Continuity, Differentiability, and Application of Derivatives

References:

• NCERT Class 11: Textbook: Chapter 13: Limits and Derivatives, Chapter 14: Applications of Derivatives
• NCERT Class 12: Textbook: Chapter 6: Continuity and Differentiability, Chapter 7: Differentiation, Chapter 8: Applications of Derivatives

Detailed notes on subtopics:

1. Continuity:

• Continuity: definition, types (pointwise and uniform continuity). Interpretation as ’no sudden jumps'.
• Theorems on Continuity: Algebra of continuous functions, continuity with respect to a function’s domain.
• Properties of Continuous Functions: Bounds, Intermediate Value Theorem (existence of roots), Extreme Value Theorem.

2. Differentiability:

• Definition of Derivative: limit definition, geometrical interpretation as tangent slope.
• Rules for Differentiation: power, sum/difference, product, quotient, inverse function rules.
• Derivative of Implicit Functions: introduction and methods.
• Higher Order Derivatives: definitions and geometrical interpretations.

3. Applications:

• Maxima and Minima: relative and absolute, first derivative test, second derivative test.
• Related Rates: rates of change, implicit differentiation.
• Optimisation: word problems, objective functions, constraints.
• Integration: introduction and connection with derivatives.

4. Differential Equations:

• First Order Differential Equations: basic solutions.
• Separable Differential Equations: technique and examples.
• Homogeneous Differential Equations: technique and examples.
• Linear Differential Equations: first order, initial conditions.

5. Limits and Special Functions:

• Limits: algebraic techniques, squeeze theorem, L’Hopital’s rule, sandwich theorem.
• Limits to Infinity: algebraic, indeterminate forms.
• Logarithmic Functions: definition, graphs, properties, equations.
• Exponential Functions: definition, graphs, properties, equations.

6. Asymptotes:

• Asymptotes: definition, vertical, horizontal, oblique.
• Asymptotic Behavior: graphs, limits.

7. Curvature & Radius:

• Curvature: measure, formula.
• Radius of Curvature: formula, geometrical interpretation.

8. Applications of Derivatives:

• Curve Sketching: extrema, asymptotes, curvature, concavity.
• Graph Transformations: shifts, reflections, stretches.
• Linearisation and Approximation: tangent line approximation.
• Normal and Tangent Lines: equations, applications.
• Newton’s Method: finding roots.

Disclaimer: The above notes provide an overview of important subtopics within the topic of Continuity, Differentiability, and Application of Derivatives. They are based on the NCERT textbooks and standard syllabus; however, the actual details and depth of coverage may vary based on the specific examination or curriculum guidelines. It’s important to refer to the recommended study materials and syllabus provided by the relevant educational institution or examination board for the most accurate and comprehensive information.