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This chapter reviews key ideas from precalculus Master this chapter and you will be well-prepared for calculus

It is important to stress that AlphaRef can be assigned a reference to a MyClass<Beta> object only because T is covariant in IMyCoVarGenIF To prove this, remove out from in IMyCoVarGenIF s declaration of T, and then attempt to recompile the program The compilation will fail because the default strict type-checking will not allow the assignment It is possible for one generic interface to be inherited by another In other words, a generic interface with a covariant type parameter can be extended For example,

public interface IMyCoVarGenIF2<out T> : IMyCoVarGenIF<T> { // }

Part I:

Notice that out is specified only in the extending interface s declaration Specifying out in the base interface clause is not necessary or legal One last point: it is legal for IMyCoVarGenIF2 to not specify T as covariant However, doing so eliminates the covariance that extending IMyCoVarGetIF could provide Of course, making IMyCoVarGenIF2 invariant may be required for some uses Here are some restrictions that apply to covariance A covariant type parameter can be applied only to a method return type Thus, out cannot be applied to a type parameter that is used to declare a method parameter Covariance works only with reference types A covariant type cannot be used as a constraint in an interface method For example, this interface is illegal:

Number lines Graphing Sets Idea of a function Trigonometry Plotting the graph of a function Composition of functions Inverse of a function

Consider the writer s objective What does Paula want Jim to do Remember to express her objective as an action step What did you decide is Paula s objective Isn t it true that Paula wants Jim to cooperate, contact her with news, and include her in meetings and updates With that objective in mind, rewrite the E-mail to ensure a positive Empathy Index It s easy to be distracted from our objective by our emotions Listen to Paula s perspective The situation was embarrassing and humiliating I had no idea whether Jim was keeping me in the dark because he was try60

public interface IMyCoVarGenIF2<out T> { void M<V>() where V:T; // Error, covariant T cannot be used as constraint }

As it applies to a generic interface, contravariance is the feature that lets a method use an argument whose type is a base class of the type specified by the type parameter for that parameter In the past, because of the strict type-checking applied to generics, a method s argument type had to match the type parameter precisely Contravariance relaxes this rule in a type-safe way A contravariant type parameter is declared by preceding the type parameter with the keyword in To understand the effects of contravariance, we will again work through an example To begin, here is a contravariant generic interface called IMyContraVarGenIF Notice that its type parameter T is contravariant and it uses T in the declaration of a method called Show( )

// This generic interface supports contravariance public interface IMyContraVarGenIF<in T> { void Show(T obj); }

Calculus is one of the most important parts of mathematics It is fundamental to all of modern science How could one part of mathematics be of such central importance It is because calculus gives us the tools to study rates of change and motion All analytical subjects, from biology to physics to chemistry to engineering to mathematics, involve studying quantities that are growing or shrinking or moving---in other words, they are changing Astronomers study the motions of the planets, chemists study the interaction of substances, physicists study the interactions of physical objects All of these involve change and motion In order to study calculus effectively, you must be familiar with cartesian geometry, with trigonometry, and with functions We will spend this first chapter reviewing the essential ideas Some readers will study this chapter selectively, merely reviewing selected sections Others will, for completeness, wish to review all the material The main point is to get started on calculus ( 2)

Notice that T is specified as contravariant by preceding it with in Also, notice that the parameter type of obj is T Next, MyClass implements IMyContraVarGenIF, as shown here: