By James Stewart

This article via best-selling writer James Stewart embodies the large ideas of calculus reform--conceptual realizing stimulated through real-world purposes and the applying of the "Rule of 4" in numerical, visible, algebraic, and verbal interpretations. while, this article keeps the simplest of conventional calculus. Stewart emphasizes visualization and challenge fixing.

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**Example text**

A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable. 1 x (input) f ƒ (output) FIGURE 2 Machine diagram for a function ƒ ƒ x a f(a) f D FOUR WAYS TO REPRESENT A FUNCTION 13 It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f ͑x͒ according to the rule of the function.

As we will see in Chapter 2, it represents the average rate of change of f ͑x͒ between x a and x a ϩ h. 2a 2 ϩ 4ah ϩ 2h 2 Ϫ 5a Ϫ 5h ϩ 1 Ϫ 2a 2 ϩ 5a Ϫ 1 h 4ah ϩ 2h 2 Ϫ 5h 4a ϩ 2h Ϫ 5 h Representations of Functions There are four possible ways to represent a function: ■ verbally (by a description in words) ■ numerically (by a table of values) ■ visually (by a graph) ■ algebraically (by an explicit formula) If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function.

In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing a¡ 1 (a) 1 lim nlϱ 1 2 3 4 5 6 7 8 1 0 n n In general, the notation (b) FIGURE 10 lim a n L nlϱ is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large. 8 A PREVIEW OF CALCULUS The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number.