Calculus Online Book

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Here's a structured list of topics for a comprehensive Calculus tutorial, covering fundamental to advanced concepts, similar to what you'd find in an academic textbook.

1. Preliminaries

1.1 Review of Algebra
1.2 Review of Trigonometry
1.3 Functions and Their Graphs
1.4 Exponential and Logarithmic Functions
1.5 Limits and Continuity: An Intuitive Approach

2. Limits and Continuity

2.1 Definition of a Limit
2.2 Techniques for Evaluating Limits
2.3 Limits Involving Infinity
2.4 Continuity
2.5 Intermediate Value Theorem
2.6 Limits and Continuity in Higher Dimensions

3. Derivatives

3.1 Definition of the Derivative
3.2 Basic Differentiation Rules (Power, Product, Quotient Rules)
3.3 The Chain Rule
3.4 Implicit Differentiation
3.5 Higher-Order Derivatives
3.6 Applications of Derivatives
3.6.1 Tangent Lines and Rates of Change
3.6.2 Related Rates
3.6.3 Linear Approximation and Differentials
3.6.4 Maxima and Minima: Optimization Problems
3.6.5 Mean Value Theorem
3.6.6 L'Hopital's Rule

4. Applications of Derivatives

4.1 Curve Sketching (Using First and Second Derivatives)
4.2 Concavity and Inflection Points
4.3 Optimization Problems
4.4 Newton's Method
4.5 Motion and Related Rates

5. Integrals

5.1 Definition of the Indefinite Integral
5.2 Techniques of Integration:
5.2.1 Integration by Substitution
5.2.2 Integration by Parts
5.2.3 Trigonometric Integrals
5.2.4 Partial Fraction Decomposition
5.2.5 Improper Integrals
5.3 Definition of the Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Applications of Definite Integrals
5.5.1 Area Under a Curve
5.5.2 Volume of Solids of Revolution
5.5.3 Arc Length
5.5.4 Surface Area
5.5.5 Work, Force, and Pressure
5.5.6 Center of Mass and Centroid

6. Applications of Integrals

6.1 Applications in Physics: Work, Energy, and Power
6.2 Applications in Economics: Consumer and Producer Surplus
6.3 Probability and Statistics: Continuous Distributions

7. Series and Sequences

7.1 Definition of a Sequence and Series
7.2 Convergence of Series
7.3 Tests for Convergence (Geometric Series, P-Series, etc.)
7.4 Alternating Series and Absolute Convergence
7.5 Power Series
7.6 Taylor and Maclaurin Series
7.7 Applications of Taylor Series

8. Multivariable Calculus

8.1 Functions of Several Variables
8.2 Partial Derivatives
8.3 Chain Rule for Multivariable Functions
8.4 Directional Derivatives and Gradient Vectors
8.5 Tangent Planes and Linear Approximations
8.6 Maxima, Minima, and Saddle Points (Lagrange Multipliers)
8.7 Double and Triple Integrals
8.7.1 Applications of Double and Triple Integrals
8.7.2 Surface Area, Volume, and Mass
8.7.3 Change of Variables: Polar, Cylindrical, and Spherical Coordinates

9. Vector Calculus

9.1 Vector Fields
9.2 Line Integrals
9.3 Green's Theorem
9.4 Surface Integrals
9.5 Divergence Theorem
9.6 Stokes' Theorem

10. Differential Equations

10.1 Basic Concepts and Terminology
10.2 First-Order Differential Equations
10.2.1 Separable Equations
10.2.2 Linear First-Order Equations
10.2.3 Exact Equations
10.3 Second-Order Differential Equations
10.4 Systems of Differential Equations
10.5 Applications to Physics, Engineering, and Biology

11. Advanced Topics (Optional)

11.1 Fourier Series and Transforms
11.2 Laplace Transforms
11.3 Partial Differential Equations
11.4 Complex Analysis: Complex Functions and Contour Integrals
11.5 Calculus of Variations