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