Question
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: .
Answer :
Word Count : 445
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 = __________ _________ ___ _________ ___ ____ ________ __________ __________ ____ ___.
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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 = __________ _________ ___ _________ ___ ____ ________ __________ __________ ____ ___.
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