If t is a real number such that
0<t<2, calculate .
Given the function , find the points
where it gets a relative maximum. (The integral function is
supposed to be extended so that it is everywhere continuous).
Solve the following inequality: .
f is a function continuous together with its first
and second derivative; the tangents to the graph of
f in the numbers a and b create an
angle whose measure is, respectively, of π/3 and π/4
with he positive x-axis. Calculate the following two
integrals: .
Given the function , find the
intervals where the function is increasing and those where
the function is convex.
Given the function , find the values
of the numbers a and b such that
f(0)=0 and . Graph the
function.
Calculate .
Given the function , calculate its
integral function with starting point -1. What can say about
the derivative of this integral function?
By direct calculation of the integral find the derivative of
the function . Find again the same
derivative without calculating the integral.
Find the domain of the following functions:
.
.
.
.
.
Prove that if f is an integrable function over
[-a,a] and f(-x)=-f(x) than . Give a geometrical interpretation of this result.
The integral of a function is unchanged if we alter the
values of a fucntion in a finite number of points. However
this is no more true if we alter the function in a countably
infinite number of points. Proof this with an example (sugg.:
use Dirichlet's function).