Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 27/en

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Warm-up-exercises

Exercise

Let

be an increasing function and . Show that the sequence , , converges to if and only if

holds, i.e. if the limit of the function for is .


Exercise

Let be an interval, a boundary point of and

a continuous function. Prove that the existence of the improper integral

does not depend on the choice of the starting point .


Exercise

Let be a bounded open interval and

a continuous function, which can be extended continuously to . Prove that the improper integral

exists.


Exercise

Formulate and prove computation rules for improper integrals (analogous to Lemma 23.5).


Exercise

Decide whether the improper integral

exists.


Exercise

Determine the improper integral


Exercise

Let be a bounded interval and let

be a continuous function. Let be a decreasing sequence in with limit and let be an increasing sequence in with limit . Assume that the improper integral exists. Prove that the sequence

converges to this improper integral.




Hand-in-exercises

Exercise (2 points)

Compute the energy that would be necessary to move the Earth, starting from the current position relative to the Sun, infinitely far away from the Sun.


Exercise (3 points)

Decide whether the improper integral

exists and compute it in case of existence.


Exercise (5 points)

Give an example of a not bounded, continuous function

such that the improper integral exists.


Exercise (2 points)

Decide whether the improper integral

exists and compute it in case of existence.


Exercise (4 points)

Decide whether the improper integral

exists.


Exercise (8 points)

Decide whether the improper integral

exists.

(Do not try to find an antiderivative for the integrand.)



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