1. Chapter 1: Functions
Our world is full of changing quantities. Almost all quantities are functions of time. For instance, the motion of planets, sounds, etc.
Functions are fundamental objects in calculus.
They are used to describe relationship between quantities in daily life.

2. We just have a few basic types of functions:
polynomial, rational, exponential, logarithmic,
trigonometric, inverse trigonometric, root functions.
They are like building blocks. Using these
building blocks, we are able to create infinitely
many different functions to meet the needs of
applications.
There are many methods to create new functions.
Two basic methods creating new functions are:
composition
combination

3. Section 1.1: Definition I. Notation
Functions arise whenever one quantity depends
another.
Example:
(1)The area A of a circle depends on the radius r
of the circle.
(2)Your height H depends on the time t.
(3)The human population of the world P depends
on the time t.
(4)The cost C of mailing a letter depends on the
weight w of the letter.

4. Many problems in science and engineering
concern the relationship between two quantities.
The purpose of each problem is to:
find the rule that how one quantity, denoted by y,
depends on another, denoted by x.
(2) find the properties of the function once the
rule is determined.

5. Example: In the vacuum, the distance S in meters
fallen after t seconds is given by S = ½ (g t 2),
where g  9.8 m/s2 is the acceleration due to gravity.
This famous formula is called Galileo’s law.
It shows how S depends on t.
It tells us how to find the value of S when the value of time t is given.

12. Example(Verbally): A rectangle has perimeter 20 meters.
Express the area of the rectangle as a function of the length
of one of its sides.
Solution: Let x be the length of one side and y the length of the other side. Let A be the area. Then
A(x, y) = xy. This is a two variable function.
We need a single variable function.
How to eliminate one variable?
The question shows us the
relationship between x and y:
2x + 2y = 20  x + y = 10
 y = 10 – x.
Then A(x) = x (10 – x) with DA = (0, 10).

13. Example(Numerically):
The population of the world is listed below
the rule of this function is given by the table
DP={1900,1930,1960,2000}
RP={1650,2070,3040,6080}
Comment: We treat t as independent variable and P as dependent variable.
Year (t)
1900
1930
1960
2000
Population in millions (P)
1650
2070
3040
6080

14. Example(Visually): The following curve
defines a function. Find its domain and range.
Comment: The curve shows us the defining rule.
Question: Can you find an algebraic expression that
generates this graph?