Ableitung des Integrals
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:
Sei ab ein kompaktes Intervall mit Endpunkten a b und sei f in Rab eine auf ab R-bare Funktion. Falls f bei x_ in ab stetig ist so ist F:x in ab mapsto _a^b ft ddt bei x_ diffbar und F'x_fx_.
Solution:
Beweis. Sei epsilon . Dann existiert ein delta mit |fx-fx_| epsilon für alle x in x_-delta x_+deltacap ab. Man verwet dies nun in Kombination mit der Dreiecksungleichung für das Riemann-Integral Satz . und der Intervalladditivität des Riemann-Integrals Satz . um die Aussage zu zeigen. Für x in x_-delta x_+deltacap ab gilt left|fracFx-Fx_x-x_-fx_right| left|fracx-x_left_a^x ft ddt-_a^x_ ft ddtright-fx_right| left|fracx-x__x_^x ft ddt-fx_right| left|fracx-x__x_^x ft ddt-fracx-x__x_^x fx_ ddtright| left|fracx-x__x_^x ft-fx_ ddtright| &leq fracx-x_ _x_^x |ft-fx_| ddt &leq fracx-x_ _x_^x epsilon ddt epsilon Analog gilt für x in x_-deltax_cap ab dass left|fracFx-Fx_x-x_-fx_right| left|fracFx_-Fxx-x_-fx_right| left|fracx-x__x^x_ ft-fx_ ddtright| &leq fracx-x__x^x_ |ft-fx_| ddt &leq epsilon Da epsilon beliebig war beweist dies lim limits_x rightarrow x_ fracFx-Fx_x-x_fx_ und damit den Satz.
Sei ab ein kompaktes Intervall mit Endpunkten a b und sei f in Rab eine auf ab R-bare Funktion. Falls f bei x_ in ab stetig ist so ist F:x in ab mapsto _a^b ft ddt bei x_ diffbar und F'x_fx_.
Solution:
Beweis. Sei epsilon . Dann existiert ein delta mit |fx-fx_| epsilon für alle x in x_-delta x_+deltacap ab. Man verwet dies nun in Kombination mit der Dreiecksungleichung für das Riemann-Integral Satz . und der Intervalladditivität des Riemann-Integrals Satz . um die Aussage zu zeigen. Für x in x_-delta x_+deltacap ab gilt left|fracFx-Fx_x-x_-fx_right| left|fracx-x_left_a^x ft ddt-_a^x_ ft ddtright-fx_right| left|fracx-x__x_^x ft ddt-fx_right| left|fracx-x__x_^x ft ddt-fracx-x__x_^x fx_ ddtright| left|fracx-x__x_^x ft-fx_ ddtright| &leq fracx-x_ _x_^x |ft-fx_| ddt &leq fracx-x_ _x_^x epsilon ddt epsilon Analog gilt für x in x_-deltax_cap ab dass left|fracFx-Fx_x-x_-fx_right| left|fracFx_-Fxx-x_-fx_right| left|fracx-x__x^x_ ft-fx_ ddtright| &leq fracx-x__x^x_ |ft-fx_| ddt &leq epsilon Da epsilon beliebig war beweist dies lim limits_x rightarrow x_ fracFx-Fx_x-x_fx_ und damit den Satz.
Meta Information
Exercise:
Sei ab ein kompaktes Intervall mit Endpunkten a b und sei f in Rab eine auf ab R-bare Funktion. Falls f bei x_ in ab stetig ist so ist F:x in ab mapsto _a^b ft ddt bei x_ diffbar und F'x_fx_.
Solution:
Beweis. Sei epsilon . Dann existiert ein delta mit |fx-fx_| epsilon für alle x in x_-delta x_+deltacap ab. Man verwet dies nun in Kombination mit der Dreiecksungleichung für das Riemann-Integral Satz . und der Intervalladditivität des Riemann-Integrals Satz . um die Aussage zu zeigen. Für x in x_-delta x_+deltacap ab gilt left|fracFx-Fx_x-x_-fx_right| left|fracx-x_left_a^x ft ddt-_a^x_ ft ddtright-fx_right| left|fracx-x__x_^x ft ddt-fx_right| left|fracx-x__x_^x ft ddt-fracx-x__x_^x fx_ ddtright| left|fracx-x__x_^x ft-fx_ ddtright| &leq fracx-x_ _x_^x |ft-fx_| ddt &leq fracx-x_ _x_^x epsilon ddt epsilon Analog gilt für x in x_-deltax_cap ab dass left|fracFx-Fx_x-x_-fx_right| left|fracFx_-Fxx-x_-fx_right| left|fracx-x__x^x_ ft-fx_ ddtright| &leq fracx-x__x^x_ |ft-fx_| ddt &leq epsilon Da epsilon beliebig war beweist dies lim limits_x rightarrow x_ fracFx-Fx_x-x_fx_ und damit den Satz.
Sei ab ein kompaktes Intervall mit Endpunkten a b und sei f in Rab eine auf ab R-bare Funktion. Falls f bei x_ in ab stetig ist so ist F:x in ab mapsto _a^b ft ddt bei x_ diffbar und F'x_fx_.
Solution:
Beweis. Sei epsilon . Dann existiert ein delta mit |fx-fx_| epsilon für alle x in x_-delta x_+deltacap ab. Man verwet dies nun in Kombination mit der Dreiecksungleichung für das Riemann-Integral Satz . und der Intervalladditivität des Riemann-Integrals Satz . um die Aussage zu zeigen. Für x in x_-delta x_+deltacap ab gilt left|fracFx-Fx_x-x_-fx_right| left|fracx-x_left_a^x ft ddt-_a^x_ ft ddtright-fx_right| left|fracx-x__x_^x ft ddt-fx_right| left|fracx-x__x_^x ft ddt-fracx-x__x_^x fx_ ddtright| left|fracx-x__x_^x ft-fx_ ddtright| &leq fracx-x_ _x_^x |ft-fx_| ddt &leq fracx-x_ _x_^x epsilon ddt epsilon Analog gilt für x in x_-deltax_cap ab dass left|fracFx-Fx_x-x_-fx_right| left|fracFx_-Fxx-x_-fx_right| left|fracx-x__x^x_ ft-fx_ ddtright| &leq fracx-x__x^x_ |ft-fx_| ddt &leq epsilon Da epsilon beliebig war beweist dies lim limits_x rightarrow x_ fracFx-Fx_x-x_fx_ und damit den Satz.
Contained in these collections
| Title | Matched on |
|---|---|
| Zwischenwerteigenschaft Ableitung | tags |
| Notwendige Bedingung für Extremum | tags |
| Beweis Integrationskonstante | tags |
| Integration von Potenzreihen | tags |
| Linearität des Riemann-Integrals | tags |