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FUNDAMENTALS OF MATHEMATICS DIFFERENTIAL CALCULUS

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.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Preface
  5. Acknowledgements
  6. Chapter 1 The Limit of a Function
    1. Introduction
      1. • Archimedes and The Problem of Area:
      2. • Neighbourhood of Point ‘a ’
      3. • Deleted Neighbourhood of a Point a
      4. • Left Deleted Neighbourhood of ‘a’
      5. • Right Deleted Neighbourhood of ‘a’
      6. • Meaning of ‘x → a’ (x Tends to a)
    2. Limit of a Function
      1. • Why Limit of a Function is Needed?
    3. Indeterminate Forms
      1. • Left Hand Limit of Function
      2. • Right Hand Limit of a Function:
      3. • Finite Limit at infinity:
      4. • Infinite Limit at a Finite Point:
      5. • Infinite Limit at Infinity:
    4. Algebra of Limits
    5. Infinitesimal Quantity
    6. Properties of Infinitesimal:
      1. • List of Equivalent Infinitesimals
    7. Sandwich Theorem and Standard Results on Limits
    8. Standard Results on Limits
    9. Evaluation of Limits
      1. • 1. By Using Factorization:
      2. • 2. By Rationalization :
      3. • 3. By Using Substitution:
      4. • By Using Standard Limits:
    10. Application of Standard limits
      1. • Transformations:
    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
      1. • Evaluation of Limit When it is of the form (1)x
      2. • Evaluat of Limit When it is of the form (0)0
      3. • Evaluate of Limit When it is of the Form (x)0
    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
      1. • Horizontal Asymptote (Asymptotes Parallel to x-axis):
      2. • Algorithm
    23. Procedure to Find Vertical Asymptote
    24. Oblique Asymptotes
      1. • Procedure to Find Oblique Asymptotes:
      2. • Method to Find Oblique Asymptotes for Algebraic Curves of Any Degree
      3. • Asymptote by Expansion
      4. • The Position of the Curve with Respect to an Asymptote
    25. Multiple-Choice Questions
    26. Tutorial Exercise
      1. Answer Keys
      2. Hints and Solutions
  7. Chapter 2 Continuity and Differentiability
    1. Continuity
    2. Introduction
      1. • Different Situations of Discontinuity at x = a
    3. Continuity of a Function at a Point
      1. • Mathematical Definition
      2. • Geometric Significance
    4. Continuity of an Even and Odd Function
    5. Discontinuity of a Function f(x) at x = a
      1. • Types of 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
      1. • Domain of Continuity of Some Standard Function:
      2. • Continuity in an Open Interval
    10. Continuity of a Function on a Closed Interval
    11. Properties of Continuous Function
      1. • P1 (Fermat’s Theorem):
      2. • P: 2: Intermediate Value Theorem:
      3. • P:3 Weierstrass Theorem (Extreme Value Theorem)
      4. • P: 4: Bolzanos Theorem:
      5. • P: 5 A Continuous Functions Whose Domain is Some Closed Interval Must Have its Range Also a Closed Interval
      6. • P: 6 Continuity of Inverse Function
      7. • P: 7 If a Function x f(x) is Integrable on [a, b] , then is Continuous Function
      8. • To a find the range of function using the properties of continuous functions
    12. Differentiability
    13. Introduction
      1. • Differentiability at a point
      2. • Physical significance
      3. • Geometrical significance
    14. Concept of Tangent and its Association with Derivability:
      1. • Theorem relating continuity and differentiability
      2. • Reasons of nondifferentiability of a function at x = a
    15. Algebra of Differentiability
      1. • Domain of differentiability
    16. Domain of Differentiability of Some Standard Functions
      1. • Differentiability in open and closed interval
    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
      1. • Alternative limit form of the derivative
      2. • Another alternative form of derivative by using centered difference quotient
    20. Differentiability of Parametric Functions
    21. Derivatives of Higher Orders and Repeatedly Differentiable Functions
    22. Functional Equation
      1. • Solution of a functional Equation
      2. • Some famous functional equations in two variable and their corresponding solutions
      3. • Jensen's functional equation
      4. • D ’ Alambert's functional equation
    23. Multiple-Choice Questions
    24. Tutorial Exercise
      1. Answer Keys
      2. Hints and Solutions
  8. Chapter 3 Method of Differentiation
    1. Introductions
      1. • Derivatives Using First Principle (Ab-initio) Method
      2. • Algorithm to Differentiate One variable w.r.t. Another
      3. • Derivatives of Some Standard Functions
      4. • Algebra of differentiation
      5. • Addition and subtraction rule
      6. • Product Rule
      7. • Quotient Rule
    2. Chain Rule
    3. Differentiation of a Function with Respect to Another Function
    4. Order of Derivative and Higher Differential Coefficient
      1. • Rules of Higher Order Derivative
    5. Logarithmic and Exponential Differentiation
      1. • Algorithm to Find Logarithmic Differentiation
    6. Differentiation of inverse functions
      1. • Geometrical Interpretation
    7. Implicit Differentiation
      1. • Shortcut for Implicit Functions
    8. Parametric Differentiation
    9. Determinant Forms of Differentiation
    10. Some Standard Substitution
      1. • Expression 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
      1. Answer Keys
      2. Hints and Solutions
  9. Chapter 4 Application of Derivatives I
    1. Rate of Change
    2. Introduction
    3. Derivative as the Rate of Change
      1. • Instantaneous rate of change of quantities
    4. Application of Derivative as a Rate of Change
      1. • Velocity and Acceleration:
    5. Application in Two Dimension
      1. • Area and perimeter of some standard two dimensional figures are listed below
    6. Application in Three Dimension Geometry
      1. • Area and perimeter of some standard three dimensional figures are listed below
    7. Problems Based on Marginal Costs and Marginal Revenue
      1. • Working Rule:
    8. Errors and Approximations
      1. • Types of errors
    9. Approximations
      1. • Algorithm
    10. Tangents and Normals
    11. Introduction
    12. Definition
      1. • Geometrical Interpretation:
    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
      1. • Algorithm:
    23. Common Tangents
      1. • Equation of normal
      2. • Number of solutions
    24. Shortest Distance
    25. Length of Tangent, Sub-Tangent, Normal, Sub-Normal
      1. • Length of tangent
      2. • Length of sub-tangent
      3. • Length of normal:
      4. • Length of sub-normal
    26. Multiple-Choice Questions
    27. Tutorial Exercise
      1. Answer Keys
      2. Hints and Solutions
  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
      1. • Monotonicity over an Interval
      2. • Monotonicity of Differentiable Functions in an Interval
      3. • Monotonicity for continuous but non-Differentiable Functions in an Interval
      4. • Monotonicity for discontinuous functions in an interval
    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
      1. • Conclusion
    20. Sign of Curvature
      1. • Concave upwards (convex downwards)
      2. • Concave downwards (convex upwards)
    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
      1. • Discussion
      2. • Conclusion
    26. Mean Value Theorem
    27. Rolle’s and Mean Value Theorem
      1. • Rolle’s Theorem
      2. • Conclusion:
    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
      1. • Necessary and sufficient conditions for local maxima and minima: (For differentiable functions)
    37. Fermat Theorem
    38. Conclusion
    39. Continuous and Non-differentiable Functions
    40. First Derivative Test (Continuous Functions)
    41. Saddle Point
    42. Boundedness
      1. • Greatest lower bound
      2. • Lowest upper bound
    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
      1. Answer Keys
      2. Hints and Solutions
  11. Copyright