Unit 2937: Comprehensive Calculus

1. Introduction to Limits

• Definition of a limit
• Intuitive understanding and graphical interpretation
• Formal epsilon-delta definition (conceptual overview)
• One-sided limits
• Limits involving infinity
• Limits and continuity
• Using limits to define continuity
• Relationship between limits, continuity, and differentiability

2. Differentiation Techniques

• Definition of the derivative
• Interpretation of the derivative as rate of change and slope of tangent
• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule
• Higher-order derivatives
• Differentiation of implicit functions

3. Applications of Derivatives

• Critical points and stationary points
• First and second derivative tests for local extrema
• Inflection points and concavity
• Curve sketching using derivatives
• Optimization problems in various contexts
• Related rates problems

4. Integration and Antiderivatives

• Concept of antiderivative
• Indefinite integrals
• The Fundamental Theorem of Calculus
• Definite integrals and their properties
• Relationship between differentiation and integration

5. Techniques of Integration

• Integration by substitution
• Integration by parts
• Trigonometric substitutions
• Partial fractions decomposition
• Improper integrals (introductory concepts)

6. Applications of Integrals

• Area under curves
• Area between curves
• Volumes of solids of revolution (disk, washer, shell methods)
• Work done by variable force
• Average value of a function over an interval
• Applications in physics and engineering contexts

7. Differential Equations

• Introduction to differential equations
• Order and degree of differential equations
• Separation of variables
• First-order linear differential equations
• Modeling real-world phenomena (growth/decay, motion, electrical circuits)

8. Multivariable Calculus

• Functions of multiple variables
• Limits and continuity in higher dimensions
• Partial derivatives
• Gradient vectors and directional derivatives
• Multiple integrals (double and triple integrals)
• Applications of multiple integrals
• Introduction to vector calculus (divergence, curl)

9. Taylor Series and Maclaurin Series

• Concept of power series
• Derivation of Taylor and Maclaurin series
• Radius and interval of convergence
• Using Taylor series for function approximation
• Error estimation and remainder terms

10. Applications of Calculus in Physics and Engineering

• Modeling motion with derivatives and integrals
• Electrical circuit analysis using calculus
• Fluid dynamics and flow rate calculations
• Optimization problems in engineering design
• Case studies involving real-world applications