Continuity and Differentiability (Chapter 5)

Introduction and Overview

This session is a one-shot video covering Chapter 5 of Mathematics - Continuity and Differentiability.
Purpose: To help in exam preparation with all topics covered as per the latest syllabus.
Tools Provided: (i) Join Arvind Academy Telegram for presentations; (ii) Download Arvind Academy App for courses.

Graphical Understanding: Continuous if the plot does not require the pencil to be lifted. Points of discontinuity occur where a teeny plot would result in lifting the pencil.
Mathematical Representation: Function f(x) is continuous at point 'a' if:
- Left Hand Limit (LHL) = Right Hand Limit (RHL) = Value of the Function at point 'a' (f(a)).
- Formulae:
  - LHL: (\lim_{{x \to a^-}} f(x)).
  - RHL: (\lim_{{x \to a^+}} f(x)).
  - Need: (\lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x) = f(a)).

If two functions f and g are continuous at x = a, then:
- f + g is continuous at x = a.
- f - g is continuous at x = a.
- f * g is continuous at x = a.
- (\frac{f}{g}) is continuous at x = a, provided g(a) != 0.
Polynomial, rational, trigonometric, exponential, and constant functions are continuous in their respective domains.*

Discussing continuity of trigonometric function sin(x).
Evaluating properties and examples of continuous functions.
Checking given functions for points of continuity and determining constant k as proofs and problems.

Function f(x) is differentiable at point 'a' if the limit exists:
- (\lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}).
Differentiability implies continuity, but not vice versa.
Key formula: (\lim_{{x \to a}} \frac{{f(x) - f(a)}}{x - a}).

Basic Rules: Power, product, quotient, and chain rules.
- Product Rule: ((u * v)' = u'v + uv').
- Quotient Rule: (\left(\frac{u}{v}\right)' = \frac{{u'v - uv'}}{v^2}).
- Chain Rule: Derivative of composite functions (f(g(x))' = f'(g(x)) * g'(x)).
Implicit Differentiation: Differentiate both sides concerning x separately.
- Then rearrange to isolate (\frac{dy}{dx}).
Parametric Differentiation: If x and y are functions of another variable t:
- (\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}).
Inverse Trigonometric Functions: Various identities and differentiation rules.
Logarithmic Differentiation: Useful for functions involving variables raised to variables.
- Apply log on both sides, then differentiate.

Second Order Derivatives: Derivative of a derivative. If (\frac{dy}{dx} = f'(x)), then (\frac{d^2y}{dx^2} = (f'(x))').
Intensive usage in problems solving involving higher orders.

Various examples from exams involving continuity and differentiability.
- Differentiating given implicit, parametric, or explicit functions.
- Proving differentiability at specific points.
- Finding second-order derivatives for given functions.

Step-by-step: Identify type (implicit, parametric, etc.), apply relevant differentiation rules, and simplify.
Practice problems involving trigonometric identities, inverse trigonometric functions, higher-order derivatives.

Importance of practice and understanding step-by-step procedures in solving different types of differentiation and continuity problems.
Availability of resources and motivation to persist in learning and practicing.
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Practice is key to mastering both continuity and differentiability.
Regularly refer to properties and differentiate between types for clarity and quick recall.
Apps and additional help resources available on Arvind Academy platforms.

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