Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 24/en

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Warm-up-exercises

Exercise

Compute the definite integral


Exercise

Determine the second derivative of the function


Exercise

An object is released at time and it falls freely without air resistance from a certain height down to the earth thanks to the (constant) gravity force. Determine the velocity and the distance as a function of time . After which time the object has traveled meters?


Exercise

Let be a differentiable function and let be a continuous function. Prove that the function

is differentiable and determine its derivative.


Exercise

Let be a continuous function. Consider the following sequence

Determine whether this sequence converges and, in case, determine its limit.


Exercise

Let be a convergent series with for all and let

be a Riemann-integrable function. Prove that the series
is absolutely convergent.


Exercise

Let be a Riemann-integrable function on with for all . Show that if is continuous at a point with , then


Exercise

Prove that the equation
has exactly one solution .


Exercise

Let

be two continuous functions such that

Prove that there exists such that .




Hand-in-exercises

Exercise (2 points)

Determine the area below[1] the graph of the sine function between and .


Exercise (3 points)

Compute the definite integral


Exercise (3 points)

Determine an antiderivative for the function


Exercise (4 points)

Compute the area of ​​the surface, which is enclosed by the graphs of the two functions and such that


Exercise (4 points)

We consider the function

with

Show, with reference to the function , that has an antiderivative.


Exercise (3 points)

Let

be two continuous functions and let for all . Prove that there exists such that




Fußnoten
  1. Here we mean the area between the graph and the -axis.



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