Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 23/en

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Warm-up-exercises

Exercise

Determine the Riemann sum (Treppenintegral) over of the staircase function


Exercise

a) Subdivide the interval in six subintervals of equal length.

b) Determine the Riemann sum of the staircase function on , which takes alternately the values and on the subdivision constructed in a).


Exercise

Give an example of a function which assumes only finitely many values, but is not a staircase function.


Exercise

Let

be a staircase function and let

be a function. Prove that the composite is also a staircase function.


Exercise

Give an example of a continuous function

and a staircase function

such that the composite is not a staircase function.


Exercise

Determine the definite integral

explicitly with upper and lower staircase functions.


Exercise

Determine the definite integral

explicitly with upper and lower staircase functions.


Exercise

Let be a compact interval and let

be a function. Consider a sequence of staircase functions  such that and a sequence of staircase functions  such that . Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that is Riemann-integrable and that


Exercise

Kompaktes Intervall/Reelle Funktion/Riemann integrierbar auf Unterteilung/Fakt/Beweis/Aufgabe/en


Exercise

Let be a compact interval and let be two Riemann-integrable functions. Prove the following statements.

  1. If for all , then .
  2. If for all , then .
  3. We have .
  4. For we have .


Exercise

Let be a compact interval and let be a Riemann-integrable function. Prove that


Exercise

Let be a compact interval and let be two Riemann-integrable functions. Prove that is also Riemann-integrable.




Hand-in-exercises

Exercise (2 points)

Let

be two staircase functions. Prove that is also a staircase function.


Exercise (3 points)

Determine the definite integral

as a function of and explicitly with lower and upper staircase functions


Exercise (4 points)

Determine the definite integral

explicitly with upper and lower staircase functions.


Exercise (3 points)

Prove that for the function

neither the lower nor the upper integral exist.


Exercise (6 points)

Prove that for the function

the lower integral exists, but the upper integral does not exist.


Exercise (5 points)

Let be a compact interval and let

be a monotone function. Prove that is Riemann-integrable.




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