文档介绍：A Preview of Calculus
Calculus is fundamentally different from the mathe- useful to have an overview of the subject before
matics that you have studied previously. Calculus is beginning its intensive study. Here we give a glimpse
less static and more dynamic. It is concerned with of some of the main ideas of calculus by showing
change and motion; it deals with quantities that how the concept of a limit arises when we attempt to
approach other quantities. For that reason it may be solve a variety of problems.
The Area Problem
A¡ The origins of calculus go back at least 2500 years to the ancient Greeks, who found
areas using the “method of exhaustion.” They knew how to find the area A of any poly-
A∞
A™ gon by dividing it into triangles as in Figure 1 and adding the areas of these triangles.
A£ A¢ It is a much more difficult problem to find the area of a curved figure. The Greek
method of exhaustion was to inscribe polygons in the figure and circumscribe poly-
gons about the figure and then let the number of sides of the polygons increase.
A=A¡+A™+A£+A¢+A∞
Figure 2 illustrates this process for the special case of a circle with inscribed regular
FIGURE 1 polygons.
A£ A¢ A∞ Aß A¶ иии A¡™иии
FIGURE 2
Let An be the area of the inscribed polygon with n sides. As n increases, it appears
that An es closer and closer to the area of the circle. We say that the area of the
The Preview Module is a numeri- circle is the limit of the areas of the inscribed polygons, and we write
cal and pictorial investigation of
A lim An
the approximation of the area of a circle n l ϱ
by inscribed and circumscribed polygons. The Greeks themselves did not use limits explicitly. However, by indirect reasoning,
Eudoxus (fifth century .) used exhaustion to prove the familiar formula for the area
of a circle: A r 2.
We will use a similar idea in Chapter 5 to find areas of regions of the type shown
in Figure 3. We will approximate the desired area A b