Fourierkoeffizienten des Baumes
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.
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 -
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 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
Die pi-periodisch fortsetzbaren Funktionen fx und -fx begrenzen den stark vereinfachten Achsenquerschnitt eines Mammutbaums. center tikzpicture mydispo tikzpicture center abcliste abc Berechne die Fourierkoeffizienten a_ und a_ von f exakt mit Lösungsweg. abc Gegeben sind die Fourierkoeffizienten b_-frac b_-frac b_-frac b_-frac von f. medskip Errate den Wert von b_ und gib eine explizite Formel an mit der jeder Term der Form b_k- für k dots berechnet werden kann. abcliste
Solution:
enumerate itema edt a_ dfracpi _loA^upA fracpix dx + dfracpi _loB^upB fracpi dx dfracpi leftfracpix^right_loA^upA + dfracpi leftfracpixright_loB^upB ex dfracpi leftfracpi fracpi^ - right + dfracpi leftfracpi pi - dfracpi dfracpiright dfracpi^ + fracpi - fracpi ex dfracpi^ + fracpi dfracpi^ + pi ed medskip edt a_ dfracpi _loA^upA fracpix cosx dx + dfracpi _loB^upB fracpi cosx dx ex dfracpi leftfracpix sinxright_loA^upA - fracpi _loA^upA dfracpi sinx dx + dfracpi leftfracpisinxright_loB^upB ex fracleftupA sinleftupAright - loA sinleftloArightright + fracbig cosx big_loA^upA + fracbigsinxbig_loB^upB ex -fracpi + fracleft cosleftupAright - cosleftloAright right + fracleftsinleftupBright - sinleftloBrightright ex -fracpi - frac + frac -fracpi ed itemb b_-dfrac b_-dfrac b_-dfrac b_-dfrac quad Rightarrow quad b_ dfrac medskip allgemein: b_k- dfrack-k-^ dfrack-k-^ enumerate
Die pi-periodisch fortsetzbaren Funktionen fx und -fx begrenzen den stark vereinfachten Achsenquerschnitt eines Mammutbaums. center tikzpicture mydispo tikzpicture center abcliste abc Berechne die Fourierkoeffizienten a_ und a_ von f exakt mit Lösungsweg. abc Gegeben sind die Fourierkoeffizienten b_-frac b_-frac b_-frac b_-frac von f. medskip Errate den Wert von b_ und gib eine explizite Formel an mit der jeder Term der Form b_k- für k dots berechnet werden kann. abcliste
Solution:
enumerate itema edt a_ dfracpi _loA^upA fracpix dx + dfracpi _loB^upB fracpi dx dfracpi leftfracpix^right_loA^upA + dfracpi leftfracpixright_loB^upB ex dfracpi leftfracpi fracpi^ - right + dfracpi leftfracpi pi - dfracpi dfracpiright dfracpi^ + fracpi - fracpi ex dfracpi^ + fracpi dfracpi^ + pi ed medskip edt a_ dfracpi _loA^upA fracpix cosx dx + dfracpi _loB^upB fracpi cosx dx ex dfracpi leftfracpix sinxright_loA^upA - fracpi _loA^upA dfracpi sinx dx + dfracpi leftfracpisinxright_loB^upB ex fracleftupA sinleftupAright - loA sinleftloArightright + fracbig cosx big_loA^upA + fracbigsinxbig_loB^upB ex -fracpi + fracleft cosleftupAright - cosleftloAright right + fracleftsinleftupBright - sinleftloBrightright ex -fracpi - frac + frac -fracpi ed itemb b_-dfrac b_-dfrac b_-dfrac b_-dfrac quad Rightarrow quad b_ dfrac medskip allgemein: b_k- dfrack-k-^ dfrack-k-^ enumerate
Meta Information
Exercise:
Die pi-periodisch fortsetzbaren Funktionen fx und -fx begrenzen den stark vereinfachten Achsenquerschnitt eines Mammutbaums. center tikzpicture mydispo tikzpicture center abcliste abc Berechne die Fourierkoeffizienten a_ und a_ von f exakt mit Lösungsweg. abc Gegeben sind die Fourierkoeffizienten b_-frac b_-frac b_-frac b_-frac von f. medskip Errate den Wert von b_ und gib eine explizite Formel an mit der jeder Term der Form b_k- für k dots berechnet werden kann. abcliste
Solution:
enumerate itema edt a_ dfracpi _loA^upA fracpix dx + dfracpi _loB^upB fracpi dx dfracpi leftfracpix^right_loA^upA + dfracpi leftfracpixright_loB^upB ex dfracpi leftfracpi fracpi^ - right + dfracpi leftfracpi pi - dfracpi dfracpiright dfracpi^ + fracpi - fracpi ex dfracpi^ + fracpi dfracpi^ + pi ed medskip edt a_ dfracpi _loA^upA fracpix cosx dx + dfracpi _loB^upB fracpi cosx dx ex dfracpi leftfracpix sinxright_loA^upA - fracpi _loA^upA dfracpi sinx dx + dfracpi leftfracpisinxright_loB^upB ex fracleftupA sinleftupAright - loA sinleftloArightright + fracbig cosx big_loA^upA + fracbigsinxbig_loB^upB ex -fracpi + fracleft cosleftupAright - cosleftloAright right + fracleftsinleftupBright - sinleftloBrightright ex -fracpi - frac + frac -fracpi ed itemb b_-dfrac b_-dfrac b_-dfrac b_-dfrac quad Rightarrow quad b_ dfrac medskip allgemein: b_k- dfrack-k-^ dfrack-k-^ enumerate
Die pi-periodisch fortsetzbaren Funktionen fx und -fx begrenzen den stark vereinfachten Achsenquerschnitt eines Mammutbaums. center tikzpicture mydispo tikzpicture center abcliste abc Berechne die Fourierkoeffizienten a_ und a_ von f exakt mit Lösungsweg. abc Gegeben sind die Fourierkoeffizienten b_-frac b_-frac b_-frac b_-frac von f. medskip Errate den Wert von b_ und gib eine explizite Formel an mit der jeder Term der Form b_k- für k dots berechnet werden kann. abcliste
Solution:
enumerate itema edt a_ dfracpi _loA^upA fracpix dx + dfracpi _loB^upB fracpi dx dfracpi leftfracpix^right_loA^upA + dfracpi leftfracpixright_loB^upB ex dfracpi leftfracpi fracpi^ - right + dfracpi leftfracpi pi - dfracpi dfracpiright dfracpi^ + fracpi - fracpi ex dfracpi^ + fracpi dfracpi^ + pi ed medskip edt a_ dfracpi _loA^upA fracpix cosx dx + dfracpi _loB^upB fracpi cosx dx ex dfracpi leftfracpix sinxright_loA^upA - fracpi _loA^upA dfracpi sinx dx + dfracpi leftfracpisinxright_loB^upB ex fracleftupA sinleftupAright - loA sinleftloArightright + fracbig cosx big_loA^upA + fracbigsinxbig_loB^upB ex -fracpi + fracleft cosleftupAright - cosleftloAright right + fracleftsinleftupBright - sinleftloBrightright ex -fracpi - frac + frac -fracpi ed itemb b_-dfrac b_-dfrac b_-dfrac b_-dfrac quad Rightarrow quad b_ dfrac medskip allgemein: b_k- dfrack-k-^ dfrack-k-^ enumerate
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