Grenzwerte und Ungleichungen
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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Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Seien a_n_n b_n_n reelle Folgen mit Grenzwerten alim limits_n rightarrow infty a_n blim limits_n rightarrow inftyb_n. abcliste abc Falls a_n leq b_n für alle n in mathbbN dann gilt auch a leq b. gilt nicht! Bsp. a_n b_nfracn abc Falls a b dann existiert ein N in mathbbN s.d. a_n b_n für alle n geq N. abcliste
Solution:
Beweis. Zuerst zweite Behauptung: Angenommen a b. dann ist epsilon fracb-a und es existiert ein N in mathbbN definiert als ein Maximum s.d. n geq N Longrightarrow a-epsilon a_n a+epsilon n geq N Longrightarrow b-epsilon b_n b+epsilon Da aber a+epsilon b-epsilon nach Wahl von epsilon ergibt sich a_n a+epsilon b-epsilon b_n für alle n geq N. Dies beweist auch die erste Behauptung. Denn falls a b wäre dann gäbe es nach obigem Folgenglieder a_nb_n mit a_n b_n.
Seien a_n_n b_n_n reelle Folgen mit Grenzwerten alim limits_n rightarrow infty a_n blim limits_n rightarrow inftyb_n. abcliste abc Falls a_n leq b_n für alle n in mathbbN dann gilt auch a leq b. gilt nicht! Bsp. a_n b_nfracn abc Falls a b dann existiert ein N in mathbbN s.d. a_n b_n für alle n geq N. abcliste
Solution:
Beweis. Zuerst zweite Behauptung: Angenommen a b. dann ist epsilon fracb-a und es existiert ein N in mathbbN definiert als ein Maximum s.d. n geq N Longrightarrow a-epsilon a_n a+epsilon n geq N Longrightarrow b-epsilon b_n b+epsilon Da aber a+epsilon b-epsilon nach Wahl von epsilon ergibt sich a_n a+epsilon b-epsilon b_n für alle n geq N. Dies beweist auch die erste Behauptung. Denn falls a b wäre dann gäbe es nach obigem Folgenglieder a_nb_n mit a_n b_n.
Meta Information
Exercise:
Seien a_n_n b_n_n reelle Folgen mit Grenzwerten alim limits_n rightarrow infty a_n blim limits_n rightarrow inftyb_n. abcliste abc Falls a_n leq b_n für alle n in mathbbN dann gilt auch a leq b. gilt nicht! Bsp. a_n b_nfracn abc Falls a b dann existiert ein N in mathbbN s.d. a_n b_n für alle n geq N. abcliste
Solution:
Beweis. Zuerst zweite Behauptung: Angenommen a b. dann ist epsilon fracb-a und es existiert ein N in mathbbN definiert als ein Maximum s.d. n geq N Longrightarrow a-epsilon a_n a+epsilon n geq N Longrightarrow b-epsilon b_n b+epsilon Da aber a+epsilon b-epsilon nach Wahl von epsilon ergibt sich a_n a+epsilon b-epsilon b_n für alle n geq N. Dies beweist auch die erste Behauptung. Denn falls a b wäre dann gäbe es nach obigem Folgenglieder a_nb_n mit a_n b_n.
Seien a_n_n b_n_n reelle Folgen mit Grenzwerten alim limits_n rightarrow infty a_n blim limits_n rightarrow inftyb_n. abcliste abc Falls a_n leq b_n für alle n in mathbbN dann gilt auch a leq b. gilt nicht! Bsp. a_n b_nfracn abc Falls a b dann existiert ein N in mathbbN s.d. a_n b_n für alle n geq N. abcliste
Solution:
Beweis. Zuerst zweite Behauptung: Angenommen a b. dann ist epsilon fracb-a und es existiert ein N in mathbbN definiert als ein Maximum s.d. n geq N Longrightarrow a-epsilon a_n a+epsilon n geq N Longrightarrow b-epsilon b_n b+epsilon Da aber a+epsilon b-epsilon nach Wahl von epsilon ergibt sich a_n a+epsilon b-epsilon b_n für alle n geq N. Dies beweist auch die erste Behauptung. Denn falls a b wäre dann gäbe es nach obigem Folgenglieder a_nb_n mit a_n b_n.
Contained in these collections
| Title | Matched on |
|---|---|
| Additive und multiplakative Eigenschaften des Grenzwerts | tags |
| Beschränktheit konvergenter Folgen | tags |
| Konvergenz von Teilfolgen | tags |
| Divergenz Folge (-1)^n | tags |
| Indexverschiebung | tags |