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

Ch. 2 Differentiation - 7

Differentiation of implicit functions

Ex: x² = y² + 2xy + 3x - 4y - 8 = 0

Ch. 2 Differentiation - 8

Differentiation of parametric functions

y is given as f(t) and x is given as g(t)

we have to find dy/dx
dy/dx = (dy/dt) *( dt/dx)

ch. 2 Derivatives - 9

Higher order derivatives


d²y/dx² etc.

Applications of Derivatives - 1

Introduction

Geometrical and Physical meaning of a derivative

Gradient (slope) of a curve

ch. 3 Applications of Derivatives - 3

Rate measure, related rates

If y is a function of x, then dy/dx, if it exists, represents the actual rate of change of y with respect to x.

If y =s displacement and x = t time
ds/dt = velocity v
dv/dt = acceleration a

Application of Derivatves - 4

Approximation and errors

Interpretation of the sign of the derivative

Increasing and decreasing functions

Ch. 3 Applications of Derivatives - 5

Maximum and minimum

a function f is said to have a minimum at x = a if we can find δ>0 such that f(x)>f(a) for all x between (x-δ and x+δ) and x≠a.

f(x) is greater than f(a) around its neighbourhood.

sufficient conditions for extreme values

Ch. 4 Integration - 1

Introduction

standard forms already learnt

∫sin x dx = - cos x + c
∫cos x dx = sin x + c

Ch. 4 Integration - 2

Indefinite integrals

Standard forms

Method of substitution
Linear substitution
Integrating powers of trigonometric functions
Integrating expressions having square roots

Ch. 4 Integration - 3

Integration by parts

∫ uv dx = u∫vdx - ∫(∫v dx)(du/dx)dx

Ch. 4 Integration - 4

Integration by partial fractions

Distinct linear factors
(3x-4)/(x-1)(x-2) = 1/(x-1) + 2/(x-2)

disguised linear factors
(x² +15)/(x² -1)(x² +2)

Repeated linear factors
(5x² - 19x - 17)/(x-1)(x-2)²

Non repeated quadratic factors
If the denominator contains a factor ax² + bx + c then the fraction corresponding to it is
(px+ q)/(ax² + bx + c)

Definite Integration - 1

Introduction
If ∫f(x)dx = g(x)+c
∫f(x)dx (a to b) = [g(x)] from a to b = g(b) - g(a)

Ch.5 Definite integration - 2

INtgral as limit of a sum

the definite integral

fundamental theorem of integral calculus

Ch 5. Definite Integration - 3

Properties of definite integration

∫f(x)dx (a to b) = ∫f(t) dt (a to b)

∫f(x) dx (a to b) = -∫f(x) dx (b to a)

∫f(x) dx ( 0 to a) = ∫f(a-x)dx (0 to a)

∫f(x)dx (a to b) = ∫f(a+b-x) dx (a to b)

Ch. 6 Applications of Definite Integral - 1

Introduction

Ch. 6 Applications of Definite Integral - 2

area under a curve

Loop of curve

Ch. 6 Applications of Definite Integral - 3

Volume of a solid of revolution

Formula

Differential Equations

Introduction

d²y/dx² + dy/dx +y = 0 is a differential equation

Differential Equations - 2

Definitions

Teh order of differential equation is the order of the highestorder differential coefficient appearing in it.

The degree is the power to which the highest order derivative is raised in the equation

Differential Equations

Formation of differential equations

Ch. 7 Differential Equations - 4

Solutions of equations of first order and first degree

M + Ndy/dx = 0

Example 2x + 3y² dy/dx = 0

dy/dx is first order direvative of y and it has a power of 1 only.

we can write the given equation as
3y² dy/dx = -2x
3y² dy = -2xdx

∫ 3y² dy = ∫-2xdx

y³ = -x² + c is the solution to this equation

Ch. 8 Applications of Differential Equations - 1

Introduction

Initial conditions are provided for some physical situations so that arbitrary constant can be given the required value.

Ch. 8 Applications of Differential Equations - 2

Growth and decay

Ch. 9 Numberical Methods - 1

Introduction

Mathematicians have systematic ways of going as near the answer as possible. This branch of mathematics is called numerical methods.

Ch. 9 Numberical Methods - 2

Finite differences

basic terms

Forward difference table

backward differences

shift operator E

Relation between E and delta
operator E^-1

Ch. 9 Numberical Methods - 3

Interpolation

Newton's (forward) interpolation formula

Newtons bacward interpolation formula

Lagrange interpolation formula

Ch. 9 Numberical Methods - 4

Numerical integration

Division of an interval

Trapezoidal rule for integral ydx from a to b

Simpson's (1/3)rd and (3/*)th rules for integral ydx from a to b

Boolean Algebra - 1

Geogre Boole developed this area of mathematics.

10 Boolean Algebra - 2

Boolean algebra as algebraic structure
B is a nonempty set
1. for all x,y in B

(i) x+y belong to B
(ii) x.y belongs to B

Ch .10 Boolean Algebra - 3

Principle of duality

Ch. 10 Boolean Algebra - 4

Boolean functions and switching circuits

Boolean functions

Ch. 10 Boolean Algebra - 5

Application of Boolean Algebra to switching circuits

A logic gate

Writing boolean expressions using gates