Complete Summary and Solutions for Continuity and Differentiability – NCERT Class XII Mathematics Part I, Chapter 5 – Limits, Continuity, Differentiability, Properties, and Applications
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Updated: 7 months ago
Continuity and Differentiability
Chapter 5: Mathematics - Ultimate Study Guide | NCERT Class 12 Notes, Solved Examples, Exercises & Quiz 2025
Full Chapter Summary & Detailed Notes - Continuity and Differentiability Class 12 NCERT
“The whole of science is nothing more than a refinement of everyday thinking.” — ALBERT EINSTEIN
5.1 Introduction
This chapter continues the study of differentiation from Class XI, covering polynomials and trig functions. Key concepts: continuity, differentiability, and their relations. Topics include derivatives of inverse trigonometric functions, exponential and logarithmic functions, and geometric applications via fundamental theorems.
Conceptual Diagram: Continuity Graph (Like Book Fig 5.1)
Step function: f(x) = 1 if x ≤ 0, 2 if x > 0. Limit mismatch at x=0: left=1, right=2. Graph jumps, pen lift needed.
\[ \begin{cases} f(x) = 1 & x \leq 0 \\ f(x) = 2 & x > 0 \end{cases} \]Visual: Horizontal lines at y=1 (left) and y=2 (right), discontinuity at origin.
Why This Guide Stands Out (Expanded for 2025 Exams)
Comprehensive coverage mirroring NCERT pages 104-150: All subtopics point-wise with evidence (e.g., Ex 1 linear continuity), full examples (e.g., piecewise checks), debates (continuous vs differentiable). Added 2025 relevance: Continuity in ML for loss functions. Processes for limits/derivs with step-by-step derivations. Proforma: Limit check → Value match → Continuous.
5.2 Continuity
Informal: Graph drawable without pen lift. Formal: lim_{x→c} f(x) = f(c). Requires existence of left/right limits equaling f(c). Discontinuous if mismatch (jump, removable, infinite).
Examples: Polynomials, constants, identity continuous everywhere. |x| continuous at 0. 1/x continuous for x≠0.
Quick Table: Continuity Checks (Expanded with Book Examples)
| Function Type | Continuous At | Example |
|---|---|---|
| Polynomial | Everywhere | f(x)=x^2, lim=x^2=f(c) |
| Rational | Except poles | 1/x, x≠0 |
| Absolute | Everywhere | |x|, left/right both |c| |
| Piecewise | Check joints | x+2 (x<1), x-2 (x>1), mismatch at 1 |
Algebra of Continuous Functions
Sum, product, quotient (denom≠0) of continuous are continuous. Composition too.
5.3 Differentiability
f differentiable at c if lim_{h→0} [f(c+h)-f(c)]/h exists (finite). Equivalent: f' (c) exists. Continuous implies differentiable? No, but differentiable ⇒ continuous.
Examples: Polynomials, trig differentiable everywhere. |x| continuous but not diff at 0 (left deriv -1, right +1).
5.4 Derivatives of Inverse Trigonometric Functions
Key formulas:
\[ \frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2}, \quad \frac{d}{dx} \cot^{-1} x = -\frac{1}{1+x^2} \] \[ \frac{d}{dx} \sec^{-1} x = \frac{1}{|x| \sqrt{x^2-1}}, \quad \frac{d}{dx} \csc^{-1} x = -\frac{1}{|x| \sqrt{x^2-1}} \]Domains: [-1,1] for arcsin/arccos, etc.
Derivation: Arcsin Derivative (Chain Rule)
Let y = sin^{-1} x, x = sin y, dx/dy = cos y = \sqrt{1 - sin^2 y} = \sqrt{1-x^2}. Thus dy/dx = 1 / \sqrt{1-x^2}. Sign positive in [-π/2, π/2].
5.5 Exponential and Logarithmic Functions
e^x: Derivative e^x (unique base with f'=f). log_a x = ln x / ln a, deriv 1/(x ln a).
\[ \frac{d}{dx} e^x = e^x, \quad \frac{d}{dx} \ln x = \frac{1}{x} \]General: d/dx a^x = a^x ln a, d/dx log_a x = 1/(x ln a).
5.6 Logarithmic Differentiation
For complex: y = f(x)^{g(x)}, ln y = g ln f, (1/y) y' = g'/g + (f'/f) g, y' = y (g'/g + f'/f g).
5.7 Derivatives of Functions in Parametric Form
x = f(t), y = g(t), dy/dx = (dy/dt) / (dx/dt).
5.8 Second Order Derivative
d²y/dx² = d/dx (dy/dx). Geometric: Concavity, inflection points.
5.9 Mean Value Theorem (Rolle’s, Lagrange’s)
Rolle’s: Continuous [a,b], diff (a,b), f(a)=f(b) ⇒ ∃c, f'(c)=0.
Lagrange’s: ∃c ∈ (a,b), f'(c) = [f(b)-f(a)]/(b-a).
Derivation: Proof Sketch (Rolle’s)
Assume max/min at c interior, f'(c)=0. At ends equal, so yes. Full: Extreme value theorem applies.
Example 1 (Integrated - Basic Check)
f(x)=2x+3 at x=1. lim=5=f(1), continuous.
Example 2 (Integrated - Polynomial)
f(x)=x^2 at 0. lim=0=f(0).
Example 3 (Integrated - Absolute)
|x| at 0. Left lim=-x→0=0, right=x→0=0=f(0).
Example 4 (Integrated - Removable)
f(x)=(x^3+3)/x if x≠0, 1 if x=0. lim=3≠1, discontinuous.
Example 5 (Integrated - Constant)
f(x)=k everywhere continuous.
Example 6 (Integrated - Identity)
f(x)=x, lim=c=f(c).
Example 7 (Integrated - |x| Full)
Continuous all reals: Left -x, right x, match.
Example 8 (Integrated - Cubic)
x^3 + x^2 -1, polynomial continuous.
Example 9 (Integrated - 1/x)
Continuous x≠0.
Example 10 (Integrated - Piecewise Jump)
x+2 (≤1), x-2 (>1), discontinuous at 1 (3 vs -1).
Example 11 (Integrated - With Zero)
Similar jump at 1.
Example 12 (Integrated - Undefined at 0)
x+2 (<0), -x+2 (>0), continuous domains.
Example 13 (Integrated - x^2 and x)
Piecewise x^2 (≥0), x (<0), continuous at 0.
Example 14 (Integrated - Polynomial Proof)
lim p(x)=p(c), continuous.
Example 15 (Integrated - Greatest Integer)
[x] discontinuous at integers (left n-ε→n-1, right n).
Summary & Exercises Tease
Key: Continuity for limits match; diff for tangent existence. Ex 5.1: Continuity checks; 5.2: Diff basics; 5.3-5.5: Inverse/exp/log derivs; 5.6-5.8: Parametric/second; Misc: Theorems apps.
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