Grenzwert
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 Funktion f: rightarrow mathbbR sei gleichmässig stetig auf . Man beweise dass der Grenzwert displaystylelim_xrightarrow + fx existiert.
Solution:
Zu zeigen ist eigentlich: forall Folgen x_n mit x_n und lim x_n : displaystylelim_nrightarrowinfty fx_n existiert und er hängt nicht von der Auswahl der Folge ab. compactitem item Sei leftfx_nright_ninmathbbN eine Cauchy-Folge. Sei weiter varepsilon . Dann exists delta so dass aus |x-x'|delta folgt |fx-fx'|varepsilon wegen der gleichmässigen Stetigkeit. Da weiter x_nrightarrow Rightarrow exists n_:forall n n_ gilt x_ndelta. Also ist forall pnge n_: |x_n+p-x_n|delta und damit |fx_n+p-fx_nright|varepsilon. item Seien nun x_nrightarrow &quad x_nquadmboxund y_nrightarrow &quad y_n mit displaystylelim_nrightarrowinftyfx_n neq lim_nrightarrowinftyfy_n. Dann exists leftfz_nright_ninmathbbN mit z_nx_n z_n+y_n divergent da HP. Somit ist der Grenzwert unabhängig von der Wahl der Folge. compactitem
Die Funktion f: rightarrow mathbbR sei gleichmässig stetig auf . Man beweise dass der Grenzwert displaystylelim_xrightarrow + fx existiert.
Solution:
Zu zeigen ist eigentlich: forall Folgen x_n mit x_n und lim x_n : displaystylelim_nrightarrowinfty fx_n existiert und er hängt nicht von der Auswahl der Folge ab. compactitem item Sei leftfx_nright_ninmathbbN eine Cauchy-Folge. Sei weiter varepsilon . Dann exists delta so dass aus |x-x'|delta folgt |fx-fx'|varepsilon wegen der gleichmässigen Stetigkeit. Da weiter x_nrightarrow Rightarrow exists n_:forall n n_ gilt x_ndelta. Also ist forall pnge n_: |x_n+p-x_n|delta und damit |fx_n+p-fx_nright|varepsilon. item Seien nun x_nrightarrow &quad x_nquadmboxund y_nrightarrow &quad y_n mit displaystylelim_nrightarrowinftyfx_n neq lim_nrightarrowinftyfy_n. Dann exists leftfz_nright_ninmathbbN mit z_nx_n z_n+y_n divergent da HP. Somit ist der Grenzwert unabhängig von der Wahl der Folge. compactitem
Meta Information
Exercise:
Die Funktion f: rightarrow mathbbR sei gleichmässig stetig auf . Man beweise dass der Grenzwert displaystylelim_xrightarrow + fx existiert.
Solution:
Zu zeigen ist eigentlich: forall Folgen x_n mit x_n und lim x_n : displaystylelim_nrightarrowinfty fx_n existiert und er hängt nicht von der Auswahl der Folge ab. compactitem item Sei leftfx_nright_ninmathbbN eine Cauchy-Folge. Sei weiter varepsilon . Dann exists delta so dass aus |x-x'|delta folgt |fx-fx'|varepsilon wegen der gleichmässigen Stetigkeit. Da weiter x_nrightarrow Rightarrow exists n_:forall n n_ gilt x_ndelta. Also ist forall pnge n_: |x_n+p-x_n|delta und damit |fx_n+p-fx_nright|varepsilon. item Seien nun x_nrightarrow &quad x_nquadmboxund y_nrightarrow &quad y_n mit displaystylelim_nrightarrowinftyfx_n neq lim_nrightarrowinftyfy_n. Dann exists leftfz_nright_ninmathbbN mit z_nx_n z_n+y_n divergent da HP. Somit ist der Grenzwert unabhängig von der Wahl der Folge. compactitem
Die Funktion f: rightarrow mathbbR sei gleichmässig stetig auf . Man beweise dass der Grenzwert displaystylelim_xrightarrow + fx existiert.
Solution:
Zu zeigen ist eigentlich: forall Folgen x_n mit x_n und lim x_n : displaystylelim_nrightarrowinfty fx_n existiert und er hängt nicht von der Auswahl der Folge ab. compactitem item Sei leftfx_nright_ninmathbbN eine Cauchy-Folge. Sei weiter varepsilon . Dann exists delta so dass aus |x-x'|delta folgt |fx-fx'|varepsilon wegen der gleichmässigen Stetigkeit. Da weiter x_nrightarrow Rightarrow exists n_:forall n n_ gilt x_ndelta. Also ist forall pnge n_: |x_n+p-x_n|delta und damit |fx_n+p-fx_nright|varepsilon. item Seien nun x_nrightarrow &quad x_nquadmboxund y_nrightarrow &quad y_n mit displaystylelim_nrightarrowinftyfx_n neq lim_nrightarrowinftyfy_n. Dann exists leftfz_nright_ninmathbbN mit z_nx_n z_n+y_n divergent da HP. Somit ist der Grenzwert unabhängig von der Wahl der Folge. compactitem
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Uniprüfung Analysis F5 by uz