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Calculus With Tech I
Instructor:
Dr. Chekad Sarami
What is calculus?
Calculus is a branch of mathematics
that deals with rates of change.
Modern Calculus began with Newton
and Leibnitz in the 17th century.
Today it is used extensively in many
areas of science. Basic ideas of
calculus include the idea of limit ,
derivative , and integral .
Examples
The derivative of a function is its
instantaneous rate of change, with respect
to something else. Thus,
the derivative of height , (with respect to
position) is slope ;
the derivative of position , (with respect to
time) is velocity ; and
the derivative of velocity (with respect to
time) is acceleration.
Why is calculus Extremely
important?
In the sciences, many processes involving change,
or related variables, are studied.
If these variables are linked in a way that involves
chance, and significant random variation, statistics
is one of the main tools used to study the
connections.
In cases where a deterministic model is at least a
good approximation, calculus is a powerful tool to
study the ways in which the variables interact.
Situations involving rates of change over time, or
rates of change from place to place, are particularly
important examples.
Applications
Physics, astronomy, mathematics, and engineering
make particularly heavy use of calculus;
it is difficult to see how any of those disciplines
could exist in anything like its modern form without
calculus.
Biology, chemistry, economics, computing science,
and other sciences use calculus too.
Many faculties of science therefore require a
calculus course from all their students; in other
cases you may be able to choose between, say,
calculus, statistics, and computer programming.
Resources
http://www.stewartcalculus.com/media/3_h
ome.php
http://archives.math.utk.edu/visual.calculus/
index.html
http://web.rollins.edu/~dchild/calcAssist/
http://www.brookscole.com/cgiwadsworth/course_products_wp.pl?fid=M20
bI&product_isbn_issn=0534409865&discipli
ne_number=1
Section 1.1
Functions
and
Models
Let X and Y be two nonempty sets of real
numbers. A function from X into Y is a
relation that associates with each element of
X a unique element of Y.
The set X is called the domain of the
function.
For each element x in X, the
corresponding element y in Y is called the
image of x. The set of all images of the
elements of the domain is called the
range of the function.
f
x
y
x
y
x
X
DOMAIN
Y
RANGE
Example: Which of the following relations
are function?
{(1, 1), (2, 4), (3, 9), (-3, 9)}
A Function
{(1, 1), (1, -1), (2, 4), (4, 9)}
Not A Function
Functions are often denoted by letters such as f,
F, g, G, and others. The symbol f(x), read “f of
x” or “f at x”, is the number that results when x is
given and the function f is applied.
Elements of the domain, x, can be though of as
input and the result obtained when the function
is applied can be though of as output.
Restrictions on this input/output machine:
1. It only accepts numbers from the
domain of the function.
2. For each input, there is exactly one
output (which may be repeated for
different inputs).
For a function y = f(x), the variable x is
called the independent variable, because it
can be assigned any of the permissible
numbers from the domain.
The variable y is called the dependent
variable, because its value depends on x.
The independent variable is also called
the argument of the function.
Example: Given the function f ( x) 2x 5
2
Find:
f (3)
f (3) 2(3) 5 23
2
f (x) is the number that results when the
number x is applied to the rule for f.
Find: f ( x h)
f ( x h) 2( x h) 5
2
2( x 2 xh h ) 5
2
2
2x 4xh 2h 5
2
2
The domain of a function f is the set of real
numbers such that the rule of the function
makes sense.
Domain can also be thought of as the set of
all possible input for the function machine.
Example: Find the domain of the following function:
g ( x ) 3x 5x 1
3
Domain: All real numbers
Example: Find the domain of the following function:
4
s( t )
t 1
Domain of s is t | t 1 .
Example: Find the domain of the following function:
h( z ) z 2
z20
z 2
Domain of h is z| z 2 .
Example: Express the area of a circle as a function of
its radius.
A(r ) r
2
The dependent variable is A and the independent
variable is r.
The domain of the function is {r | r 0}
Four ways to represent a
Function
verbal
numerical
visual
algebraic
http://archives.math.utk.edu/visual.calcu
lus/0/functions.11/index.html
The graph of f(x) is given below.
y
4
(2, 3)
(4, 0)
0
(0, -3)
-4
(1, 0)
(10, 0)
x
What is the domain and range of f ?
Domain: [0,10]
Range: [-3,3]
Find f(0), f(4), and f(12)
f(0) = -3
f(4) = 0
f(12) does not exist since 12 isn’t in
the domain of f
A function f is even if for every number x in
its domain the number -x is also in the
domain and f(x) = f(-x).
A function is even if and only if its graph is
symmetric with respect to the y-axis.
A function f is odd if for every number x in
its domain the number -x is also in the
domain and -f(x) = f(-x).
A function is odd if and only if its graph
is symmetric with respect to the origin.
y
Example of an Even
Function. It is symmetric
about the y-axis
(0,0)
x
y
Example of an Odd
Function. It is symmetric
about the origin
(0,0)
x
Determine whether each of the following functions
is even, odd, or neither. Then determine whether
the graph is symmetric with respect to the y-axis or
with respect to the origin.
a.)
g (z) z 2 2
g ( z) ( z) 2 z 2
2
2
g ( z) g ( z)
Even function, graph symmetric with
respect to the y-axis.
b.)
f ( x ) 4 x 5 3 x
f ( x ) 4( x ) 5 3( x ) 4 x 5 3x
f ( x) f ( x)
Not an even function.
f ( x ) ( 4 x 5 3 x ) 4 x 5 3 x
f ( x ) 4( x ) 3( x ) 4 x 3x
5
5
f ( x) f ( x)
Odd function, and the graph is
symmetric with respect to the origin.
A function f is increasing on an open
interval I if, for any choice of x1 and x2 in I,
with x1 < x2, we have f(x1) < f(x2).
A function f is decreasing on an open
interval I if, for any choice of x1 and x2 in I,
with x1 < x2, we have f(x1) > f(x2).
A function f is constant on an open interval I
if, for any choice of x in I, the values of f(x)
are equal.
Determine where the following graph is
increasing, decreasing and constant.
Increasing on (0,2)
y
4
Decreasing on (2,7)
(2, 3)
Constant on (7,10)
(4, 0)
0
(0, -3)
-4
(1, 0)
x
(7, -3)
(10, -3)
Section 1.2
Mathematical
Models:
A Catalog of
Essential
Functions
The following library of functions will be used
throughout the text. Be able to recognize the shape
of each graph and associate that shape with the
given function.
y (0,c)
The Constant Function
f ( x) c
The Identity Function
f ( x) x
x
y
(0,0)
x
y
The Square Function
f (x) x2
(0,0)
x
y
The Cube Function
f (x) x3
(0,0)
x
y
The Square Root Function
f ( x) x
x
(0,0)
y
The Reciprocal Function
1
f (x)
x
(1,1)
x
(-1,-1)
The Absolute
Value Function
f ( x) x
y
(0,0)
x
y
The Cube Root
Function
f ( x)
3
x
x
When functions are defined by more than
one equation, they are called piecewisedefined functions.
Example: The function f is defined as:
x 3 - 2 x 1
f ( x) 3
x 1
x 3 x 1
a.) Find f (1) = 3
Find f (-1) = (-1) + 3 = 2
Find f (4) = - (4) + 3 = -1
x 3 - 2 x 1
f ( x) 3
x 1
x 3 x 1
b.) Determine the domain of f
Domain: [2, ) in interval notation
or {x | x 2} in set builder notation
c.) Graph f
y
3
2
1
1 2 3
x
d.) Find the range of f from the graph
found in part c.
y
3
2
1
1 2 3
x
Range: ( , 4 ) in interval notation
or { y | y 4} in set builder notation
Power Functions
A polynomial function is a function of the
form
f ( x ) a n x a n 1 x
n
n 1
a 1 x a 0
where an , an-1 ,…, a1 , a0 are real numbers
and n is a nonnegative integer. The domain
consists of all real numbers. The degree of
the polynomial is the largest power of x that
appears.
Example: Determine which of the following are
polynomials. For those that are, state the degree.
(a) f ( x ) 3x 4 x 5
2
Polynomial of degree 2
(b) h( x ) 3 x 5
Not a polynomial
5
3x
(c) F ( x )
5 2x
Not a polynomial