Continuity and Differentiability – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

A function is continuous if its graph has no breaks; it is differentiable if it has a well-defined tangent. Rules of differentiation handle composite, implicit, inverse, exponential and logarithmic functions.

Continuity and differentiability

• f is continuous at a if lim(x→a) f(x) = f(a).
• Differentiable ⇒ continuous, but not conversely (|x| at 0 is continuous but not differentiable).
• Polynomials, sin x, cos x, eˣ are continuous everywhere.

Rules

• Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).
• d/dx eˣ = eˣ; d/dx log x = 1/x; d/dx sin⁻¹x = 1/√(1 − x²); d/dx tan⁻¹x = 1/(1 + x²).
• Logarithmic differentiation for xˣ: d/dx xˣ = xˣ(1 + log x); parametric dy/dx = (dy/dt)/(dx/dt).
• Rolle's and Mean Value theorems link derivative values with function values.

Key terms

Continuous function

function whose limit equals its value at every point of the domain

Derivative

instantaneous rate of change, dy/dx

Rolle's theorem

if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for some c in (a, b)

Formulas

• d/dx (uv) = u′v + uv′
• d/dx (u/v) = (u′v − uv′)/v²
• d/dx xˣ = xˣ(1 + log x)

Important questions with answers

Quick revision

• Differentiable ⇒ continuous.
• |x| is not differentiable at 0.
• Chain rule for composite functions.

Exam tips

• Check left-hand and right-hand derivatives at corner points.
• For implicit differentiation, differentiate both sides with respect to x and treat y as a function of x (d/dx y² = 2y dy/dx).

Practice MCQs

1. d/dx (sin x) is

   1. −cos x
   2. −sin x
   3. sin x
   4. cos x
   Show answer

   Answer: (d) cos x. Standard derivative.

2. f(x) = |x| at x = 0 is

   1. not continuous
   2. differentiable
   3. continuous but not differentiable
   4. undefined
   Show answer

   Answer: (c) continuous but not differentiable. Corner point.

3. d/dx (e^(3x)) is

   1. e^(3x)
   2. 3e^(3x)
   3. 3x e^(3x−1)
   4. e³
   Show answer

   Answer: (b) 3e^(3x). Chain rule.

4. d/dx (log(sin x)) is

   1. 1/sin x
   2. tan x
   3. cot x
   4. cos x
   Show answer

   Answer: (c) cot x. cos x / sin x.

5. d/dx [sin⁻¹(2x√(1 − x²))] for −1/√2 < x < 1/√2 is

   1. 1/√(1 − x²)
   2. 2x
   3. −2/√(1 − x²)
   4. 2/√(1 − x²)
   Show answer

   Answer: (d) 2/√(1 − x²). Put x = sin θ.

6. If x = a cos t, y = a sin t, then dy/dx is

   1. tan t
   2. −cot t
   3. cot t
   4. −tan t
   Show answer

   Answer: (b) −cot t. (a cos t)/(−a sin t).

7. d/dx (xˣ) is

   1. x·xˣ⁻¹
   2. xˣ log x
   3. xˣ
   4. xˣ(1 + log x)
   Show answer

   Answer: (d) xˣ(1 + log x). Logarithmic differentiation.

8. For f(x) = x² on [1, 3], the c of the Mean Value Theorem is

   1. 1.5
   2. 2
   3. 2.5
   4. 3
   Show answer

   Answer: (b) 2. f′(c) = (9 − 1)/2 = 4 = 2c.