Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 24



Warm-up-exercises

Compute the definite integral

 



Determine the second derivative of the function

 



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?



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

 

is differentiable and determine its derivative.



Let   be a continuous function. Consider the following sequence

 

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



Let   be a convergent series with   for all   and let  

be a Riemann-integrable function. Prove that the series

 

is absolutely convergent.



Let   be a Riemann-integrable function on   with

 

for all  . Show that if   is continuous at a point   with  , then

 



Prove that the equation

 

has exactly one solution  .



Let

 

be two continuous functions such that

 

Prove that there exists   such that  .





Hand-in-exercises

Exercise (2 marks)

Determine the area below the graph of the sine function between   and  .



Exercise (3 marks)

Compute the definite integral

 



Exercise (3 marks)

Determine an antiderivative for the function

 



Exercise (4 marks)

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

 



Exercise (4 marks)

We consider the function

 

with

 

Show, with reference to the function

 

that   has an antiderivative.



Exercise (3 marks)

Let

 

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