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A predicate is a delegate of type SystemPredicate that returns either true or false, based upon some condition It is declared as shown here: public delegate bool Predicate<T> (T obj) The object to be tested against the condition is passed in obj If obj satisfies that condition, the predicate must return true Otherwise, it must return false Predicates are used by several methods in Array, including Exists( ), Find( ), FindIndex( ), and FindAll( ) The following program demonstrates using a predicate to determine if an array of integers contains a negative value If a negative value is found, the program then obtains the first negative value in the array To accomplish this, the program uses Exists( ) and Find( )

// Demonstrate Predicate delegate using System; class PredDemo { // A predicate method // It returns true if v is negative static bool IsNeg(int v) { if(v < 0) return true; return false; } static void Main() { int[] nums = { 1, 4, -1, 5, -9 }; ConsoleWrite("Contents of nums: "); foreach(int i in nums) ConsoleWrite(i + " "); ConsoleWriteLine(); // First see if nums contains a negative value if(ArrayExists(nums, PredDemoIsNeg)) { ConsoleWriteLine("nums contains a negative value"); // Now, find first negative value int x = ArrayFind(nums, PredDemoIsNeg); ConsoleWriteLine("First negative value is : " + x); }

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The inverse of the function f is the function g which assigns to x the power to which you need to raise 3 to obtain x For instance, g(9) = 2 because f (2) = 9 g(1/27) = 3 because f ( 3) = 1/27 g(1) = 0 because f (0) = 1 We usually call the function g the logarithm to the base 3 and we write g(x) = log3 x Logarithms to other bases are de ned similarly While this approach to logarithms has heuristic appeal, it has many drawbacks: we do not really know what 3x means when x is not a rational number; we have no way to determine the derivative of f or of g; we have no way to determine the integral of f or of g Because of these dif culties, we are going to use an entirely new method for studying logarithms It turns out to be equivalent to the intuitive method described above, and leads rapidly to the calculus results that we need

else ConsoleWriteLine("nums contains no negative values"); } }

The output is shown here:

Displacement (cm) Velocity (cm/rad) Acceleration (cm/rad/rad) Constraint 45 90 Cam rotation angle (deg) 135 180

Contents of nums: 1 4 -1 5 -9 nums contains a negative value First negative value is : -1

this says that logs convert multiplication to addition It turns out that this property alone uniquely determines the logarithm function Let (x) be a differentiable function with domain the positive real numbers and whose derivative function (x) is continuous Assume that satis es the multiplicative law (x y) = (x) + (y) ( ) for all positive x and y Then it must be that (1) = 0 and there is a constant C such that C (x) = x In other words x C dt (x) = 1 t A function (x) that satis es these properties is called a logarithm function The particular logarithm function which satis es (1) = 1 is called the natural

In the program, the method passed to Exists( ) and Find( ) for the predicate is IsNeg( ) Notice that IsNeg( ) is declared like this:

FIGURE 57 Cam follower displacement, velocity, and accleration obtained by spline synthesis with adjusted knot spacing in Example 3

static bool IsNeg(int v) {

The methods Exists( ) and Find( ) will automatically pass the elements of the array (in sequence) to v Thus, each time IsNeg( ) is called, v will contain the next element in the array

TABLE 54 Cam angle (degrees) 000 450 900 1350 1500 1800 2100 2250 2700

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