent corresponds to the total size of the graph.  For the function itself, the `percent error' is multiplied by the ymax-ymin; for the first derivative, it's multiplied by (ymax-ymin)/(xmax-xmin); for the second derivative, it's multiplied by (ymax-ymin)/(xmax-xmin)$^2$; and for the integral, it's multiplied by (ymax-ymin)*(xmax-xmin).
        !            23: \end{itemize}
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1.5     ! www        25: The figure below shows some examples of rules.
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1.5     ! www        27: \includegraphics[width=0.8\paperwidth]{FPRexamples}
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        !            29: \begin{enumerate}
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        !            31: \item Checks that the derivative at $x=3$ is negative.
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        !            33: \item Checks that the second derivative at $x=$\$time1 is $0$.
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        !            35: \item Checks that the value of the function $>7$ from ``Start of Graph" to $x=5$.
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        !            37: \item Checks that the function is $0$ from $x=0$ until the function is no longer $0$, labeling this new point `notzero'.
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        !            39: \item Checks that the first derivative is positive between the point `notzero' and the end of the graph.
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        !            41: \item Checks to see that the value of the function at $x=4$ is the same as the value of the function at \$time2. The Value here is \&fpr\_f(\$time2).
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        !            43: \item Checks that the value of the function at point `notzero' is equal to the value of the function at $x=$ \$time3.  The Value here is \&fpr\_f(\&fpr\_val(`notzero')).
        !            44: 
        !            45: \end{enumerate}