# Tutorials (Question Examples)

The following tutorials are written to illustrate concrete questions that could be given to students in higher education. We aim to discuss the STACK and JSXGraph features used as well as the motivation for the question, but there is room for improvement, so please give feedback.

Each tutorial will provide the Moodle XML code for the question, so that you can easily import it in your own server and test it, as well as the discussing of the design.

The questions are not really indended to be ready for use.
We reckon that most teachers will want to tailor the questions to their students and use the questions primarily as examples to build on.

## Visualising functions in two variables

• Question match-properties. Give an arbitrary function with given properties, with visual aid.
• Question Tune 3D Function Tune parameters of a function to match another function visually
• Question Tune 3D polynomoial Minimal working example of question using jsxgraph

# Sketches (questions under construction)

## Calculus of functions in two variables

### Functions of two variables

In the following questions the tangent plane is shown

In optimization the local 2nd order behavior of a function hast to be analysed. This is addressed by these questions

## Visualising curves in 3D

• Question Circular curve: Find the parameters of $$t \mapsto \begin{pmatrix} r \cdot \cos(t) \\ r \cdot \sin(t) \\ h \cdot \cos(n \cdot t - \phi) \end{pmatrix}.$$
• Question Elliptical curve (helix): Find the parameters of the curve given by $$t \mapsto \begin{pmatrix} x_0+ a \cdot \cos(t) \cdot \cos(\alpha) - b \cdot \sin(t) \cdot \sin(\alpha) \\ y_0+ a \cdot \cos(t) \cdot \sin(\alpha) + b \cdot \sin(t) \cdot \cos(\alpha) \\ h \cdot t \end{pmatrix}.$$

• Question Elliptical curve: Find the parameters of the curve given by $$t \mapsto \begin{pmatrix} x_0+ a \cdot \cos(t) \cdot \cos(\alpha) - b \cdot \sin(t) \cdot \sin(\alpha) \\ y_0+ a \cdot \cos(t) \cdot \sin(\alpha) + b \cdot \sin(t) \cdot \cos(\alpha) \\ h \cdot \cos(t)\end{pmatrix}.$$

• Question Rotating a curve about two axis This question trains the imagination of rotating objects around two axis of the coordinate system.

## Vector fields in 3D

• Question Curl of a vector field Given is a vector field (as formula and plotted). The student has to choose the right representation for the curl. The filed is selected from a list of given fields.

• Question Vector field of given curl Given is a curl of a vector field (as formula and plotted). The student has to choose the right representation for a vector field, that is fits to the curl. The filed is selected from a list of given fields.

• Question Rotating a curve about two axis This question trains the imagination of rotating objects and guides the student through the topic.
• Question Curl of a given vector field This question trains the computation of the curl of a vector field. This implementation demonstrates the usaga of a single JSXGraph object in an adaptive setting.