The
fundamental objects that we deal with in calculus are functions. Functions
arise whenever one quantity depends on another. Consider the following three
situations:
1.
The
area A of a circle depends on the radius r of the circle. The rule that
connects r and A is given by the equation A=πr2. With each positive
number r there is associated one value of A, and we say that A is a function
of r.
Year
|
Population
(billions)
|
1900
|
1.65
|
1910
|
1.75
|
1920
|
1.86
|
1930
|
2.07
|
1940
|
2.30
|
1950
|
2.56
|
1960
|
3.04
|
1970
|
3.71
|
1980
|
4.45
|
1990
|
5.28
|
2000
|
6.07
|
2.
The
human population of the world P depends on the time t. The table below gives
estimates of the world population P(t) at a time t, for certain years.
For instance,
P(1950)
= 2,560,000,000
But for each value of t there is
a corresponding value of P, and we say that P is a
function of t.
3.
The
cost C of mailing a first-class letter depends on the weight w of the letter.
Although there is no simple formula that connects w and c, the post office has
a rule for determining C when w is known.
Each
of these examples describes a rule whereby, given a number, another number is
assigned. In each case we say that the second number is a function of the first
number.
Definition: A
function f is a rule that assigns to each element x in a set A exactly one
element, called f(x), in a set B.
We
usually consider functions for which the sets A and B are sets of real numbers.
The
set A is called the domain of the
function.
The
number f(x) is the value of f at x
and is read “f of x.”
The
range of f is the set of all
possible values of f(x) as x varies throughout the domain.
A
symbol that represents an arbitrary number in the domain is called an independent variable.
A
symbol that represents a number in the range of f is called a dependent variable.
There
are four ways to represent a function:
1.
Verbally
(by a description in words)
2.
Numerically
( by a table of values)
3.
Visually
(by a graph)
4.
Algebraically
(by an explicit formula)
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ReplyDeleteThis is the first of a series of lectures on calculus that I'll be posting. The sequels are to follow, so stay tuned.
ReplyDeleteFurthermore, you may notice that some parts do not have examples. I normally give examples as illustrations during classes, so if you happen to be one of my apprentices(students), then you should be paying attention to what I say.
ReplyDelete