One of the difficulties that arise in teaching mathematics is related to the identification of the target and the most appropriate teaching methods for the people who are part of it.
This aspect, true for all disciplines, applies to mathematics in particular. In fact, for example, an axiomatic approach is certainly suitable for Mathematical, Physical and Engineering Sciences, while students of many applied sciences, such as Agricultural and Life Sciences, need to focus on calculation tools and methodologies useful for their professional development rather than in dealing with the theoretical foundations of mathematics.
The peculiarity of this book is not so much in setting classical approach “Theorem: Hypothesis, Thesis” with relative proofs, but in adopting a more pragmatic approach that renounce classical demonstrations, while maintaining a formal coherence in the topics dealt with.
In this perspective, considering the approach required by the target to which it is addressed, the objective of this book is to provide methods to studying the variation of a phenomenon and its cumulative effects and consequently the study of the functions and the calculation of integrals respectively.
One of the qualifying features is given by a series of completely resolved problems, occupying two-thirds of the volume, in which each mathematical step is detailed to understand “step by step” how to obtain the solution.
Author(s): Claudio Caprara
Series: Mathematics Research Developments
Publisher: Nova Science Publishers
Year: 2022
Language: English
Pages: 403
City: New York
Contents
Preface
Chapter 1
Principles of Set Theory
1.1. Introduction and Definitions
1.2. Set Theory
1.3. Notion of Relation
Chapter 2
Real Numbers
2.1. Set of Real Numbers
2.2. Sets of Natural, Integer, Rational and Irrational Numbers
2.3. Limit Superior and Limit Inferior of a Set
2.3.1. Upper Bound of a Set
2.3.2. Limit Superior of a Set
2.3.3. Lower Bound of a Set
2.3.4. Limit Inferior of a Set
2.3.5. Maximum and Minimum of a Set
Chapter 3
Functions of Real Variable
3.1. Definition of Function
3.2. Surjection, Injection and Bijection of a Function
3.3. Examples of Functions
3.4. Composite Function
3.5. Inverse Function
3.6. Some Considerations on the Domain of a Function
3.7. Absolute Value
Chapter 4
Limit of a Function
4.1. Neighborhood of a Point
4.2. Accumulation Point
4.3. Definition of Limit
4.3.1. Continuous Functions
4.3.2. Definition of Limit for Particular Values
4.4. Calculus of Function Limits
4.4.1. Application Examples
4.4.2. Methodological Considerations
4.5. Properties of Limits
4.6. Limit for x that Tends to x0 from the Right or from the Left
4.7. Application Examples of Limit from Right and Left
4.7.1. Example 1
4.7.2. Example 2
4.8. Exponential Function
4.9. Logarithm Function
4.10. Properties of the Logarithm Function
4.11. Natural Logarithms and Logarithms to the Base 10
4.11.1. Napier’s Number and Logarithms Definition
4.11.2. Conversion of Natural Logarithm to Common Logarithm
4.12. Power Function
4.13. Trigonometric Functions
4.14. Sine and Cosine Functions
4.15. Tangent and Cotangent Functions
4.15.1. Tangent Function
4.15.2. Cotangent Function
Chapter 5
Derivative of a Function
5.1. Incremental Ratio and Derivative of a Function at a Point
5.1.1. Example
5.1.2. Continuity and Derivability of a Function
5.2. Considerations on the Incremental Ratio and Derivative of a Function
5.3. Properties of Derivatives and Summary Table of Derivation Rules
5.4. Derivative of the Exponential Function
5.5. Derivative of the Logarithm Function
5.6. Derivative of the Power Function
5.7. Derivative of Trigonometric Functions
5.7.1. Derivative of the Sine and Cosine Functions
5.7.2. Derivative of the Tangent Function
5.7.3. Derivative of the Cotangent Function
5.8. De L’Hôpital’s Rule
Chapter 6
Study of a Function: Points of Maximum and Minimum, Points of Inflection
6.1. Definition of Second Derivative
6.2. Monotone Increasing Functions and Monotone Decreasing Functions in an Interval
6.3. Increasing and Decreasing Functions at a Point
6.4. Convex and Concave Functions
6.5. Maximum and Minimum of a Function
6.6. Points of Maximum and Minimum
6.7. Points of Inflection
6.8. Asymptotes of a Function
6.8.1. Vertical Asymptotes
6.8.2. Horizontal Asymptotes
6.8.3. Oblique Asymptotes
Chapter 7
Indefinite Integral
7.1. Infinitesimals and Differentials
7.1.1. Infinitesimals
7.1.1.1. Order of an Infinitesimal
7.1.2. Differentials
7.2. Indefinite Integral
7.2.1. Examples of Calculation of Some Indefinite Integrals
7.2.1.1. Example 1
7.2.1.2. Example 2
7.2.1.3. Example 3
7.3. Properties of Integrals
7.4. Integration by Substitution
7.4.1. Example of Calculation of an Integral by Substitution
7.5. Integration by Parts
7.5.1. Examples of Calculation of Some Integrals by Parts
7.5.1.1. Example 1.
7.5.1.2. Example 2
7.5.1.3. Example 3
Chapter 8
Definite Integral
8.1. Definition of the Definite Integral
8.2. Properties of Definite Integrals
8.2.1. Integral Mean
8.3. Improper Integral
8.3.1. Unbounded Function and Bounded Integration Interval
8.3.2. Bounded Function and Unbounded Integration Interval
8.3.3. Unbounded Function and Unbounded Integration Interval
Chapter 9
Calculation of Function Limits
9.1. Problems
9.1.1. Problem 9.1
9.1.2. Problem 9.2
9.1.3. Problem 9.3
9.1.4. Problem 9.4
9.1.5. Problem 9.5
9.1.6. Problem 9.6
9.1.7. Problem 9.7
9.1.8. Problem 9.8
9.1.9. Problem 9.9
9.1.10. Problem 9.10
9.1.11. Problem 9.11
9.1.12. Problem 9.12
9.1.13. Problem 9.13
9.1.14. Problem 9.14
9.1.15. Problem 9.15
9.1.16. Problem 9.16
9.1.17. Problem 9.17
9.2. Solutions
9.2.1. Problem 9.1
9.2.2. Problem 9.2
9.2.3. Problem 9.3
9.2.4. Problem 9.4.
9.2.5. Problem 9.5
9.2.6. Problem 9.6
9.2.7. Problem 9.7
9.2.8. Problem 9.8
9.2.9. Problem 9.9
9.2.10. Problem 9.10
9.2.11. Problem 9.11
9.2.12. Problem 9.12
9.2.13. Problem 9.13
9.2.14. Problem 9.14
9.2.15. Problem 9.15
9.2.16. Problem 9.16
9.2.17. Problem 9.17
Chapter 10
Calculation of Function Derivatives
10.1. Problems
10.2. Solutions
10.2.1. Problem 10.1
10.2.2. Problem 10.2
10.2.3. Problem 10.3
10.2.4. Problem 10.4
10.2.5. Problem 10.5
10.2.6. Problem 10.6
10.2.7. Problem 10.7
10.2.8. Problem 10.8
10.2.9. Problem 10.9
10.2.10. Problem 10.10
10.2.11. Problem 10.11
10.2.12. Problem 10.12
10.2.13. Problem 10.13
10.2.14. Problem 10.14
10.2.15. Problem 10.15
10.2.16. Problem 10.16
10.2.17. Problem 10.17
10.2.18. Problem 10.18
Chapter 11
Problems Related to the Study of Functions
11.1. Problems
11.2. Solutions
11.2.1. Problem 11.1
11.2.1.1. Domain of the Function
11.2.1.2. Intersection with the Axes
11.2.1.3. Asymptotes
11.2.1.4. Maximum and/or Minimum Points
11.2.1.5. Inflection Points
11.2.2. Problem 11.2
11.2.2.1. Intersection with the Axes
11.2.2.2. Asymptotes
11.2.2.3. Maximum and/or Minimum Points
11.2.2.4. Inflection Points
11.2.2.5. Comments on the Functions of the Problems 11.1 and 11.2
11.2.3. Problem 11.3
11.2.3.1. Domain of the Function
11.2.3.2. Intersection with the Axes
11.2.3.3. Asymptotes
11.2.3.4. Maximum and/or Minimum Points
11.2.3.5. Inflection Points
11.2.4. Problem 11.4
11.2.4.1. Domain of the Function
11.2.4.2. Intersection with the Axes
11.2.4.3. Asymptotes
11.2.4.4. Maximum and/or Minimum Points
11.2.4.5. Inflection Points
11.2.5. Problem 11.5
11.2.5.1. Domain of the Function
11.2.5.2. Intersection with the Axes
11.2.5.3. Asymptotes
11.2.5.4. Maximum and/or Minimum Points
11.2.5.5. Inflection Points
11.2.5.6. Appendix
11.2.6. Problem 11.6
11.2.6.1. Domain of the Function
11.2.6.2. Intersection with the Axes
11.2.6.3. Asymptotes
11.2.6.4. Maximum and/or Minimum Points
11.2.6.5. Inflection Points
11.2.7. Problem 11.7
11.2.7.1. Domain of the Function
11.2.7.2. Intersection with the Axes
11.2.7.3. Asymptotes
11.2.7.4. Maximum and/or Minimum Points
11.2.7.5. Inflection Points
11.2.8. Problem 11.8
11.2.8.1. Domain of the Function
11.2.8.2. Intersection with the Axes
11.2.8.3. Asymptotes
11.2.8.4. Maximum and/or Minimum Points
11.2.8.5. Inflection Points
11.2.8.6. Additional Considerations on the Maximum Points
11.2.9. Problem 11.9
11.2.9.1. Domain of the Function
11.2.9.2. Intersection with the Axes
11.2.9.3. Asymptotes
11.2.9.4. Maximum and/or Minimum Points
11.2.9.5. Inflection Points
11.2.10. Problem 11.10
11.2.10.1. Domain of the Function
11.2.10.2. Intersection with the Axes
11.2.10.3. Asymptotes
11.2.10.4. Maximum and/or Minimum Points
11.2.10.5. Inflection Points
11.2.11. Problem 11.11
11.2.11.1. Domain of the Function
11.2.11.2. Intersection with the Axes
11.2.11.3. Asymptotes
11.2.11.4. Maximum and/or Minimum Points
11.2.11.5. Inflection Points
11.1.12. Problem 11.12
11.2.12.1. Domain of the Function
11.2.12.2. Intersection with the Axes
11.2.12.3. Asymptotes
11.2.12.4. Maximum and/or Minimum Points
11.2.12.5. Inflection Points
11.1.13. Problem 11.13
11.2.13.1. Domain of the Function
11.2.13.2. Intersection with the Axes
11.2.13.3. Asymptotes
11.2.13.4. Maximum and/or Minimum Points
11.2.13.5. Inflection Points
11.2.14. Problem 11.14
11.2.14.1. Domain of the Function
11.2.14.2. Intersection with the Axes
11.2.14.3. Asymptotes
11.2.14.4. Maximum and/or Minimum Points
11.2.14.5. Inflection Points
11.2.15. Problem 11.15
11.2.15.1. Domain of the Function
11.2.15.2. Intersection with the Axes
11.2.15.3. Asymptotes
11.2.15.4. Maximum and/or Minimum Points
11.2.15.5. Inflection Points
11.2.16. Problem 11.16
11.2.16.1. Domain of the Function
11.2.16.2. Intersection with the Axes
11.2.16.3. Asymptotes
11.2.16.4. Maximum and/or Minimum Points
11.2.16.5. Inflection Points
11.2.17. Problem 11.17
11.2.17.1. Domain of the Function
11.2.17.2. Intersection with the Axes
11.2.17.3. Asymptotes
11.2.17.4. Maximum and/or Minimum Points
11.2.17.5. Inflection Points
11.2.18. Problem 11.18
11.2.18.1. Domain of the Function
11.2.18.2. Intersection with the Axes
11.2.18.3. Asymptotes
11.2.18.4. Maximum and/or Minimum Points
11.2.18.5. Inflection Points
11.2.19. Problem 11.19
11.2.19.1. Domain of the Function
11.2.19.2. Intersection with the Axes
11.2.19.3. Asymptotes
11.2.19.4. Maximum and/or Minimum Points
11.2.19.5. Inflection Points
11.2.20. Problem 11.20
11.2.20.1. Domain of the Function
11.2.20.2. Intersection with the Axes
11.2.20.3. Asymptotes
11.2.20.4. Maximum and/or Minimum Points
11.2.20.5. Inflection Points
11.1.21. Problem 11.21
11.2.21.1. Domain of the Function
11.2.21.2. Intersection with the Axes
11.2.21.3. Asymptotes
11.2.21.4. Maximum and/or Minimum Points
11.2.21.5. Inflection Points
11.2.22. Problem 11.22
11.2.22.1. Domain of the Function
11.2.22.2. Intersection with the Axes
11.2.22.3 Asymptotes
11.2.22.4. Maximum and/or Minimum Points
11.2.22.5. Inflection Points
11.2.23. Problem 11.23
11.2.23.1. Domain of the Function
11.2.23.2. Intersection with the Axes
11.2.23.3. Asymptotes
11.2.23.4. Maximum and/or Minimum Points
11.2.23.5. Inflection Points
11.2.24. Problem 11.24
11.2.24.1. Domain of the Function
11.2.24.2. Intersection with the Axes
11.2.24.3. Asymptotes
11.2.24.4. Maximum and/or Minimum Point
11.2.24.5. Inflection Points
11.1.25. Problem 11.25
11.2.25.1. Domain of the Function
11.2.25.2. Intersection with the Axes
11.2.25.3. Asymptotes
11.2.25.4. Maximum and/or Minimum Points
11.2.25.5. Inflection Points
11.2.26. Problem 11.26
11.2.26.1. Domain of the Function
11.2.26.2. Intersection with the Axes
11.2.26.3. Asymptotes
11.2.26.4. Maximum and/or Minimum Points
11.2.26.5. Inflection Points
11.1.27. Problem 11.27
11.2.27.1. Domain of the Function
11.2.27.2. Intersection with the Axes
11.2.27.3. Asymptotes
11.2.27.4. Maximum and/or Minimum Points
11.2.27.5. Inflection Points
11.2.28. Problem 11.28
11.2.28.1. Domain of the Function
11.2.28.2. Intersection with the Axes
11.2.28.3. Asymptotes
11.2.28.4. Maximum and/or Minimum Points
11.2.28.5. Inflection Points
11.2.29. Problem 11.29
11.2.29.1. Domain of the Function
11.2.29.2. Intersection with the Axes
11.2.29.3. Asymptotes
11.2.29.4. Maximum and/or Minimum Points
11.2.29.5. Inflection Points
11.2.30. Problem 11.30
11.2.30.1. Intersection with the Axes
11.2.30.2. Asymptotes
11.2.30.3. Maximum and/or Minimum Points
11.2.30.4. Inflection Points
11.2.31. Problem 11.31
11.2.31.1. Domain of the Function
11.2.31.2. Intersection with the Axes
11.2.31.3. Asymptotes
11.2.31.4. Maximum and/or Minimum Points
11.2.31.5. Inflection Points
Chapter 12
Calculation of Integrals
12.1. Problems
12.2. Solutions
12.2.1. Problem 12.1
12.2.2. Problem 12.2
12.2.3. Problem 12.3
12.2.4. Problem 12.4
12.2.5. Problem 12.5
12.2.6. Problem 12.6
12.2.7. Problem 12.7
12.2.8. Problem 12.8
12.2.9. Problem 12.9
12.2.10. Problem 12.10
12.2.11. Problem 12.11
12.2.12. Problem 12.12
12.2.13. Problem 12.13
12.2.14. Problem 12.14
12.2.15. Problem 12.15
12.2.16. Problem 12.16
12.2.17. Problem 12.17
12.2.18. Problem 12.18
12.2.19. Problem 12.19
12.2.20. Problem 12.20
12.2.21. Problem 12.21
12.2.22. Problem 12.22
12.2.23. Problem 12.23
12.2.24. Problem 12.24
12.2.25. Problem 12.25
12.2.26. Problem 12.26
12.2.27. Problem 12.27
12.2.28. Problem 12.28
12.2.29. Problem 29
12.2.30. Problem 30
12.2.31. Problem 31
12.2.32. Problem 32
12.2.33. Problem 33
12.2.34. Problem 34
12.2.35. Problem 35
About the Author
Index
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