Check whether the intervals [5,9] and [6,12] are equivalent or not.
In real analysis, two intervals are considered equivalent if they have the same length or measure. The length of an interval [a, b] is given by the absolute difference between its endpoints, i.e., |b -
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Check whether the limit exists or not?
If the partition P2 is a refinement of the partition P1 of [a,b] then L(P1,f) < L(P2,f) and U(P2,f) < U(p1,f) Verify this result for the function f(x) = 2cos x defined over the interval and the partitions p1 =
and p2 =
Prove that a strictly decreasing function is always one-one
Show that Rn(x), the lagrange’s form of remainder in the Maclaurin series expansion of , e4s tends to zero as n fi ¥. Hence obtain the Maclaurin’s infinite expansion for e4s .
Determine the local minimum and local maximum values of the function f defined by f(x) = 3 - 5x3 + 5x4 - x5.
Evaluate:
Check whether the set of integers is countable or not.