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



Warm-up-exercises

Determine the Taylor polynomial of degree   of the function

 

at the zero point.



Determine all the Taylor polynomials of the function

 

at the point  .



Let   be a convergent power series. Determine the derivatives  .



Let   be a polynomial and

 

Prove that the derivative   has also the shape

 

where   is a polynomial.



We consider the function

 

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

 



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).



Determine the Taylor polynomial of degree   of the function

 

at point

 



Let

 

be a differentiable function with the property

 

Prove that   for all  .



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 (4 marks)

Find the Taylor polynomials in   up to degree   of the function

 



Exercise (4 marks)

Discuss the behavior of the function

 

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



Exercise (4 marks)

Discuss the behavior of the function

 

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



Exercise (4 marks)

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 (6 marks)

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