Unit 3071: Comprehensive Calculus

1. Introduction to Calculus

• Overview of calculus: definition and scope
• Historical development: Newton, Leibniz, and evolution of calculus
• Importance and applications in physics, engineering, economics, biology, etc.

2. Limits and Continuity

• Concept of limits: intuitive and formal definition
• Calculating limits algebraically
• One-sided limits and infinite limits
• Continuity: definition and types of discontinuities
• Techniques for evaluating limits:
  • Direct substitution
  • Factoring and simplifying
  • Rationalizing
  • Squeeze theorem
• Advanced limit techniques:
  • L'Hôpital's Rule (conditions and applications)
  • Limits involving infinity

3. Derivatives

• Definition of the derivative as a limit
• Interpretation of derivatives: rate of change, slope of tangent
• Differentiation rules:
  • Power rule
  • Product and quotient rules
  • Chain rule
• Higher-order derivatives
• Implicit differentiation

4. Applications of Derivatives

• Finding maxima and minima (local and global)
• Critical points and inflection points
• Curve sketching:
  • Increasing/decreasing functions
  • Concavity and points of inflection
• Related rates problems
• Optimization problems in real-world contexts
• Application examples from physics (motion), economics (cost, revenue), and engineering

5. Integration

• Concept of integration as area under a curve
• Indefinite vs definite integrals
• Fundamental Theorem of Calculus
• Basic integration techniques:
  • Substitution method
  • Integration by parts
  • Partial fractions
• Improper integrals (introduction)

6. Applications of Integration

• Computing areas between curves
• Volumes of revolution (disk, washer, and shell methods)
• Work done by a variable force
• Fluid pressure and force calculations
• Solving practical problems using definite integrals
• Geometric interpretation of integration

7. Differential Equations

• Introduction to differential equations
• First-order differential equations:
  • Separable equations
  • Linear first-order equations
• Second-order differential equations (homogeneous and non-homogeneous)
• Applications:
  • Exponential growth and decay
  • Simple harmonic motion

8. Taylor Series and Maclaurin Series

• Concept of series expansions
• Deriving Taylor and Maclaurin series
• Radius and interval of convergence
• Approximating functions with polynomials
• Applications in calculus and physics

9. Multivariable Calculus

• Functions of several variables
• Partial derivatives and higher-order partial derivatives
• Gradient vectors and directional derivatives
• Multiple integrals:
  • Double integrals
  • Triple integrals
• Vector calculus basics:
  • Vector fields
  • Divergence and curl (introductory)
• Applications in optimization, physics, and engineering in 3D space