Differential Calculus - NU

1. Limits and ContinuityIntuitive & Rigorous Limits: Evaluating mathematical limits graphically, algebraically, and exploring the formal ε-δ framework.Limit Laws: Computing limits involving indeterminate forms (\(\frac{0}{0}\), \(\frac{\infty }{\infty }\)) using algebraic factoring, rationalization, and the Squeeze Theorem.Continuity: Verifying the three continuous criteria at a point, analyzing types of discontinuities, and applying the Intermediate Value Theorem (IVT).Asymptotes: Investigating infinite limits and limits at infinity to find vertical and horizontal asymptotes.2. The Derivative and Basic Differentiation RulesThe Limit Definition: Defining the derivative as a tangent slope or instantaneous rate of change via the limit difference quotient:\(f^{\prime }(x)=\lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)Core Differentiation Rules: Implementing the Power, Product, Quotient, and Constant Multiple Rules.Transcendental Functions: Finding derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions.The Chain Rule: Differentiating composite functions step-by-step through nested outer and inner operations.3. Advanced Differentiation TechniquesImplicit Differentiation: Differentiating equations where y cannot be explicitly isolated as a function of x (e.g., circles or ellipses).Logarithmic Differentiation: Utilizing properties of natural logarithms to simplify the differentiation of highly complex variable-exponent expressions (\(y = x^x\)).Higher-Order Derivatives: Computing acceleration or structural concavity parameters using second, third, and n-th derivatives.4. Applications of DerivativesRelated Rates: Solving dynamic real-world problems involving quantities changing over time (e.g., expanding volumes, moving shadows).Extreme Values: Locating absolute and relative extrema on closed intervals using the Extreme Value Theorem and critical points.Mean Value Theorem (MVT): Explaining and applying the foundational theoretical properties of the MVT and Rolle's Theorem.Curve Sketching: Using the First Derivative Test (for increasing/decreasing zones) and Second Derivative Test (for concavity and inflection points) to map functions.Optimization: Modeling and solving mathematical minimization or maximization scenarios (e.g., maximizing area under cost restraints).L’Hôpital’s Rule: Resolving advanced indeterminate forms using top-and-bottom fractional derivatives.