a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
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Show that is an algebraic number.
Verify the second mean value theorem for the function and
in the interval
.
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
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.
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.
Using the principle of mathematical induction, show that
Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?