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1 (a) (b) (c) (d) (e) (f) (a) F (x) = x 3 /3 + cos x + C F (x) = e3x /3 + x 5 /5 2x + C F (t) = t 3 + [ln t]2 /2 F (x) = ln(cos x) sin x [cos 3x]/3 + C F (x) = [sin 3x]/3 [cos 4x]/4 + x + C F (x) = esin x + C x sin x 2 dx = cos x 2 +C 2

With each iteration, an object reference called anotherOb is created, and it is assigned a reference to the object constructed by the factory At the end of each iteration of the loop, anotherOb goes out of scope, and the object to which it refers is recycled

Since in C# arrays are implemented as objects, a method can also return an array (This differs from C++ in which arrays are not valid as return types) For example, in the following program, the method FindFactors( ) returns an array that holds the factors of the argument that it is passed:

(235)

// Return an array using System; class Factor { /* Return an array containing the factors of num On return, numfactors will contain the number of factors found */ public int[] FindFactors(int num, out int numfactors) { int[] facts = new int[80]; // size of 80 is arbitrary int i, j;

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4

Part I:

(b - q )

// Find factors and put them in the facts array for(i=2, j=0; i < num/2 + 1; i++) if( (num%i)==0 ) { facts[j] = i; j++; } numfactors = j; return facts; } } class FindFactors { static void Main() { Factor f = new Factor(); int numfactors; int[] factors; factors = fFindFactors(1000, out numfactors); ConsoleWriteLine("Factors for 1000 are: "); for(int i=0; i < numfactors; i++) ConsoleWrite(factors[i] + " "); ConsoleWriteLine(); } }

The output is shown here:

(236)

(b) (c) (d) (e) (f) 3 (a) 3 3 ln x 2 dx = ln2 x 2 + C x 4 1 sin x cos x dx = sin2 x + C 2 1 tan x ln cos x dx = ln2 cos x + C 2 sec2 x etan x dx = etan x + C (2x + 1) (x 2 + x + 7)43 dx = We have

Factors for 1000 are: 2 4 5 8 10 20 25 40 50 100 125 200 250 500

In Factor, FindFactors( ) is declared like this:

(237)

public int[] FindFactors(int num, out int numfactors) {

Notice how the int array return type is specified This syntax can be generalized Whenever a method returns an array, specify it in a similar fashion, adjusting the type and dimensions as needed For example, the following declares a method called someMeth( ) that returns a two-dimensional array of double:

2 1 k

Note that the jerk values equal zero except where that value is in nite at the points of acceleration discontinuity

public double[,] someMeth() { //

In C#, two or more methods within the same class can share the same name, as long as their parameter declarations are different When this is the case, the methods are said to be overloaded, and the process is referred to as method overloading Method overloading is one of the ways that C# implements polymorphism In general, to overload a method, simply declare different versions of it The compiler takes care of the rest You must observe one important restriction: The type and/or number of the parameters of each overloaded method must differ It is not sufficient for two methods to differ only in their return types They must differ in the types or number of their parameters

This curve (follower motion) of the polynomial family has a triangular acceleration curve It is a modi cation of the parabolic curve, eliminating the abrupt change in acceleration at the beginning and the end of the stroke However it does have an acceleration discontinuity at the midpoint This curve has limited applications since it also has high velocities and high acceleration values, giving excessive pressure angle and inertia forces It is not very practical except when combined with other curves The construction of this curve (Fig 26) is similar to that of the acceleration curve previously with the increments being 1, 7, 19, and so forth, which is a cubic relationship in lieu of 1, 3, 5 as with the constant acceleration curve Thus it will not be shown here By methods shown for the parabolic curve, we can solve the cubic curve for 0 q b 2

8:

j =1

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