Teorema di Rolle 6
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
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
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:
Considera la funzione continua e due volte differenziabile f:tomathbbR tale che f f e f''xgeq per xin leftright. Dimostra che fxleq x per xin leftright.
Solution:
Considera la funzione ausiliaria gxfx-x. Allora abbiamo che g g. Ci fosse un massimo locale tra e avremmo un punto in cui la derivata seconda dovrebbe essere strettamente negativa ma siccome g''xf''xgeq questo non è possibile e dunque la funzione è costante oppure ha solo un minimo tra e . Di conseguenza la funzione gx non supera mai lo .
Considera la funzione continua e due volte differenziabile f:tomathbbR tale che f f e f''xgeq per xin leftright. Dimostra che fxleq x per xin leftright.
Solution:
Considera la funzione ausiliaria gxfx-x. Allora abbiamo che g g. Ci fosse un massimo locale tra e avremmo un punto in cui la derivata seconda dovrebbe essere strettamente negativa ma siccome g''xf''xgeq questo non è possibile e dunque la funzione è costante oppure ha solo un minimo tra e . Di conseguenza la funzione gx non supera mai lo .
Meta Information
Exercise:
Considera la funzione continua e due volte differenziabile f:tomathbbR tale che f f e f''xgeq per xin leftright. Dimostra che fxleq x per xin leftright.
Solution:
Considera la funzione ausiliaria gxfx-x. Allora abbiamo che g g. Ci fosse un massimo locale tra e avremmo un punto in cui la derivata seconda dovrebbe essere strettamente negativa ma siccome g''xf''xgeq questo non è possibile e dunque la funzione è costante oppure ha solo un minimo tra e . Di conseguenza la funzione gx non supera mai lo .
Considera la funzione continua e due volte differenziabile f:tomathbbR tale che f f e f''xgeq per xin leftright. Dimostra che fxleq x per xin leftright.
Solution:
Considera la funzione ausiliaria gxfx-x. Allora abbiamo che g g. Ci fosse un massimo locale tra e avremmo un punto in cui la derivata seconda dovrebbe essere strettamente negativa ma siccome g''xf''xgeq questo non è possibile e dunque la funzione è costante oppure ha solo un minimo tra e . Di conseguenza la funzione gx non supera mai lo .
Contained in these collections
| Title | Matched on |
|---|---|
| Teorema di Rolle 2 | tagstitle |
| Teorema di Rolle 3 | tagstitle |
| Teorema di Rolle 4 | tagstitle |
| Teorema di Rolle 5 | tagstitle |
| Teorema di Rolle 1 | tagstitle |
Similar exercises (6)
| Title | Matched on |
|---|---|
| Teorema di Rolle 2 | tagstitle |
| Teorema di Rolle 3 | tagstitle |
| Teorema di Rolle 4 | tagstitle |
| Teorema di Rolle 5 | tagstitle |
| Teorema di Rolle 1 | tagstitle |
| Teorema di Rolle 1 | tagstitle |