Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 22

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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 derivatives .



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 an remark.





Hand-in-exercises

Exercise

Find the Taylor polynomials in up to degree of the function



Exercise

Discuss the behavior of the function

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



Exercise

Discuss the behavior of the function

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



Exercise

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



Exercise

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