Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Two intervals are said to be equivalent if there exists a bijection (one-to-one and onto function) between them. A simpler criterion for intervals _________ _______ _______ ____ ___ __________ _____ _________.
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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: .
Using the principle of mathematical induction, show that
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Using Weiestrass M-test, show that the following series converges uniformly.
Prove that a strictly decreasing function is always one-one.
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.
Apply Bonnet Mean Value Theorem for integrals to show that