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7 If a car has position p(t) = 3t 2 2t + 10 feet, where t is measured in seconds, then what is the velocity of that car at time t = 4 What is the average velocity of that car from t = 2 to t = 6 What is the greatest velocity over the time interval [4, 10] 8 In each of these problems, use the formula for the derivative of an inverse function to find [ f 1 ] (1) (a) f (0) = 1, f (0) = 2 (b) f (2) = 1, f (2) = 6 (c) f (3) = 1, f (3) = (d) f (1) = 1, f (1) = 20

18:

assignment would not have been valid and the compile would have failed This is one case in which the difference between value types and reference types might be important to a generic routine The value type constraint is the complement of the reference type constraint It simply ensures that any type argument is a value type, including a struct or an enum (In this context, a nullable type is not considered a value type) Here is an example:

// Demonstrate a value type constraint using System; struct MyStruct { // } class MyClass { // } class Test<T> where T : struct { T obj; public Test(T x) { obj = x; } // } class ValueConstraintDemo { static void Main() { // Both of these declarations are legal Test<MyStruct> x = new Test<MyStruct>(new MyStruct()); Test<int> y = new Test<int>(10); // But, the following declaration is illegal! Test<MyClass> z = new Test<MyClass>(new MyClass()); } }

One of the things that makes the derivative so important is its many applications to the study of functions and the study of physical processes Isaac Newton was motivated in his work by the applications of calculus, not by its theory Leibniz, on the other hand, cared mostly about the theory In this chapter we get our first exposure to some of these key ideas Each type of problem discussed in this chapter has its own special features Physics plays a significant role in most of them Certainly, throughout history, calculus and physics have been inextricably intertwined

In this program, Test is declared as shown here:

class Test<T> where T : struct {

Because T of Test now has the struct constraint, T can be bound only to value type arguments This means that Test<MyStruct> and Test<int> are valid, but Test<MyClass> is not To prove this, try removing the comment symbols from the start of the last line in the program and recompiling An error will be reported

Part I:

We know that the value of the derivative of a function f at a point x represents the slope of the tangent line to the graph of f at the point (x, f (x)) If that slope is positive, then the tangent line rises as x increases from left to right, hence so does the curve (we say that the function is increasing) If instead the slope of the tangent line is negative, then the tangent line falls as x increases from left to right, hence so does the curve (we say that the function is decreasing) We summarize: On an interval where f > 0 the graph of f goes uphill On an interval where f < 0 the graph of f goes downhill See Figure 31 With some additional thought, we can also get useful information from the second derivative If f = ( f ) > 0 at a point, then f is increasing Hence the slope of the tangent line is getting ever greater (the graph is concave up) The picture must be as in Figure 32(a) or 32(b) If instead f = ( f ) < 0 at a point then f is decreasing Hence the slope of the tangent line is getting ever less (the graph is concave down) The picture must be as in Figure 33(a) or 33(b) Using information about the first and second derivatives, we can render rather accurate graphs of functions We now illustrate with some examples

You ll recall that the four ways you can refer to your readers are by: pronouns shared interest use of their name or names inference

There is a variation of the base class constraint that allows you to establish a relationship between two type parameters For example, consider the following generic class declaration:

class Gen<T, V> where V : T {