Fourier-Reihe
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:
Gegeben ist die Funktion displaystyle fx cases & text für -pi leq x -fracpi -fracpisqrt & text für -fracpi leq x -fracpi & text für -fracpi leq x fracpi fracpisqrt & text für fracpileq x fracpi & text für fracpi leq x pi fx+kpi & text sonst kin mathbbZ cases enumerate itema Zeichne eine Skizze des Graphen der Funktion fx. itemb Bilde die Fourierreihe zu fx d.h. bestimme die ersten sechs von null verschiedenen Glieder und beschreibe wie die Reihe weitergeht. itemc Tastet man die Funktion fx auf leq x pi Mal ab so entsteht die folge Zerlegung: zxa_ cosx -fracpicos x + pi frac+sqrt sinx +pi frac-sqrtsinx Berechne a_ enumerate
Solution:
In die Skizze eingezeichnet ist der Graph von fx sowie die Zerlegung von Aufgabe c. minipageb.linewidth enumerate item Siehe rechts item a_k forall k b_k falls k gerade fxsin x-dots dots -fracsin leftxright-fracsin left xright+dots dots +fracsin leftxright+fracsin leftxright -ldots ldots -fracsin leftxright-frac sin leftxright + ldots ldots +ldots +ldots -ldots -ldots enumerate minipage% enumerate itemc vec c_pmatrix cos cos fracpi cos fracpi cos fracpi cos pi cos fracpi cos fracpi cos fracpi pmatrixpmatrix fracsqrt -fracsqrt - -fracsqrt fracsqrt pmatrix vec ppmatrix fracpisqrt fracpisqrt -fracpisqrt -fracpisqrt pmatrix a_fracvec p vec c_vec c_ vec c_fracpi enumerate
Gegeben ist die Funktion displaystyle fx cases & text für -pi leq x -fracpi -fracpisqrt & text für -fracpi leq x -fracpi & text für -fracpi leq x fracpi fracpisqrt & text für fracpileq x fracpi & text für fracpi leq x pi fx+kpi & text sonst kin mathbbZ cases enumerate itema Zeichne eine Skizze des Graphen der Funktion fx. itemb Bilde die Fourierreihe zu fx d.h. bestimme die ersten sechs von null verschiedenen Glieder und beschreibe wie die Reihe weitergeht. itemc Tastet man die Funktion fx auf leq x pi Mal ab so entsteht die folge Zerlegung: zxa_ cosx -fracpicos x + pi frac+sqrt sinx +pi frac-sqrtsinx Berechne a_ enumerate
Solution:
In die Skizze eingezeichnet ist der Graph von fx sowie die Zerlegung von Aufgabe c. minipageb.linewidth enumerate item Siehe rechts item a_k forall k b_k falls k gerade fxsin x-dots dots -fracsin leftxright-fracsin left xright+dots dots +fracsin leftxright+fracsin leftxright -ldots ldots -fracsin leftxright-frac sin leftxright + ldots ldots +ldots +ldots -ldots -ldots enumerate minipage% enumerate itemc vec c_pmatrix cos cos fracpi cos fracpi cos fracpi cos pi cos fracpi cos fracpi cos fracpi pmatrixpmatrix fracsqrt -fracsqrt - -fracsqrt fracsqrt pmatrix vec ppmatrix fracpisqrt fracpisqrt -fracpisqrt -fracpisqrt pmatrix a_fracvec p vec c_vec c_ vec c_fracpi enumerate
Meta Information
Exercise:
Gegeben ist die Funktion displaystyle fx cases & text für -pi leq x -fracpi -fracpisqrt & text für -fracpi leq x -fracpi & text für -fracpi leq x fracpi fracpisqrt & text für fracpileq x fracpi & text für fracpi leq x pi fx+kpi & text sonst kin mathbbZ cases enumerate itema Zeichne eine Skizze des Graphen der Funktion fx. itemb Bilde die Fourierreihe zu fx d.h. bestimme die ersten sechs von null verschiedenen Glieder und beschreibe wie die Reihe weitergeht. itemc Tastet man die Funktion fx auf leq x pi Mal ab so entsteht die folge Zerlegung: zxa_ cosx -fracpicos x + pi frac+sqrt sinx +pi frac-sqrtsinx Berechne a_ enumerate
Solution:
In die Skizze eingezeichnet ist der Graph von fx sowie die Zerlegung von Aufgabe c. minipageb.linewidth enumerate item Siehe rechts item a_k forall k b_k falls k gerade fxsin x-dots dots -fracsin leftxright-fracsin left xright+dots dots +fracsin leftxright+fracsin leftxright -ldots ldots -fracsin leftxright-frac sin leftxright + ldots ldots +ldots +ldots -ldots -ldots enumerate minipage% enumerate itemc vec c_pmatrix cos cos fracpi cos fracpi cos fracpi cos pi cos fracpi cos fracpi cos fracpi pmatrixpmatrix fracsqrt -fracsqrt - -fracsqrt fracsqrt pmatrix vec ppmatrix fracpisqrt fracpisqrt -fracpisqrt -fracpisqrt pmatrix a_fracvec p vec c_vec c_ vec c_fracpi enumerate
Gegeben ist die Funktion displaystyle fx cases & text für -pi leq x -fracpi -fracpisqrt & text für -fracpi leq x -fracpi & text für -fracpi leq x fracpi fracpisqrt & text für fracpileq x fracpi & text für fracpi leq x pi fx+kpi & text sonst kin mathbbZ cases enumerate itema Zeichne eine Skizze des Graphen der Funktion fx. itemb Bilde die Fourierreihe zu fx d.h. bestimme die ersten sechs von null verschiedenen Glieder und beschreibe wie die Reihe weitergeht. itemc Tastet man die Funktion fx auf leq x pi Mal ab so entsteht die folge Zerlegung: zxa_ cosx -fracpicos x + pi frac+sqrt sinx +pi frac-sqrtsinx Berechne a_ enumerate
Solution:
In die Skizze eingezeichnet ist der Graph von fx sowie die Zerlegung von Aufgabe c. minipageb.linewidth enumerate item Siehe rechts item a_k forall k b_k falls k gerade fxsin x-dots dots -fracsin leftxright-fracsin left xright+dots dots +fracsin leftxright+fracsin leftxright -ldots ldots -fracsin leftxright-frac sin leftxright + ldots ldots +ldots +ldots -ldots -ldots enumerate minipage% enumerate itemc vec c_pmatrix cos cos fracpi cos fracpi cos fracpi cos pi cos fracpi cos fracpi cos fracpi pmatrixpmatrix fracsqrt -fracsqrt - -fracsqrt fracsqrt pmatrix vec ppmatrix fracpisqrt fracpisqrt -fracpisqrt -fracpisqrt pmatrix a_fracvec p vec c_vec c_ vec c_fracpi enumerate
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