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# Elementary Functions and Calculus (Digital book, pp. 307)

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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 emphasis is on disentangling and explaining all 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.

## 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)

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
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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