MATH- 111 Instructor: Dr.
Naila Amir
Why mathematics?
•Mathematics is the
study of how to create,
manipulate & understand
abstract structures.
•We study math because it
teaches us a way of thinking.
•It provides us with a
method of solving a whole
host of life’s problems away
from the classroom
What is calculus??
And the Answer is……
Calculus (from Latin calculus,
literally “small pebble”, used
for counting and calculations)
is the study of how things
change. It provides a
framework for modeling
systems in which there is
change, and a way to
deduce the predictions of
such models.
History:
Calculus was developed out of a need to
understand continuously changing
quantities.
Newton, for example, was trying to
understand the effect of gravity which
causes falling objects to constantly Isaac Newton
accelerate.
How can one, for example, determine the
speed of a falling object at a frozen instant
in time, such as its speed when it strikes
the ground?
No mathematician prior to Newton and
Leibnitz's time could answer such a
Gottfried Wilhelm
question. Leibniz
Calculus is everywhere
The differentiation and
integration have many
real-world applications
from sports to
engineering to astronomy
and space travel.
Calculus in the engineering field
Calculus was initially developed for
better navigation system.
Engineers use calculus for building
skyscrapers, bridges.
In robotics calculus is used how
robotic parts will work on given
command.
Electrical and Computer engineers
use calculus for system design.
Calculus is used to improve safety
of vehicles.
Calculus for computer science
Calculus in computer science is just like
having the right tools for the job.
So many computer programs require calculus.
Computer scientists use calculus in creating
visuals or graphs. The graphs/visuals are
usually 3-dimensional.
These are used often for video games,
especially physics engines.
Physics engines define the physics in the
game such as gravity, friction, etc.
The military uses these visuals for
simulations, flight and artillery paths, maps,
satellite images, etc.
Architects use them for graphing buildings,
outlines, etc.
Any optimized software algorithm is
optimized through calculus methods. This
basically what calculus is - finding optimized
solutions/methods.
Calculus provides the foundation to physics, engineering, and many
higher math courses. It is also important to chemistry, astronomy,
economics and statistics. Medical schools and pharmacy schools use
it as a screening tool.
I can go on and on but let me stop with this observation:
“ today we can safely declare that there is no
branch of science which does not use
calculus. ”
The real world applications need calculus. That is, perhaps, one
reason why we demand calculus as a pre-requisite for all programs
in different fields of studies.
Enjoy the course!
Course Description:
The course reviews the concepts of basic
calculus; including Limits, continuity,
differentiation and integration.
A brief account of three dimensional geometry
and complex numbers is also included as pre-
calculus review.
Stress is laid on applications of differentiation
and integration to practical/engineering
problems.
Convergence/divergence of the sequence and
series are included towards the end of the
syllabus.
Course Objectives:
Upon the successful completion of course
students should develop understanding of the
basic concepts of analytical geometry involving
limits, continuity, differentiation and integration
for solving the real world problems and analyzing
the convergence/divergence of sequence and
series.
Course Learning Outcomes
(CLOs)
At the end of the course the students will be able to:
CLO1: Understand the concept of limit, continuity and
derivative with its application to find extrema.
CLO2: Understand integration and use it to compute areas,
volumes and arc length.
CLO3: Comprehend sequence, series and their convergence
using miscellaneous tests.
Marks Distribution
Functions
Domain and Range
Functions vs. Relations
• A "relation" is just a relationship
between sets of information.
• A “function” is a well-behaved
relation, that is, given a starting
point we know exactly where
to go.
Example
• Students and their heights, i.e. the
pairing of students and heights.
• We can think of this relation as
ordered pair:
• (height, student)
Or
• (student, height)
Name Height
Ali=1 6’=6
Usman=2 5’9”=5.75
Hina=3 5’=5
Alia=4 5’=5
Salar=5 6’6”=6.5
Salar
Alia
Hina
Usman
Ali
Ali Usman Hina Alia Salar
• Both graphs are relations
• (height, student) is not well-behaved .
• Given a height there might be several students corresponding to that
height.
• For a relation to be a function, there must be exactly one y value
that corresponds to a given x value.
Conclusion and Definition
• Not every relation is a function.
• Every function is a relation.
Definition:
Let X and Y be two nonempty sets.
A function from X into Y is a relation or rule
that assigns a unique (single) element y Є Y to
each element of x Є X .
Function
A relation such that there is no more than one
output for each input
A
W
B
Z
C
Algebraic Can be written as finite sums,
Function differences, multiples, quotients,
and radicals involving xn.
f x 3 x 2 x 10, g x 2 x1
Examples: 4 x4
Transcendental A function that is not Algebraic.
Function Examples:h x sin x , g x ln x
• In our example, the pairing of students
and heights.
x=student and y=height
• Variable x is called independent variable
• Variable y is called dependent variable
• For convenience, we use f(x) instead of y.
• The ordered pair in new notation becomes:
(x, y) = (x, f(x))
Domain and Range
• Suppose, we are given a function f from X into Y.
• Recall, for each element x in X there is exactly
one corresponding element y=f(x) in Y.
• This element y=f(x) in Y we call the image of x.
All possible input values (x) which allows
Domain the function to work. The is a collection of
all possible x-values.
All possible output values (y) which result
Range from using the function. The set of all
images as x varies throughout the domain.
f
x y
Note: The domain and range help determine
the window of a graph.
Our Example
• Domain = {Ali, Usman, Hina, Alia, Salar}
• Range = {6, 5.75, 5, 6.5}
Representations of
Functions
• Verbally
• Numerically, i.e. by a table
• Visually, i.e. by a graph
• Algebraically, i.e. by an explicit
formula
• Once we have decided on the
representation of a function, we ask
the following question:
• What are the possible x-values
(names of students from our
example) and y-values (their
corresponding heights) for our
function we can have?
• Recall, our example: the pairing of students and
heights.
x=student and y=height
• We can have many students for our x-value, but
what about heights?
• For our y-values we should not have 0 feet or 11
feet, since both are impossible.
• Thus, our collection of heights will be greater
than 0 and less that 11.
Graph Of Functions
Interval Notation
The interval does NOT include the endpoint(s)
Interval Notation Inequality Notation Graph
Parentheses < Less than Open Dot
( ) > Greater than
The interval does include the endpoint(s)
Interval Notation Inequality Notation Graph
Square Bracket ≤ Less than Closed Dot
[ ] ≥ Greater than
• Recall, the graph of (height, name):
Salar
Alia
Hina
Usman
Ali
What happens at the height = 5?
Vertical-Line Test
• A set of points in the xy-plane is the
graph of a function if and only if
every vertical line intersects the
graph in at most one point.
