## Description

This **digital book contains** **19 lectures on **practically everything STEM students need to know about **Elementary Functions and Calculus** to succeed in their courses! The Lecture Notes contain useful Summaries (Cheat Sheets) and description of necessary Study Skills, including tips for preparing for tests and exams. The Notes are also supplied with answers to Socratic questions dispersed throughout and Solutions to suggested Home Exercises (Worked Examples).

The emphasis is on disentangling and explaining the necessary concepts. Teachers using these notes would enhance the quality of their teaching and by implication, students success. The Notes can be used on their own or to complement standard classroom materials.

**Product Summary:**

- Comprehensive Lecture Notes on College Algebra and Basic Calculus
- Socratic Questions and Suggested Answers
- Useful Summaries (also known as Cheat Sheets)
- A mathematics-specific Study Skills Guide, including tips for exam preparation
- Solutions to Exercises suggested for Self-study (Worked Examples)

**Table of Contents**

I. INTRODUCTION

II. CONCEPT MAPS

III. LECTURES

Lecture 4. FUNCTIONS

4.1 Variables

4.2 Functions

4.3 Elementary operations on functions

4.4 Order of Operations (extended)

4.5 Applications of real functions of real variable and operations on such functions

4.6 A historical note

4.7 Instructions for self-study

Lecture 5. Real FUNCTIONS of One Real Variable: Graphs, Polynomials

5.1 Graphical representation of real functions of one real variable

5.2 How to use graphs of real functions y = f(x) of one real variable

5.3 Applications of graphs

5.4 Elementary functions: monomials (natural powers) and polynomials

5.5 Instructions for self-study

Lecture 6. Real FUNCTIONS of One Real Variable: Exponential Functions, Logarithmic Functions, Inverse Functions

6.1 Exponential functions

6.2 Logarithmic functions

6.3 Revision: Trigonometry

6.4 A historical note

6.5 Instructions of self-study

Lecture 7. Real FUNCTIONS of One Real Variable: Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions

7.1 Trigonometric functions

7.2 Inverse trigonometric functions

7.3 Hyperbolic functions and their inverses

7.4 Instructions for self-study

Lecture 8. Real FUNCTIONS of One Real Variable: Sketching and Using Graphs, Simple Transformations

8.1 Sketching graphs using a table

8.2 Using graphs to find y1 given x1

8.3 Using graphs to find x1 given y1

8.4 Sketching graphs using simple transformations

8.5 Sketching graphs using several simple transformations

8.6 Sketching a parabola by simple transformations

8.7 Applications of simple transformations

8.8 Instructions for self-study

Lecture 9. Real FUNCTIONS of One Real Variable: Sketching Graphs by Simple Transformations (ctd.)

9.1 Sketching graphs using several simple transformations (ctd.)

9.2 Sketching graphs using pointwise operations

9.3 Instructions for self-study

Lecture 10. ALGEBRA: Addition of Complex Numbers, the Argand Diagram, Forms of Complex Numbers

10.1 Imaginary unity j

10.2 Applications of complex numbers: solving quadratic equations

10.3 Operations: the whole powers of j

10.4 Variables: definition of a complex number z

10.5 Operations: addition of complex numbers z1 and z2

10.6 Operations: subtraction of complex numbers

10.7 The Argand diagram and Cartesian form of a complex number

10.8 The Argand diagram and polar form of a complex number

10.9 The Cartesian-polar and polar-Cartesian transformations

10.10 The trigonometric form of a complex number

10.11 Euler’s formula

10.12 The exponential form of a complex number

10.13 Applications

10.14 A historical note

10.15 Instructions for self-study

Lecture 11. ALGEBRA: Multiplication and Division of Complex Numbers

11.1 Multiplication of a complex number z by a real number

11.2 Operations: multiplication of complex numbers z1 and z2

11.3 Operations: complex conjugate z* of a complex number z

11.4 Operations: division of complex numbers z1 and z2

11.5 Integer powers of complex numbers

11.6 The fractional powers (k-th roots) of complex numbers

11.7 Instructions for self-study

Lecture 12. ALGEBRA: Fractional Powers, Logs and Loci of Complex Numbers

12.1 The fractional powers (whole roots k) of complex numbers (ctd.)

12.2 Logs of complex numbers

12.3 Loci on the Argand diagram

12.4 Applications

12.5 Instructions for self-study

Lecture 13. CALCULUS: Sequences, Limits and Series

13.1 Sequences

13.2 Limit of a sequence

13.3 Series

13.4 Instructions for self-study

Lecture 14. DIFFERENTIAL CALCULUS: Limits, Continuity and Differentiation of Real Functions of One Real Variable

14.1 Limits

14.2 Continuity of a function

14.3 Differentiation of a real function of one real variable

14.4 A historical note

14.5 Instructions for self-study

Lecture 15. DIFFERENTIAL CALCULUS: Differentiation (ctd.)

15.1 Differentiation Table

15.2 Differentiation Rules

15.3 Decision Tree for Differentiation

15.4 The higher order derivatives

15.5 The partial derivatives

15.6 Applications of differentiation

15.7 Instructions for self-study

Lecture 16. DIFFERENTIAL CALCULUS: Sketching Graphs Using Analysis

16.1 Stationary points

16.2 Increasing and decreasing functions

16.3 Maxima and minima

16.5 Convex and concave functions

16.6 Inflexion points

16.7 Sketching rational functions using analysis

16.8 Applications of rational functions and their graphs

16.9 Instructions for self-study

Lecture 17. Application of DIFFERENTIAL CALCULUS to Approximation of Functions: the Taylor and Maclaurin Series

17.1 Approximating a real function of a real variable using its first derivative

17.2 The Maclaurin polynomials

17.3 The Taylor polynomials

17.4 The Taylor series

17.5 The Maclaurin series

17.8 Instructions for self-study

Lecture 18. INTEGRAL CALCULUS: Integration of Real Functions of One Real Variable (Definite Integrals)

18.1 A definite integral

18.2 Notation

18.3 Comparison of a definite Integral and derivative

18.4 Examples of integrable functions

18.5 The Mean Value Theorem

18.6 A definite integral with a variable upper limit – function Φ(x)

18.7 The Fundamental Theorem of Calculus

18.8 Applications of integration

18.9 A historical note

18.10 Instructions for self-study

Lecture 19. INTEGRAL CALCULUS: Integration of Real Functions of One Real Variable (Indefinite Integrals)

19.1 The relationship between differentiation and integration

19.2 An indefinite integral

19.3 Finding an indefinite integral

19.4 The Integration Table

19.5 Elementary integration rules

19.6 Integration Decision Tree

19.7 Integration method (rule) of change of variable (substitution)

19.8 Instructions for self-study

Lecture 20. INTEGRAL CALCULUS: Advanced Integration Methods

20.1 Integration of products of trigonometric functions

20.2 Integration by parts (integration of products of different types of functions)

20.2 Partial fractions

20.3 Integration of rational functions

20.4 Decision Tree for Integration

20.5 Instructions for self-study

Lecture 21. INTEGRAL CALCULUS: Applications of Integration

21.1 Mean value of a function

21.2 Electrical systems

21.3 Mechanical systems

21.4 Rotational systems

21.5 Instructions for self-study

Lecture 22. Ordinary Differential Equations

22.1 Basic concepts

22.2 Order of a differential equation

22.3 Linearity or non-linearity of a differential equation

22.4 Differential equations with constant coefficients

22.5 Homogeneous and inhomogeneous ODEs

22.6 The first order linear homogeneous equation with constant coefficients

22.7 The initial value problems

22.8 Balance equations in chemical engineering

22.9 Ordinary differential equations with complex coefficients

IV. SUMMARIES

Algebra Summary

Functions Summary

Order of Operations Summary

Quadratics Summary

Trigonometry Summary

Complex Numbers

Decision Tree For Solving Simple Equations

Sketching Graphs by Simple Transformations

Finding a Limit of a Sequence

Differentiation Summary

Integration Summary

V. GLOSSARY

VI. STUDY SKILLS

VII. TEACHING METHODOLOGY (FAQs)

VIII. SOLUTIONS TO EXERCISES FOR SELF-STUDY (WORKED EXAMPLES)

Solutions to Exercises in Lecture

Solutions to Exercises in Lecture 4

Solutions to Exercises in Lecture 5

Solutions to Exercises in Lecture 6

Solutions to Exercises in Lecture 7

Solutions to Exercises in Lecture 8

Solutions to Exercises in Lecture 9

Solutions to Exercises in Lecture 10

Solutions to Exercises in Lecture 11

Solutions to Exercises in Lecture 12

Solutions to Exercises in Lecture 13

Solutions to Exercises in Lecture 14

Solutions to Exercises in Lecture 15

Solutions to Exercises in Lecture 16

Solutions to Exercises in Lecture 17

Solutions to Exercises in Lecture 18

Solutions to Exercises in Lecture 19

Solutions to Exercises in Lecture 20

Solutions to Exercises in Lecture 21

Solutions to Exercises in Lecture 22

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