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Mittelwertsatz für reellwertige Funktionen

About points...

We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.

That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.

About difficulty...

We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.

That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:

Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise

Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise

Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise

Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise

Level 5 -
Level 6 -

Exercise

https://texercises.com/exercise/mittelwertsatz-fur-reellwertige-funktionen/

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Exercise:
Sei U subseteq mathbbR^n offen und f:U rightarrow mathbbR diffbar. Sei x_ in U und h in mathbbR^n. Falls x_+th in U für alle t in dann gilt fx_+h-fx_textD_xifhpartial_hfxi für ein xix_+t_xih mit t_xi in . In Worten ausgedrückt existiert also entlang des geraden Weges zwischen x_ und x_+h ein Punkt wo die Ableitung entlang des durch den geraden Weg gegebenen Vektors gerade die Differenz der Funktionswerte an den Randpunkten des Weges ist.

Solution:
Beweis. Man bemerkt dass die Ableitung des geraden Weges t in mathbbR mapsto x_+th für vorgegebene x_h in mathbbR^n bei jedem t gleich h ist. Daher erfüllt die Funktion g:t in mapsto fx_+th in mathbbR auf Grund der Kettenregel in Satz . alle Voraussetungen des eindimensionalen MWS. Also existiert t_xi in mit g-gg't_xitextD_x_+t_xihfh nach der Kettenregel und somit fx_+h-fxg-gg't_xitextD_xifh für xix_+t_xih.

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Tags	analysis, beweis, eth, fs23, proof
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Difficulty	(4, default)
Points	0 (default)
Language	(Deutsch)
Type	Proof
Creator	rk
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