Contents

General Info

Summary and Analysis

Study Tools

Functions, Limits, and Continuity

Terms

Problems

Basic Terminology

We will frequently use the formal concept of a set, which is just a collection of
objects, called elements. Examples of sets include the real numbers
R
, the
integers, the set of names of the days in a week, and the set of letters in the
alphabet. One kind of set that we will encounter fairly often is called an
interval. The open interval
(a, b)
consists of the real numbers
x
such that
a < x < b
, while the closed interval
[a, b]
consists of the real numbers
x
such that
a≤x≤b
. If
x
is an element of the set
S
, we write
xâààS
. Thus
Πâààrealnumbers
,
1âàà(0, 2)
, and Tuesday
âàà
\. A function
f
from a set
S
to a set
T
is a rule that takes an
element of the set
S
and gives back an element of the set
T
. We denote this by
f : S→T
. The set
S
is called the domain of the function
f
and the
set
T
is called its range.

Suppose we have a function
f : S→T
, with
xâààS
. If
f
takes an element
xâààS
to
yâààT
, we write
f : xy
or
f (x) = y
, and say that "
f
maps
x
to
y
." We often call this element
y
the image of
x
under
f
, and
denote it by
f (x)
. This is illustrated in the figure below.

Figure %: Plot of a Function
f : S→T

If
f : S→T
and
g : T→U
, then we can define a new function
g
o
f : S→U
by
(g
o
f )(x) = g(f (x))
for each element
xâààS
. The
function
g
o
f
is called the composition of the functions
g
and
f

The graph of a function is the set of all points of the form
(x, f (x))
. One can draw
this by plotting points on a pair of coordinate axes, with the horizontal axis
corresponding to
x
, and the vertical corresponding to
f (x)
.

A function
f : S→T
is called invertible if there exists a function
g : T→S
such that
(g
o
f )(x) = x
for each element
xâààS
. If
f
is
invertible, then this function
g
is called the inverse of
f
. One way to tell if a
function is invertible is to look at its graph. A function is invertible if and only
if no horizontal line intersects the graph in more than one point. Take a moment to
convince yourself that this is true.

Examples

(1) The most familiar functions map the set of real numbers to itself. That is,
f : R→R
. An example is the function
f
such that for each
real number
x
,
f (x) = 2x
, i.e. the image of each element
x
is the element
2x
.
We may graph this function as follows:

Figure %: Plot of
f (x) = 2x

This graph is a line with
y
-intercept
0
and slope
2
. The function
f
has the
inverse
g : R→R
defined by
g(x) = x/2
.

The function denoted by
f (x) = 2x
may also be thought of as a function from the
integers to the integers. It is not, however, a function from the real numbers to the
integers, because when you put in a real number, you do not always get out an integer.
For example,
f (1/4) = 1/2
, and
1/2
is not an integer.

(2) As an example of a more exotic function, let us construct a function from the set
of names of the days in a week to the set of letters in the alphabet. We define the
function
g
to take in the name of a day in the week and to give out the first letter
in that name. For example,
g(Wednesday) = W
, and
g(Sunday) = g(Saturday) = S
. While this example shows how general the
concept of a function is, for the rest of this course we will focus on functions from
some subset of the real numbers to the real numbers.

Elementary Functions

In this section, we review the basic properties of the elementary functions
studied in pre-calculus courses. These functions will be our main focus when applying
the tools of differentiation and integration, so it is crucial to be familiar with
them. The elementary functions include the linear, polynomial, rational, power, and
trigonometric functions.

Linear Functions

We already saw one example of a linear function above,
f (x) = 2x
. A general linear
function (so called because its graph is a line) has the form
f (x) = ax + b
, where
a
and
b
are real numbers. The number
a
is called the slope of
f
and indicates
how steeply inclined is the graph of
f
. The number
b
is called the
$y$-intercept of
f
and is equal to
f (0)
, the value of the function when its
graph intersects the vertical axis, or the
y
-axis. This is illustrated in the
figure below:

Figure %: Plot of
f (x) = ax + b
and
y
-intercept at
b

All linear functions are invertible. The inverse of
f (x) = ax + b
is the function
g(x) = (1/a)x + (- b/a)
, which also happens to be linear. Check that
g
is indeed an
inverse for
f
.

There is an easy way to write down a linear function whose graph passes through two
given points with different
x
-coordinates. If
(x1, y1)
and
(x2, y2)
are two
points, the line through them has equation
(x2 - x1)(y - y1) = (y2 - y1)(x - x1)
. If
x1≠x2
, we may divide through by
(x2 - x1)
and add
y1
to each side to get
the function:

f (x) = y = (x - x1) + y1

This can be expanded into the standard form for linear functions, and doing so we find
the slope to be
and the
y
-intercept
y1 - x1
.

Linear functions are associated with constant rates of change. For example, suppose
you are pouring iced tea into a glass at a constant rate of
50
milliliters per
second. If the glass contains
65
milliliters of iced tea at time
t = 0
(where
t
is measured in seconds), then the number of milliliters of tea in the glass at time
t
is equal to
f (t) = 50x + 65
. The slope of the function
f
is equal to
50
and the
y
-intercept is equal to
65
.

We immediately see, by the horizontal line test, that this function
f
is not
invertible.

Polynomial functions arise in many physical situations. Suppose I drop a bowling ball
off the top of a 300-foot tall building. Then according to the principles of
Newtonian mechanics, the height (in feet) of the bowling ball
above ground, at time
t
seconds after the ball is dropped, is given by
h(t) = - g/2t2 + 300
, where g is a constant of acceleration (due to gravity). In order
to find out when the bowling ball hits the ground, we could solve the equation
h(t) = 0
for
t
.

Rational Functions

Rational functions are the functions obtained by taking the quotient of one
polynomial by another polynomial. A general rational function is therefore given by

f (x) = ,

where the
polynomial in the denominator must not be identically zero. Note that all polynomial
functions are also rational functions. Because the denominator may equal
0
for
certain values of
x
, the domain of a rational function
f
is not the entire set of
real numbers. An example of a rational function is
f (x) = (x - 2)/(x - 1)
, shown below for
0≤x≤2
. Note that this function is defined for all real
numbers
x
except for
x = 1
.

Figure %: Plot of
f (x) = (x - 2)/(x - 1)
for
0≤x≤2

Power Functions

Power functions are functions of the form
f (t) = Crt
, where
C
and
r
are real
numbers. The number
C
is called the initial value, and is equal to the value of the
function
f (t)
at
t = 0
. The number
r
is called the growth rate, the amount by
which the value of
f
is multiplied for each increase of
1
in the value of
t
.
Recall some properties of exponents:
r0 = 1
for any
r≠ 0
, and
rarb = ra+b
for any real number
r
. A special power function is the exponential function
f (t) = et
, where
e
is a constant approximately equal to
2.71828
. Such functions
often arise in calculating compound interest, and in many natural phenomena. We will
see another reason later on for why the number
e
is so special. The power function
f (t) = - 2(1/2)t
is shown below for
-2≤t≤2
.

Figure %: Plot of
f (t) = - 2(1/2)t
for
-2≤t≤2

By the horizontal line test, power functions (with
t≠ 0
) are invertible. Note,
however, that power functions take values only in either the positive or negative real
numbers (but not both), so the inverse function will not be defined for all real
numbers. Since the inverse function is not among the functions we have introduced so
far, we give it a new name. We define the logarithm function
g(x) = logr(x)
(with
the base
r
) to be the inverse function of
f (x) = rx
. Then if
y = f (x) = rx
, we have
x = g(y) = logr(y)
. The inverse functions of all power functions can be expressed in
terms of these logarithm functions.

Suppose there are
10
college students at a party at time
t = 0
and the number of
students at the party doubles every hour. Then the number of students at the party
t
hours after it starts is given by the function
s(t) = 10*2t
.

Trigonometric Functions

Though one first learns about the trigonometric functions while studying
triangles, perhaps the easiest way to define them is with a circle. We define the
cosine of a real number
t
,
cos(t)
, to be the
x
-coordinate of the point on the
unit circle that is
t
radians counterclockwise from the positive
x
-axis.
Similarly, the sine of
t
,
sin(t)
, is defined to be the
y
-coordinate of the
same point. The tangent of
t
is defined by taking a quotient of these two
functions:
tan(t) = sin(t)/cos(t)
. The graphs of the sine and cosine functions
behave in a periodic, wave-like, manner, since in traveling around the unit circle,
one eventually arrives back at the place where one started. The graph of
f (t) = sin(t)
is displayed below for
-2Π≤t≤2Π
.

Figure %: Plot of
f (t) = sin(t)
for
-2Π≤t≤2Π

Note that since the definition of the tangent function includes dividing by
cos(t)
, it is not defined when
cos(t) = 0
. The graph of
g(t) = tan(t)
is shown below for
-2Π≤t≤2Π
.

Figure %: Plot of
g(t) = tan(t)
for
-2Π≤t≤2Π

If we want to find inverses for the trigonometric functions, we must restrict their
domains so that they will pass the horizontal line test. Customarily, the domain of
the sine and tangent functions is restricted to
- Π/2≤t≤Π/2
and that of
the cosine function to
0≤t≤Π
. The inverse functions for the sine and
cosine will then have domain
-1≤t≤1
. We write the inverse functions of
sine, cosine, and tangent as
sin-1(t)
,
cos-1(t)
, and
tan-1(t)
,
respectively.

Trigonometric functions arise in many periodic physical phenomena, such as tides,
times of sunrise, and the motion of a pendulum or a mass on the end of a spring.