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
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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 =
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval [1,0] is an open set.
b) Every bounded sequences is not convergent.
c) The function f : [- 2, 2] fi R defined by f (x) = 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 f :[0,2] fi R defined f (x) = x + [x] is not integrable
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Write the inequality 4 < 2x + 3 < 6 in the modulus form.
Show that is an algebraic number.
Check whether the limit exists or not?
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 .