Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 19



Warm-up-exercises

Determine the derivative of the functions

 

for all  .



Determine the derivative of the function

 

for all  .



Determine the derivative of the function

 

for all  .



Determine directly (without the use of derivation rules) the derivative of the function

 

at any point  .



Prove that the real absolute value

 

is not differentiable at the point zero.



Determine the derivative of the function

 



Prove that the derivative of a rational function is also a rational function.



Consider   and  . Determine the derivative of the composite function   directly and by the chain rule.



Prove that a polynomial   has degree   (or it is  ), if and only if the  -th derivative of   is the zero poynomial.



Let

 

be two differentiable functions and consider

 

a) Determine the derivative   from the derivatives of   and  . b) Let now

 

Compute   in two ways, one directly from   and the other by the formula of part  .



Let   be a field and let   be a  -vector space. Prove that given two vectors   there exists exactly one affine-linear map

 

sucht that   and  .



Determine the affine-linear map

 

such that   and  .





Hand-in-exercises

Exercise (3 marks)

Determine the derivative of the function

 

where   is the set where the denominator does not vanish.



Exercise (4 marks)

Determine the tangents to the graph of the function  , which are parallel to  .



Exercise (7 (2+2+3) marks)

Let

 

and

 

Determine the derivative of the composite

 

directly and by the chain rule.



Exercise (2 marks)

Determine the affine-linear map

 

whose graph passes through the two points

  and  .



Exercise (3 marks)

Let   be a subset and let

 

be differentiable functions. Prove the formula