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: .
a) Consider the function (f(x) = 2 \cos x) over ([0, \pi/2]) with partitions [ P_1 = {0, \pi/3, \pi/2}, \quad P_2 = {0, \pi/6, \pi/3, \pi/2}. ] Step 1: Lower and Upper sums for (P_1) * Interval ([0, \pi/3]): (f_{\min} = \min_{[0, \pi/3]} 2\cos x = 2 \cos(\pi/3) = 2 \cdot 1/2 = 1) (f_{\max} = \max_{[0, \pi/3]} 2\cos x = 2 \cos 0 = 2) (\Delta x = \pi/3 - 0 = \pi/3) Contribution to sums: (L_1^{[0,\pi/3]} = 1 \cdot \pi/3 = \pi/3) (U_1^{[0,\pi/3]} = 2 \cdot _________ ___ ____ _________ __________ ___ ___ ________ _________ ____.
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Find the following limit
Use Cauchy's mean value theorem to prove that:
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Show that the equation has a real root other than
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: .
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
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Also check whether the function f is derivable at x = 1.
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.