Problem Instance

• Write down the relation between F and f:   F'(x)=f(x).
  
  
• Fill in the value table

  x	0	1	2	3
  F(x)	3.6	1.8	0	1.7
  F'(x)	0	-2.5	0	2.5

• Fill in the sign table

  x	]0,1[	]1,2[	]2,3[
  F'(x)	-	-	+
  F''(x)	-	+	+

• Fill in Monotonicity's table

  x	]0,1[	]1,2[	]2,3[
  F monotonicity	decr.	decr.	incr.

• Fill in Concavities' table

  x	]0,1[	]1,2[	]2,3[
  F concavities	down	up	up

• Sketch the graph of F(x)
  

Student Interaction

• The main task Cal_Problem123. This contains the problem statement and issues the challange. It has a Graph-Drawing-Pad (click here to see how this tool works) which the student can use to sketch the requested graph. This tool is only available after all five subtasks have been successfully completed. Whenever the student submits his sketch, the drawn graph is evaluated, and the student is informed about all its incorrect features.
  
  
• The subtask relate f and F. In this subtask the student uses a pallet to edit expressions involving f, F, x, t, their derivatives and integrals. This task 'knows' which relations are right, and wrong. It also knows what is the expected answer. After some failed attempts, hints can be provided.
  
  
• The subtask fillin values. Here the student has to fillin the table

  x	0	1	2	3
  F(x)	3.6	1.8	0	1.7
  F'(x)	0	-2.5	0	2.5

  On wrong entries, the task reminds the student of the established relation, and suggests him to look in the given graph for the correct values.
  
  
• The subtask fillin signs. Here the student has to fillin the table

  x	]0,1[	]1,2[	]2,3[
  F'(x)	-	-	+
  F''(x)	-	+	+

  On wrong entries, the task reminds the student of the established relation, and suggests him to look in the given graph for the correct interpretation.
  
  
• The subtask monotonicity. Here the student has to fillin the table

  x	]0,1[	]1,2[	]2,3[
  F monotonicity	decr.	decr.	incr.

  On wrong entries, the task 'asks' the student about the relation between derivative signs and monotonicity, and reminds him of the signs established in subtask fillin signs.
  
  
• The subtask concavities. Here the student has to fillin the table

  x	]0,1[	]1,2[	]2,3[
  F concavities	down	up	up

  On wrong entries, the task 'asks' the student about the relation between second derivative signs and concavities, and reminds him of the signs established in subtask fillin signs.
  
  

Construction

• Context   (One Variable Differential and Integral) Calculus.
  
  
• Parameters  

  Name	Type	Attributes	Description
  f	FUNCTION	free	The given function with domain [0,3]
  a	NUMBER(0|1|2|3)	free	point inside f domain
  C	NUMBER	free	constant
  F	FUNCTION	dependent	The function to be sketched
  g	FUNCTION_GRAPH	dependent	The graph of function f
  A1	NUMBER	dependent	Area of the first region
  A2	NUMBER	dependent	Area of the second region
  A3	NUMBER	dependent	Area of the third region

  
  
  
• Context Extension  
  
  
• Statement  
  
  
• Solution Set   This task is an extension of the draw-the-graph task mentioned below, sharing with it the same Solution Set.
  
  
• Subtasks   All subtasks are instances of general task templates.

  relate f and F	This subtask is an instance of task template to write down specific relations. In such tasks, the student can input algebraic expressions and relations, e.g., through some pallet, which the task evaluates, using its own context, to be right, or wrong. These tasks are configured selecting a particular algebraic context, and a specific goal relation.
  fill the values	This subtask is an instance of a task template to fillin function value tables. Given a list of functions, together with a list of points, the student has to fillin the table with the respective values. In this case the teacher has only to consider the list of functions {F,F'}, and the list of points {0,1,2,3}, when he configures this task from the template.
  fill the signs	This subtask is an instance of a task template to fillin function sign tables. Given a list of functions, together with a list of intervals, the student has to fillin the table with the respective signs: '+', '-', or 'undefined'. In this case the teacher has only to consider the list of functions {F',F''}, and the list of intervals { ]0,1[,  ]1,2[, ]2,3[ }, when he sets up this task from the template.
  monotonicity	This subtask is an instance of a task template to fillin a monotonicity table. Given a function, together with a list of intervals, the student has to fillin a table with the monotonicity character of that function: 'incr.', 'decr.', 'const.', or 'undefined'. In this case the teacher has only to consider the function F, and the list of intervals { ]0,1[,  ]1,2[, ]2,3[ }, when he sets up this task from the template.
  concavities	This subtask is an instance of a task template to fillin a concavity table. Given a function, together with a list of intervals, the student has to fillin a table with the concavity character of that function: 'up', 'down', 'flat', or 'undefined'. In this case the teacher has only to consider the function F, and the list of intervals { ]0,1[,  ]1,2[, ]2,3[ }, when he sets up this task from the template.
  draw-the-graph	This subtask is an instance of a task template to draw the graph of a function, configurable by setting up the value of a FUNCTION type parameter. This task has a Graph-Drawing-Pad where the student can sketch the requested graph. The student is informed about incorrect features in every submited graph sketch. The "Solution Set" of this task is based on a qualitative representation of a graph, which gathers information on: monotonicity, concavitivies, critical points and values, inflexion points and values, etc. The "Solution Set" condition of this task compares the qualitative representation of the sketched graph with the qualitative representation of the its FUNCTION type parameter. However, teachers who use these tasks in their own problems do not need to be aware of these internal details.