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## Book Description

Fundamentals of Mathematics' is a series of seven books, which are designed to provide comprehensive study material on specific areas in mathematics. It is an ideal companion for students who would like to master a particular subject area based on their individual requirements. All books in this series provide extensive coverage of the topics supported by numerous solved examples. The concepts are explained in a meticulously manner with ample illustrations and practice exercises (with answers). Overall these books enable quick learning and aid thorough preparation to crack the various engineering entrance examinations.

1. Cover
2. Title Page
3. Contents
4. Preface
5. Acknowledgements
6. Chapter 1 The Limit of a Function
1. Introduction
2. Limit of a Function
3. Indeterminate Forms
4. Algebra of Limits
5. Infinitesimal Quantity
6. Properties of Infinitesimal:
7. Sandwich Theorem and Standard Results on Limits
8. Standard Results on Limits
9. Evaluation of Limits
10. Application of Standard limits
11. Application of Standard Limits
12. Application of Standard Limits
13. Application of Standard Limit
14. Evaluation of Limit Using Expansions
15. Limit at Infinity
16. Using L-Hospital Rule
17. Evaluation of Limits by using Logarithm
18. Leibinitiz Rules:
19. Limit of Summmation of Series Using Definite Integral
20. Method to Evaluate the Limit Using Definite Integral
21. Geometrical Application:
22. Asymptotes
23. Procedure to Find Vertical Asymptote
24. Oblique Asymptotes
25. Multiple-Choice Questions
26. Tutorial Exercise
7. Chapter 2 Continuity and Differentiability
1. Continuity
2. Introduction
3. Continuity of a Function at a Point
4. Continuity of an Even and Odd Function
5. Discontinuity of a Function f(x) at x = a
6. Pole Discontinuity
7. Discontinuity of First and Second Kind
8. Algebra of Continuity
9. Continuity of a Function on a Set
10. Continuity of a Function on a Closed Interval
11. Properties of Continuous Function
12. Differentiability
13. Introduction
14. Concept of Tangent and its Association with Derivability:
15. Algebra of Differentiability
16. Domain of Differentiability of Some Standard Functions
17. Method to Check the Differentiability of a Given Function on a Set or to Find Domain of Differentiability
18. Miscellaneous Results on Differentiability
19. Miscellaneous Concepts About Differentiability and Derivative of Function
20. Differentiability of Parametric Functions
21. Derivatives of Higher Orders and Repeatedly Differentiable Functions
22. Functional Equation
23. Multiple-Choice Questions
24. Tutorial Exercise
8. Chapter 3 Method of Differentiation
1. Introductions
2. Chain Rule
3. Differentiation of a Function with Respect to Another Function
4. Order of Derivative and Higher Differential Coefficient
5. Logarithmic and Exponential Differentiation
6. Differentiation of inverse functions
7. Implicit Differentiation
8. Parametric Differentiation
9. Determinant Forms of Differentiation
10. Some Standard Substitution
11. Successive Differentiation:
12. Leibnitz's Theorem for the nth Derivative of the Product of Two Functions of x
13. Formation of Differential Equation
14. Multiple-Choice Questions
15. Tutorial Exercise
9. Chapter 4 Application of Derivatives I
1. Rate of Change
2. Introduction
3. Derivative as the Rate of Change
4. Application of Derivative as a Rate of Change
5. Application in Two Dimension
6. Application in Three Dimension Geometry
7. Problems Based on Marginal Costs and Marginal Revenue
8. Errors and Approximations
9. Approximations
10. Tangents and Normals
11. Introduction
12. Definition
13. Graphs with Vertical Tangents
14. Caution
15. Slope of Normal
16. Condition for a Given Line to be Tangent to a Curve
17. Tangents from an External Point
18. Tangents/Normals to Second Degree Curve
19. Tangent to Parametric Functions
20. Tangents Intersecting the Curve Itself
21. Tangent at Origin
22. Angles of Intersection of Two Curves
23. Common Tangents
24. Shortest Distance
25. Length of Tangent, Sub-Tangent, Normal, Sub-Normal
26. Multiple-Choice Questions
27. Tutorial Exercise
10. Chapter 5 Application of Derivatives II
1. Mo no tonicity
2. Introduction
3. Monotonicity
4. Monotonicity at a Point
5. Test of Monotonicity at a Point
6. Non-differentiable but Continuous Function at x = a
7. Non-differentiable and Discontinuous Function at x = a
8. Monotonicity at the End Point of Interval
9. Conclusion
10. Monotonic Functions
11. Interval of Monotonicity
12. Critical Points
13. Conclusion
14. Properties of Monotonic Function
15. Application of Monotonicity
16. Method of Proving Inequality (Using Monotonicity)
17. Curvature of Function
18. Curvature of a Circle
19. If the function is Given in Cartesian Form
20. Sign of Curvature
21. Hyper critical Point
22. Points of Inflextion
23. Method to Find the Points of Inflexion of the Curve y = f(x)
24. Solving Inequalities Using Curvature
25. Jenson’s Functional Equation
26. Mean Value Theorem
27. Rolle’s and Mean Value Theorem
28. Algebraic Interpretation of Rolle’s Theorem
29. Application of Rolle’s Theorem
30. Lagrange's Mean Value Theorem
31. Physical Significance:
32. Alternative form of Lmvt
33. Maxima and Minima
34. Introduction
35. Maxima and Minima
36. Relative (Local) Maxima and Minima
37. Fermat Theorem
38. Conclusion
39. Continuous and Non-differentiable Functions
40. First Derivative Test (Continuous Functions)
42. Boundedness
43. Global Maxima and Global Minima
44. Caution:
45. Algebra of Global Extrema
46. Even/Odd Function
47. Miscellaneous Method
48. Second/Higher order Derivative Test
49. Extrema of Parametric Function
50. First Derivative Test for Parametric Functions
51. Second Derivative Test for Parametric Function
52. Darboux Theorem
53. Fork Extremum Theorem
54. Extrema of Discontinuous Functions
55. Maxima and Minima of Functions of Several Variables
56. Maximum and Minimum for Discrete Valued Functions
57. Area and Perimeter of Some Standard Two Dimensional Figures are Listed Below:
58. Area and Perimeter of Some Standard Three Dimensional Figures are Listed Below
59. Some Important Cases
60. Inscribed Figures
61. Excribed Figures
62. General Concept (Shortest Distance of a Point from a Curve)
63. Multiple-Choice Questions
64. Tutorial Exercise