\label{Function_Plot_Response_Evaluation_Rule}

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.)

\begin{itemize}
\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.

\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.

\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).

\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'.

\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.

\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.

\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'.

\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.

\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).
\end{itemize}

The figure below shows some examples of rules.

\includegraphics[width=0.8\paperwidth]{FPRexamples}

\begin{enumerate}

\item Checks that the derivative at $x=3$ is negative.

\item Checks that the second derivative at $x=$\$time1 is $0$.

\item Checks that the value of the function $>7$ from ``Start of Graph" to $x=5$.

\item Checks that the function is $0$ from $x=0$ until the function is no longer $0$, labeling this new point `notzero'.

\item Checks that the first derivative is positive between the point `notzero' and the end of the graph.

\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).

\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')).

\end{enumerate}