Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 22/en

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Warm-up-exercises

Exercise

Determine the Taylor polynomial of degree of the function

at the zero point.


Exercise

Determine all the Taylor polynomials of the function

at the point .


Exercise

Let be a convergent power series. Determine the derivative .


Exercise

Let be a polynomial and

Prove that the derivative has also the shape

where is a polynomial.


Exercise

We consider the function

Prove that for all the -th derivative satisfies the following property


Exercise

Determine the Taylor series of the function at point up to order (Give also the Taylor polynomial of degree at point , where the coefficients must be stated in the most simple form).


Exercise

Determine the Taylor polynomial of degree of the function

at point .


Exercise

Let

be a differentiable function with the property

Prove that for all .


Exercise

Determine the Taylor polynomial up to fourth order of the inverse of the sine function at the point with the power series approach described in Remark 22.8.




Hand-in-exercises

Exercise (4 points)

Find the Taylor polynomials in up to degree of the function


Exercise (4 points)

Discuss the behavior of the function

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.


Exercise (4 points)

Discuss the behavior of the function

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.


Exercise (4 points)

Determine the Taylor polynomial up to fourth order of the natural logarithm at point with the power series approach described in Remark 22.8 from the power series of the exponential function.


Exercise (8 points)

For let be the area of ​​a circle inscribed in the unit regular -gon. Prove that .




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