Calculus

1. Differentiation
   1. Concept of a Derivative
      1. Derivative as a rate of change
         1. Instantaneous rate of change versus average rate of change
         2. Real-life examples: Speed in travel, population growth
      2. Geometric interpretation (tangent line)
         1. Tangent compared to secant line
         2. Slopes and angles in a coordinate system
         3. Applications in determining extreme points
      3. Notation (Leibniz and Lagrange)
         1. Historical context and usage
         2. Differences and similarities between notations
         3. Extensions to higher dimensions (partial derivatives)
   2. Techniques of Differentiation
      1. Derivatives of basic functions
         1. Polynomials and constants
         2. Exponential functions
         3. Logarithmic functions
         4. Trigonometric functions
         5. Inverse trigonometric functions
      2. Product rule
         1. Derivation and explanation
         2. Examples with polynomials and trigonometric functions
         3. Extensions to multiple factors
      3. Quotient rule
         1. Derivation and explanation
         2. Applications in rational functions
         3. Potential pitfalls and misconceptions
      4. Chain rule
         1. Understanding function composition
         2. Practical applications with implicit differentiation
         3. Extending to multiple nested functions
   3. Applications of Derivatives
      1. Motion: velocity and acceleration
         1. Definitions from derivatives
         2. Graphical interpretation of motion
         3. Application in physics and engineering
      2. Optimization problems
         1. Finding critical points and inflection points
         2. Applications in economics, finance, and engineering
         3. Constraints and Lagrange multipliers
      3. Related rates
         1. Setting up and solving related rates problems
         2. Real-world examples like ladder problems and shadow problems
         3. Importance in physics and engineering dynamics
      4. Curve sketching
         1. Analyzing behavior of functions using first and second derivatives
         2. Identifying maxima, minima, and points of inflection
         3. Building graphs with asymptotic behavior
      5. Linear approximations and differentials
         1. Taylor's theorem introduction
         2. Applications in estimating values of functions
         3. Error analysis and management in scientific computations
   4. Higher Order Derivatives
      1. Second derivative and concavity
         1. Testing for concavity and convexity
         2. The role of the second derivative test in confirmation of extrema
         3. Application in economics: concave/convex utility functions
      2. nth derivatives
         1. Derivation rules and patterns in specific functions
         2. Real-world applications: physics (jerk in motion, etc.)
         3. Practical challenges in computation and limitation of nth order power series