PRE-CALCULUS intro.pdf by Dr.Mamoona Anam

Pre-Calculus is a mathematics course (or set of topics) that prepares students for Calculus. It acts as a bridge between algebra, geometry, and calculus, bringing together various concepts that are essential for understanding the more advanced ideas in calculus.

📚 Main Topics in Pre-Calculus:
Functions and Graphs

Linear, quadratic, polynomial, rational, exponential, logarithmic, and piecewise functions

Transformations of functions

Inverse functions

Trigonometry

Trigonometric ratios and identities

Unit circle

Graphs of sine, cosine, tangent, etc.

Solving trigonometric equations

Analytic Geometry

Conic sections: circles, ellipses, parabolas, hyperbolas

Distance and midpoint formulas

Complex Numbers

Imaginary unit
𝑖
i, complex plane, operations with complex numbers

Sequences and Series

Arithmetic and geometric sequences

Sigma notation

Mathematical induction (sometimes)

Matrices and Determinants (optional in some curriculums)

Vectors (optional in some curriculums)

Limits (Introductory) (sometimes included as a transition to calculus)

1. PRE-CALCULUS
By
Dr. Mamoona Anam

2. COURSE INTRODUCTION
•This course is basically designed to develop and inculcate the basic
concepts needed for the social sciences students at undergraduate
level.
•Since Pre-calculus courses vary from one institution to the next, we
have attempted to meet the needs of as broad an audience as possible,
including all of the contents that might be covered in any particular
course.
•An effort has been made to make these contents easy and
understandable.
•It is the need of the hour to understand the applications of
Mathematics in studying different subjects of social sciences and other
fields too.

3. OBJECTIVES
The course aims to teach students, the basic concepts of
Mathematics and create value for their applications and uses
in practical situations.
The objectives of the course are:
To develop an understanding and desire of Mathematics in
students
To present the material in a way which is helpful in motivating
the students to study these subjects at a higher level
To develop an approach of application of Math in social
sciences subjects

4. HOW IS THE
COURSE
ORGANIZED?
The text is divided into nine units in one volume.
An attempt has been made to present the
material in an informal way. Only those topics of
Mathematics are covered which are thoughtful to
be useful for everyone, e.g., we started studying
Mathematics when we were in grade one and we
continue its studies.
In spite of this, most of us do not have full
command on everyday arithmetic.

5. Unit 1:
This unit is focused on the study of real numbers.
The aim is to introduce the set of real numbers and
their subsets.
The algebraic and order properties of real numbers
are also studied.
Also, the completeness property of the real
numbers is presented.
At the end, absolute value is defined for real
number and associated properties are shown.

6. Unit 2:
In unit two the notion of sets is introduced.
Basic concepts of union, intersection and
complement of a set is discussed.
All properties relevant to sets are also
studied as well.
The last part is devoted to the introduction
of Venn’s diagram and its applications.

7. Unit 3:
In this unit, the notion of functions is introduced.
To start with, methods to compute domain and range
are also discussed.
The use of arrow diagram for functions is also
discussed.
In the latter half of the unit, the types of functions are
introduced along with the associated examples.
These type include trigonometric, exponential, even
and odd functions.

8. Unit 4:
In this particular unit, the concept of limit
is introduced in a very natural way.
Limit theorems are discussed which are
beneficial for computing limits.
The concept of limit for infinite case is also
studied along with appropriate examples
and methods to compute the limits.

9. Unit 5:
Unit-5 dwelt upon the concepts of continuity.
This unit consist of definition of continuity as well
graphical picture of continuous functions.
The unit also explores relationship between limit
and continuity as well the properties of continuous
functions.
At last, composition of continuous function is
studied with appropriate examples.

10. Unit 6:
In this unit, general notion of the sequence and series is discussed in
detail.
The unit is divided into two parts.
In first part, definition of sequence is presented.
Well known sequence such as arithmetic sequence, geometric sequences
are introduced.
Some examples are listed to develop understanding of these sequences.
In the second part, series is introduced.
The arithmetic series as well geometric series is also studied.
At the end, application of series is also studied.

11. Unit 7:
The focuse of Unit 7 is on the trigonometric
functions.
This unit contains the definition of the trigonometric
functions, how to compute the domains and their
ranges.
Furthermore, the graphs of the trigonometric
functions are constructed.
At the end, some applications of trigonometric
functions are given.

12. Unit 8:
In this unit, the idea of differentiation is presented.
The idea is introduced in connection with slope of tangent lines.
The initial part is devoted to computing the derivative of a function.
The higher order derivatives as well as theorem of derivatives are also
discussed.
The unit also explored the computing derivative via chain rule as well as
implicit differentiation.
The derivatives of inverse trigonometric function are also studied.
The application of derivative for computing indeterminate form is also
studied along with examples.
At last, derivatives are applied to solve maxima and minima problems.

13. Unit 9:
This unit studies the idea of integration.
At start the idea is introduced in accordance with anti-derivative.
Integrals of some well-known functions are also listed.
Then some theorems associated with integrals are studied.
The second half is focused on technique of integration.
The method of substitution and integration by parts is illustrated with examples.
In last potion, definite integrals and fundamental theorem of calculus is studied.
Appropriate examples are listed to illustrate the theorem.
Moreover, application of definite integrals is studied for computing area under the curve.