Overview
This lecture covers the concepts of continuity and differentiability in calculus, including key definitions, graphical understanding, algebraic rules, and a variety of question-solving techniques as per the class 12 mathematics syllabus.
Review of Limits and Derivatives
- Limits and derivatives were introduced in class 11, covering basics like finding derivatives of polynomials and trigonometric functions.
- Understanding limits is essential for grasping continuity and differentiability.
Continuity
- A function is continuous at a point if the value at that point equals the limit from both sides.
- Graphically, a continuous function can be drawn without lifting the pen.
- Discontinuous functions have breaks or jumps at certain points.
- Continuity is always checked at specific points, and definitions involve the use of limits.
Types and Examples of Continuity
- Piecewise functions often need continuity checked at junction points.
- Mathematical tests: If limit as x approaches c (from both sides) equals f(c), the function is continuous at x = c.
- Algebraic combinations (sum, difference, product, quotient) of continuous functions are also continuous if the denominator is not zero.
Differentiability
- A function is differentiable at a point if its derivative exists there.
- The basic derivative formula: f'(c) = lim(h→0) [f(c+h) - f(c)] / h.
- Differentiability implies continuity, but not all continuous functions are differentiable everywhere.
- The derivative at a point represents the slope of the tangent at that point.
Rules of Differentiation
- (u+v)' = u' + v', (u−v)' = u'−v', (uv)' = u'v + uv', (u/v)' = (u'v − uv')/v² (v ≠ 0).
- Common derivatives: d/dx (xⁿ) = n xⁿ⁻¹; d/dx (sin x) = cos x; d/dx (cos x) = −sin x.
Composite and Implicit Functions
- Chain rule: For composite functions, d/dx f(g(x)) = f'(g(x))·g'(x).
- For implicit functions, differentiate both sides with respect to x and solve for dy/dx.
Special Functions
- Exponential: d/dx (eˣ) = eˣ.
- Logarithmic: d/dx (ln x) = 1/x.
- Inverse trigonometric functions have standard derivatives, e.g., d/dx (sin⁻¹x) = 1/√(1−x²).
Higher Order Derivatives
- Second order derivative: d²y/dx² is the derivative of the first derivative.
- Calculated by differentiating the first derivative again with respect to x.
Parametric Differentiation
- If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt).
Theorems in Calculus
- Rolle’s Theorem: If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) such that f'(c) = 0.
- Mean Value Theorem: If a function is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = [f(b)−f(a)] / (b−a).
Key Terms & Definitions
- Continuous Function — A function where small changes in input cause small changes in output, without jumps or breaks.
- Discontinuous Function — A function with at least one jump, break, or gap.
- Derivative — The instantaneous rate of change of a function, or the slope of its tangent.
- Differentiable Function — A function with a derivative at a given point.
- Composite Function — A function made by applying one function to the results of another.
- Implicit Function — A function where y is not isolated in terms of x.
- Parametric Function — Both x and y are functions of a third parameter, usually t.
Action Items / Next Steps
- Practice textbook exercises on continuity and differentiability, especially NCERT Exercise 5.1–5.8.
- Memorize the standard derivatives for basic, exponential, logarithmic, and inverse trigonometric functions.
- Revise and solve questions based on Rolle's and Mean Value Theorems.
- Attempt unsolved problems from your textbook independently.