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This is the composition of functions, so we must apply the Chain Rule It is essential to recognize what function will play the role of f and what function will play the role of g Notice that, if x is the variable, then x 3 x 2 is applied first and sin applied next So it must be that g( x) = x 3 x 2 and f ( s) = sin s Notice that d d f ( s) = cos s and dx g( x) = 3x 2 2x Then ds sin( x 3 x 2 ) = f g( x)

Because the return type of GetOb( ) is T, which was replaced by int when iOb was declared, the return type of GetOb( ) is also int Thus, this value can be assigned to an int variable

18:

Next, GenericsDemo declares an object of type Gen<string>:

Gen<string> strOb = new Gen<string>("Generics add power");

= cos( g( x) ) ( 3x 2 2x) = cos( x 3 x 2 ) ( 3x 2 2x)

Because the type argument is string, string is substituted for T inside Gen This creates a string version of Gen, as the remaining lines in the program demonstrate Before moving on, a few terms need to be defined When you specify a type argument such as int or string for Gen, you are creating what is referred to in C# as a closed constructed type Thus, Gen<int> is a closed constructed type In essence, a generic type, such as Gen<T>, is an abstraction It is only after a specific version, such as Gen<int>, has been constructed that a concrete type has been created In C# terminology, a construct such as Gen<T> is called an open constructed type, because the type parameter T (rather than an actual type, such as int) is specified More generally, C# defines the concepts of an open type and a closed type An open type is a type parameter or any generic type whose type argument is (or involves) a type parameter Any type that is not an open type is a closed type A constructed type is a generic type for which all type arguments have been supplied If those type arguments are all closed types, then it is a closed constructed type If one or more of those type arguments are open types, it is an open constructed type

A key point to understand about generic types is that a reference of one specific version of a generic type is not type-compatible with another version of the same generic type For example, assuming the program just shown, the following line of code is in error and will not compile:

iOb = strOb; // Wrong!

Create a Draft on Paper (or on Your Computer)

h = f g,

Even though both iOb and strOb are of type Gen<T>, they are references to different types because their type arguments differ

At this point, you might be asking yourself the following question Given that the same functionality found in the generic Gen class can be achieved without generics, by simply specifying object as the data type and employing the proper casts, what is the benefit of making Gen generic The answer is that generics automatically ensure the type safety of all operations involving Gen In the process, generics eliminate the need for you to use casts and type-check code by hand To understand the benefits of generics, first consider the following program that creates a non-generic equivalent of Gen:

// NonGen is functionally equivalent to Gen but does not use generics using System; class NonGen { object ob; // ob is now of type object // Pass the constructor a reference of type object public NonGen(object o) {

Part I:

( x 2) 2x x 1 ( x 2) 2