Linearität des Integrals von Treppenfunktionen
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
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Exercise:
Beweisen Sie folge Aussage: Die nicht-leere Menge mathcalTFab f in mathcalF ab | f textist eine Treppenfunktion der Treppenfunktionen auf dem Intervall ab ist ein Unterraum des Vektorraums mathcalFab der reellwertigen Funktionen auf ab. Des Weiteren ist die Abbildung : mathcalTFabrightarrow mathbbR linear. Das heisst für alle fg in mathcalTFab und s in mathbbR f+g in mathcalTFab sf in mathcalTFab und es gilt _a^b f+g ddx _a^b f ddx+_a^b f ddx _a^b sf ddx s _a^b f ddx
Solution:
Beweis. Falls zeta_f eine Zerlegung in Konstanzervalle von f und zeta_g eine Zerlegung in Konstanzervalle von g ist dann existiert eine gemeinsame Verfeinerung zeta ax_ x_ ... x_n- x_n b von zeta_f und zeta_g. Dies ist eine Zerlegung in Konstanzervalle von f und g. Seien c_...c_n respektive d_...d_n in mathbbR die Konstanzwerte von f respektive f bezüglich der Zerlegung zeta das heisst es gilt forall x in x_k-x_k:fxc_k textund gxd_k für alle k in ...n. Insbesondere ergibt dies für alle k in ...n forall x in x_k-x_k:fx+gxc_k textund sfxsc_k und erhalten f+g sf in mathcalTFab. Des Weiteren gilt _a^b f+g ddx If+gzeta _i^n c_k+d_kDelta x_k _i^n c_k+d_kDelta x_k Ifzeta+Igzeta _a^b f ddx+_a^b g ddx und ebenso _a^b sf ddx Isfzeta _i^n sc_kDelta x_k s_i^n c_kDelta x_k sIfzeta s_a^b f ddx
Beweisen Sie folge Aussage: Die nicht-leere Menge mathcalTFab f in mathcalF ab | f textist eine Treppenfunktion der Treppenfunktionen auf dem Intervall ab ist ein Unterraum des Vektorraums mathcalFab der reellwertigen Funktionen auf ab. Des Weiteren ist die Abbildung : mathcalTFabrightarrow mathbbR linear. Das heisst für alle fg in mathcalTFab und s in mathbbR f+g in mathcalTFab sf in mathcalTFab und es gilt _a^b f+g ddx _a^b f ddx+_a^b f ddx _a^b sf ddx s _a^b f ddx
Solution:
Beweis. Falls zeta_f eine Zerlegung in Konstanzervalle von f und zeta_g eine Zerlegung in Konstanzervalle von g ist dann existiert eine gemeinsame Verfeinerung zeta ax_ x_ ... x_n- x_n b von zeta_f und zeta_g. Dies ist eine Zerlegung in Konstanzervalle von f und g. Seien c_...c_n respektive d_...d_n in mathbbR die Konstanzwerte von f respektive f bezüglich der Zerlegung zeta das heisst es gilt forall x in x_k-x_k:fxc_k textund gxd_k für alle k in ...n. Insbesondere ergibt dies für alle k in ...n forall x in x_k-x_k:fx+gxc_k textund sfxsc_k und erhalten f+g sf in mathcalTFab. Des Weiteren gilt _a^b f+g ddx If+gzeta _i^n c_k+d_kDelta x_k _i^n c_k+d_kDelta x_k Ifzeta+Igzeta _a^b f ddx+_a^b g ddx und ebenso _a^b sf ddx Isfzeta _i^n sc_kDelta x_k s_i^n c_kDelta x_k sIfzeta s_a^b f ddx
Meta Information
Exercise:
Beweisen Sie folge Aussage: Die nicht-leere Menge mathcalTFab f in mathcalF ab | f textist eine Treppenfunktion der Treppenfunktionen auf dem Intervall ab ist ein Unterraum des Vektorraums mathcalFab der reellwertigen Funktionen auf ab. Des Weiteren ist die Abbildung : mathcalTFabrightarrow mathbbR linear. Das heisst für alle fg in mathcalTFab und s in mathbbR f+g in mathcalTFab sf in mathcalTFab und es gilt _a^b f+g ddx _a^b f ddx+_a^b f ddx _a^b sf ddx s _a^b f ddx
Solution:
Beweis. Falls zeta_f eine Zerlegung in Konstanzervalle von f und zeta_g eine Zerlegung in Konstanzervalle von g ist dann existiert eine gemeinsame Verfeinerung zeta ax_ x_ ... x_n- x_n b von zeta_f und zeta_g. Dies ist eine Zerlegung in Konstanzervalle von f und g. Seien c_...c_n respektive d_...d_n in mathbbR die Konstanzwerte von f respektive f bezüglich der Zerlegung zeta das heisst es gilt forall x in x_k-x_k:fxc_k textund gxd_k für alle k in ...n. Insbesondere ergibt dies für alle k in ...n forall x in x_k-x_k:fx+gxc_k textund sfxsc_k und erhalten f+g sf in mathcalTFab. Des Weiteren gilt _a^b f+g ddx If+gzeta _i^n c_k+d_kDelta x_k _i^n c_k+d_kDelta x_k Ifzeta+Igzeta _a^b f ddx+_a^b g ddx und ebenso _a^b sf ddx Isfzeta _i^n sc_kDelta x_k s_i^n c_kDelta x_k sIfzeta s_a^b f ddx
Beweisen Sie folge Aussage: Die nicht-leere Menge mathcalTFab f in mathcalF ab | f textist eine Treppenfunktion der Treppenfunktionen auf dem Intervall ab ist ein Unterraum des Vektorraums mathcalFab der reellwertigen Funktionen auf ab. Des Weiteren ist die Abbildung : mathcalTFabrightarrow mathbbR linear. Das heisst für alle fg in mathcalTFab und s in mathbbR f+g in mathcalTFab sf in mathcalTFab und es gilt _a^b f+g ddx _a^b f ddx+_a^b f ddx _a^b sf ddx s _a^b f ddx
Solution:
Beweis. Falls zeta_f eine Zerlegung in Konstanzervalle von f und zeta_g eine Zerlegung in Konstanzervalle von g ist dann existiert eine gemeinsame Verfeinerung zeta ax_ x_ ... x_n- x_n b von zeta_f und zeta_g. Dies ist eine Zerlegung in Konstanzervalle von f und g. Seien c_...c_n respektive d_...d_n in mathbbR die Konstanzwerte von f respektive f bezüglich der Zerlegung zeta das heisst es gilt forall x in x_k-x_k:fxc_k textund gxd_k für alle k in ...n. Insbesondere ergibt dies für alle k in ...n forall x in x_k-x_k:fx+gxc_k textund sfxsc_k und erhalten f+g sf in mathcalTFab. Des Weiteren gilt _a^b f+g ddx If+gzeta _i^n c_k+d_kDelta x_k _i^n c_k+d_kDelta x_k Ifzeta+Igzeta _a^b f ddx+_a^b g ddx und ebenso _a^b sf ddx Isfzeta _i^n sc_kDelta x_k s_i^n c_kDelta x_k sIfzeta s_a^b f ddx
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