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// Reverse an array using System; class RevCopy { static void Main() { int i,j; int[] nums1 = new int[10]; int[] nums2 = new int[10]; for(i=0; i < nums1Length; i++) nums1[i] = i; ConsoleWrite("Original contents: "); for(i=0; i < nums2Length; i++) ConsoleWrite(nums1[i] + " "); ConsoleWriteLine(); // Reverse copy nums1 to nums2 if(nums2Length >= nums1Length) // make sure nums2 is long enough for(i=0, j=nums1Length-1; i < nums1Length; i++, j--) nums2[j] = nums1[i]; ConsoleWrite("Reversed contents: "); for(i=0; i < nums2Length; i++) ConsoleWrite(nums2[i] + " "); ConsoleWriteLine(); } }

Here is the output:

Original contents: 0 1 2 3 4 5 6 7 8 9 Reversed contents: 9 8 7 6 5 4 3 2 1 0

We first calculate that if f ( x) = e x then f ( x) = ( 4x 2 2) e x and therefore | f ( x) | 2 = M for 0 x 1 In order to control the error, and to have two decimal places of accuracy, we need to have

Here, Length helps perform two important functions First, it is used to confirm that the target array is large enough to hold the contents of the source array Second, it provides the termination condition of the for loop that performs the reverse copy Of course, in this simple example, the size of the arrays is easily known, but this same approach can be applied to a wide range of more challenging situations

Part I:

A special case occurs when Length is used with jagged arrays In this situation, it is possible to obtain the length of each individual array For example, consider the following program, which simulates the CPU activity on a network with four nodes:

M ( b a) 3 < 0005 12k2 or 2 13 < 0005 12k2 Rearranging this inequality gives 100 < k2 3 Obviously k = 6 will do So we will use the partition P = {0, 1/6, 1/3, 1/2, 2/3, 5/6, 1} The corresponding trapezoidal sum is S= 1/6 2 2 2 2 e 0 + 2e ( 1/6) + 2e ( 1/3) + 2e ( 1/2) 2 +2e ( 2/3) + 2e ( 5/6) + e 1

// Demonstrate Length with jagged arrays using System; class Jagged { static void Main() { int[][] network_nodes = new int[4][]; network_nodes[0] = new int[3]; network_nodes[1] = new int[7]; network_nodes[2] = new int[2]; network_nodes[3] = new int[5];

is similar to the s-s-s-s sound in the word continuous And just as a little stream is always flowing, the word continuous means always in action Continual, by contrast, indicates stops and starts Continual refers to an action that occurs with pauses Continuous refers to an action that occurs without pauses

int i, j; // Fabricate some fake CPU usage data for(i=0; i < network_nodesLength; i++) for(j=0; j < network_nodes[i]Length; j++) network_nodes[i][j] = i * j + 70;

ConsoleWriteLine("Total number of network nodes: " + network_nodesLength + "\n"); for(i=0; i < network_nodesLength; i++) { for(j=0; j < network_nodes[i]Length; j++) { ConsoleWrite("CPU usage at node " + i + " CPU " + j + ": "); ConsoleWrite(network_nodes[i][j] + "% "); ConsoleWriteLine(); } ConsoleWriteLine(); } } }

The output is shown here:

2 2 2

Total number of network nodes: 4 CPU usage at node 0 CPU 0: 70% CPU usage at node 0 CPU 1: 70% CPU usage at node 0 CPU 2: 70% CPU CPU CPU CPU CPU usage usage usage usage usage at at at at at node node node node node 1 1 1 1 1 CPU CPU CPU CPU CPU 0: 1: 2: 3: 4: 70% 71% 72% 73% 74%

7:

CPU usage at node 1 CPU 5: 75% CPU usage at node 1 CPU 6: 76%

Some tedious but feasible calculation yields then that S= 1 {1 + 2 9726 + 2 8948 + 2 7880 + 2 6412 12 +2 4994 + 3679} = 89599 = 7451 12

CPU usage at node 2 CPU 0: 70% CPU usage at node 2 CPU 1: 72% CPU CPU CPU CPU CPU usage usage usage usage usage at at at at at node node node node node 3 3 3 3 3 CPU CPU CPU CPU CPU 0: 1: 2: 3: 4: 70% 73% 76% 79% 82%

Pay special attention to the way Length is used on the jagged array network_nodes Recall, a two-dimensional jagged array is an array of arrays Thus, when the expression