Part A) Write down explixitly the following partitions. 1)The partition that div
ID: 2867537 • Letter: P
Question
Part A)
Write down explixitly the following partitions.
1)The partition that divides [0,2] into seven sub intervals of equal length.
2). The partition that divides [-1,1] into five sub-intervals, the first of which (from left to right) has length 1/5, the 2nd 1/7, the third 1/2, and the fourth 1/2.
3) The partition that divides [0,3] into n sub-intervals of equal length, for an arbitrary n.
Part B)
Write down the norms of the three partitions in Problem 3. You may name the partitions as P1,P2,P3.
Explanation / Answer
1)The partition that divides [0,2] into seven sub intervals of equal length.
Answer :
For equal partitions,
Partition length = (b - a) / n = (2 - 0) / 7 = 2/7
So, the partitions are :
[0 , 2/7) , [2/7 , 4/7) , [4/7 , 6/7) , [6/7 , 8/7) , [8/7 , 10/7) , [10/7 , 12/7) , [12/7 , 2] ---> ANSWER
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2). The partition that divides [-1,1] into five sub-intervals, the first of which (from left to right) has length 1/5, the 2nd 1/7, the third 1/2, and the fourth 1/2.
Answer :
Let first partition be [-1 , x]
x - (-1) = 1/5
x + 1 = 1/5
x = 1/5 - 1
x = -4/5
So, first partition is [-1 , -4/5]
Let second partition be [-4/5 , y]
y - (-4/5) = 1/7
y + 4/5 = 1/7
y = 1/7 - 4/5
y = -23/35
So, second partition is [-4/5 , -23/35]
Third part has length 1/2
So, (z - (-23/35)) = 1/2
z + 23/35 = 1/2
z = 1/2 -23/35
z = -11/70
So, third partition is [-23/35 , -11/70]
Fourth has length 1/2
So, (v - (-11/70)) = 1/2
v + 11/70 = 1/2
v = 1/2 - 11/70
v = 12/35
So, fourth partition is [-11/70 , 12/35]
And fifth is of course [12/35 , 1]
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3) The partition that divides [0,3] into n sub-intervals of equal length, for an arbitrary n.
partition length = (b - a) / n
= (3 - 0) / n
= 3 / n
So, this splits the partition as :
[0 , 3/n) , [3/n , 6/n) , [6/n , 9/n) , ....... ---> ANSWER