Standard 12 Mathematics Paper II Chapters Revision Facilitator

Section headings

1. Limits
concept of limits
Algebra of limits
Standard limits
Theorem lim x →0 (sin x)/x = 1
Exponential limits
Infinity
Definition of lim x→∞ f(x) = l
Continuity
Left and right hand limits
Concept of continuity
Removable discontinuity
Algebra of continuous functions
Continuity in an interval
Continuity of standard functions

2. Differentiation
Definition of derivative
Leibnitz’s notation
Definition of dy/dx
Standard derivatives (table)
The algebra of derivatives
Differentiability and continuity
Derivative of composite functions
Derivatives of some standard composite functions (table)
Derivatives of inverse functions
Derivatives of inverse circular functions
Logarithmic differentiation
Differentiation of implicit functions
Differentiation of parametric functions
Higher order derivatives



3. Applications of derivatives
Geometrical and physical meaning of derivative
Gradient of a slope
Rate measures, related rates
Approximation and errors
Interpretation and the sign of the derivative
Increasing and decreasing functions
Maximum and minimum


4. Integration
Revision of XI class
Defintion
Standard froms
Indefinite integrals

∫(a sinx + bcos x)dx/(c sin x + d cos x)
Some more standard forms

∫ dx/ (x²+a²)
∫dx/(x²-a²)
∫dx/(a²-x²)

Method of substitution
Linear substitution
Integration of odd powers of sin x or cos x
Integration of even powers of sin x or cos x
Integration of any power of tan x or cot x
Integration of even powers of sec x or cosec x
Integration of functions involving square roots
Integration by parts

∫ √ (x²+a²)dx
∫√ (x²-a²)dx
∫√ (a²-x²)dx

∫e^x[f9x0+f ‘(x0]dx = e^xf(x) + c
Integration by partial fractions
Distinct linear factors
Disguised linear factors
Repeated linear factors
Nonrepeated quadratic factors



5. Definite Integration
Definition
Change of variables
Integral as a limit of sum
The definite integral
Fundamental theorem of integral calculus
Properties of definite integration

6. Applications of definite integral
Introduction
Area under a curve
Loop of curve
Volume of a solid of revolution

7. Differential equations
Introduction
Definitions
Formation of differential equations
Solutions of equations of first order and first degree
Equations with separable variables
Homogeneous differential equations


8. Applications of differential equations
Introduction
Growth and decay


9. Numerical methods
Introduction
Finite differences
Basic terms
Forward difference table
Backward differences
Shift operator
Relation between E and Δ
Interpolation
Newton’s backward interpolation formula
Lagrange interpolation formula
Numerical integration
Division of an interval
Trapezoid rule for ∫a to b ydx
Simpson’s (1/3)rd and (3/8)th rules for for ∫a to b ydx

10. Boolean Algebra
Introduction
Boolean algebra as algebraic structure
Principles of duality
The duality principle
Theorem: The identities 0 and 1 are unique in B.
Working with distributive laws in Boolean algebra.
Some more useful results of Boolean algebra
Idempotent laws
Absorption laws
De Morgan’s laws
Boolean functions and switching circuit
Boolean functions
Application of Boolean algebra to switching circuits
Writing Boolean expressions using gates

Standard 12 Mathematics Paper II Textbook Chapters Paper II

1. Limits

2. Differentiation

3. Applications of derivatives

4. Integration

5. Definite Integration

6. Applications of definite integral

7. Differential equations

8. Applications of differential equations

9. Numerical methods

10. Boolean Algebra

Differentiation - 1

Introduction


2. Derivative from first principless

Definition of derivative
Lebnitz/s notation
Standard forms
Algebra of derivatives

Ch. 2 Differentiation - 3

Differentiability and continuity

Derivatives - 4

Derivatives of composite functions

If y = f(u) and u = g9x) then y = f[g(x)] is a composite function of x.

example y = u³ and u = 3x+4

y = (3x+4)³ is a composite function

Differentiation - 5

Derivatives of inverse functions

y = f(x)

x = g(y)

Differenentiation - 6

Logarithmic differentiation

y = [f(x)]^g(x)

take logarithms on both sides

log y = g(x) log [f(x)]
Differntiate