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Here Here Here Here is is is is nums1: 0 1 2 3 4 5 6 7 8 9 nums2: 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 nums2 after assignment: 0 1 2 3 4 5 6 7 8 9 nums1 after change through nums2: 0 1 2 99 4 5 6 7 8 9

x/2, and so forth In sum, the aggregate of the areas of all the trapezoids is 1 f (x0 ) + f (x1 ) 2 + = x+ 1 f (x1 ) + f (x2 ) 2 x ( ) x +

As the output shows, after the assignment of nums1 to nums2, both array reference variables refer to the same object

Part I:

among/between Note the tw in the word between The tw in between is like the tw in the word two The reference to the number two makes it easy to remember when to use between and when to use among Among is used for more than two Between is used in connection with two persons or things

A number of benefits result because C# implements arrays as objects One comes from the fact that each array has associated with it a Length property that contains the number of elements that an array can hold Thus, each array provides a means by which its length can be determined Here is a program that demonstrates this property:

1 f (xk 1 ) + f (xk ) 2

// Use the Length array property using System; class LengthDemo { static void Main() { int[] nums = new int[10]; ConsoleWriteLine("Length of nums is " + numsLength); // Use Length to initialize nums for(int i=0; i < numsLength; i++) nums[i] = i * i; // Now use Length to display nums ConsoleWrite("Here is nums: "); for(int i=0; i < numsLength; i++) ConsoleWrite(nums[i] + " "); ConsoleWriteLine(); } }

This program displays the following output:

Length of nums is 10 Here is nums: 0 1 4 9 16 25 36 49 64 81

In LengthDemo notice the way that numsLength is used by the for loops to govern the number of iterations that take place Since each array carries with it its own length, you can use this information rather than manually keeping track of an array s size Keep in mind that the value of Length has nothing to do with the number of elements that are actually in use Length contains the number of elements that the array is capable of holding When the length of a multidimensional array is obtained, the total number of elements that can be held by the array is returned For example:

x { f (x0 ) + 2 f (x1 ) + 2 f (x2 ) + + 2 f (xk 1 ) + f (xk )} 2

// Use the Length array property on a 3D array using System; class LengthDemo3D { static void Main() { int[,,] nums = new int[10, 5, 6]; ConsoleWriteLine("Length of nums is " + numsLength); } }

7:

The output is shown here:

He divided the assignment among the three managers He divided the assignment between the two managers

It is known that, if the second derivative of f on the interval [a, b] does not exceed M then the approximation given by the sum ( ) is accurate to within M (b a)3 12k2 [By contrast, the accuracy of the method of rectangles is generally not better than N (b a)2 , 2k where N is an upper bound for the rst derivative of f We see that the method of trapezoids introduces an extra power of (b a) in the numerator of the error estimate and, perhaps more importantly, an extra factor of k in the denominator]

As the output verifies, Length obtains the number of elements that nums can hold, which is 300 (10 5 6) in this case It is not possible to use Length to obtain the length of a specific dimension The inclusion of the Length property simplifies many algorithms by making certain types of array operations easier and safer to perform For example, the following program uses Length to reverse the contents of an array by copying it back-to-front into another array: