Check whether the set of integers is countable or not.
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Apply Bonnet Mean Value Theorem for integrals to show that
Verify the second mean value theorem for the function and
in the interval
.
Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
Show that is an algebraic number.
Determine the local minimum and local maximum values of the function f defined by .
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.