Unit 3092: Advanced Calculus

1. Limits and Continuity

1.1 Concepts of Limits

• Definition of limits
• One-sided limits (left-hand and right-hand limits)
• Limits at infinity and infinite limits

1.2 Evaluating Limits

• Algebraic techniques
• Squeeze theorem
• Limits involving trigonometric, exponential, and logarithmic functions

1.3 Continuity

• Definition of continuity at a point and on intervals
• Types of discontinuities (removable, jump, infinite)
• Intermediate Value Theorem and its applications

2. Differentiation Techniques

2.1 Implicit Differentiation

• Differentiating implicit functions
• Applications and examples

2.2 Logarithmic Differentiation

• When and how to use logarithmic differentiation
• Differentiating complex products, quotients, and powers

2.3 Parametric Differentiation

• Differentiating parametric equations
• Finding slopes and tangents for parametric curves

2.4 Differentiating Complex Functions

• Differentiation of trigonometric, exponential, and logarithmic functions
• Higher-order derivatives and Leibniz rule

3. Applications of Derivatives

3.1 Optimization Problems

• Local and global extrema
• Critical points and second derivative test
• Real-world optimization examples

3.2 Related Rates

• Setting up and solving related rates problems
• Applications in physics and engineering

3.3 Curve Sketching

• Using first and second derivatives to analyze functions
• Identifying intervals of increase/decrease, concavity, inflection points

3.4 L'Hôpital's Rule

• Conditions for applying L'Hôpital's Rule
• Evaluating indeterminate forms

4. Integration Methods

4.1 Antiderivatives and Indefinite Integrals

• Concept and notation
• Basic integration rules

4.2 Integration by Parts

• Formula and derivation
• Applications and examples

4.3 Trigonometric Substitution

• When to use trigonometric substitution
• Solving integrals involving sqrt expressions

4.4 Partial Fractions

• Decomposing rational functions
• Integrating using partial fractions

4.5 Improper Integrals

• Definition and classification
• Techniques for evaluation and convergence tests

5. Differential Equations

5.1 First-Order Differential Equations

• Separable equations
• Linear first-order equations
• Applications

5.2 Second-Order Linear Differential Equations

• Homogeneous and non-homogeneous equations
• Characteristic equation and solution methods

5.3 Systems of Differential Equations

• Introduction to systems
• Analytical and numerical solution methods (Euler, Runge-Kutta)

6. Multivariable Calculus

6.1 Functions of Several Variables

• Domain, range, and graphs
• Level curves and surfaces

6.2 Partial Derivatives

• Definition and computation
• Higher-order partial derivatives

6.3 Gradient and Directional Derivatives

• Gradient vector
• Computing directional derivatives

6.4 Multiple Integrals

• Double and triple integrals
• Applications in volume and mass calculations

6.5 Vector Calculus Introduction

• Vector fields
• Line integrals and surface integrals

6.6 Theorems of Green, Stokes, and Gauss

• Statement and interpretation
• Applications to vector fields

7. Taylor Series and Maclaurin Series

7.1 Power Series

• Definition and examples
• Interval and radius of convergence

7.2 Taylor Series

• Derivation and formula
• Taylor polynomials and approximation

7.3 Maclaurin Series

• Special case of Taylor series at zero
• Common Maclaurin expansions

7.4 Applications

• Approximating functions
• Error estimation and convergence criteria

8. Vector Calculus

8.1 Vector-Valued Functions

• Definition and differentiation
• Arc length and curvature

8.2 Vector Fields

• Definitions and examples
• Gradient, divergence, and curl

8.3 Theorems of Green, Stokes, and Gauss

• Detailed exploration and proofs
• Physical and geometric interpretations

8.4 Applications

• Applications in physics (fluid flow, electromagnetism)
• Engineering and other sciences