4. Iterative Method
1.4 Calculus .1 Limit of a Function
Let y = f(x)
Then lim (x)= i.e, “ (x) as x a” implies or any (>0), (>0) such that whenever 0<|x a|< ,| (x) |<
Some Standard Expansions
( x) = nx ( ) n(n )(n )
x . . . x x a
x a = x x a x a . . . a e = 1 + x + + . . .
log( x) = x + . . . log( x) = x . . . Sin x = x
. . . Cos x = 1
+
. . . Sinh x = x
. . . Cosh x = 1 + . . . Some Important Limits
lim sinx
x =
lim(
x) =
lim( x) =
lim a
x = log a
lim e
x =
lim
log( x)
x =
lim x a
x a = a
limlog|x| = L – Hospital’s ule
When function is of or form, differentiate numerator &denominator and then apply limit.
Existence of Limits and Continuity:
1. f(x) is defined at a, i.e, f(a) exists.
2. If lim
f(x) = lim
f(x) = L ,then the lim
f(x) exists and equal to L.
3. lim (x) = lim (x)= f(a) then the function f(x) is said to be continuous.
Properties of Continuity
If f and g are two continuous functions at a; then a. (f+g), (f.g), (f-g) are continuous at a b. is continuous at a, provided g(a) 0 c. | | or |g| is continuous at a
olle’s theorem
If (i) f(x) is continuous in closed interval [a,b]
(ii) ’(x) exists or every value o x in open interval (a,b) (iii) f(a) = f(b)
Then there exists at least one point c between (a, b) such that ( ) = 0
Geometrically: There exists at least one point c between (a, b) such that tangent at c is parallel to x axis
C1
a C
C
2
b
Lagrange’sMean Value Theorem
If (i) f(x) is continuous in the closed interval [a,b] and
(ii) ’(x) exists in the open interval (a,b), then atleast one value c o x exist in (a,b) such that
( ) ( )
= (c).
Geometrically, it means that at point c, tangent is parallel to the chord line.
Cauchy’sMean Value Theorem
If (i) f(x) is continuous in the closed interval [a,a+h] and
(ii) (x) exists in the open interval (a,a+h), then there is at least one number (0< <1) such that
f(a+h) = f(a) + h f(a+ h) Let f1 and f2 be two functions:
i) f1,f2 both are continuous in [a,b]
ii) f1, f2 both are differentiable in (a,b) iii) f2’ 0 in (a,b)
then, for a
( ) ( ) ( ) ( ) = ( )( ) 1.4.2 Derivative:
’( ) =lim ( ) ( )
Provided the limit exists ’( ) is called the rate of change of f at x.
Algebra of derivative:-
i. ( g) = g ii. ( g) = – g iii. ( . g) = . g . g iv. ( /g) = . .
Homogenous Function
Any function f(x, y) which can be expressed in from xn ( ) is called homogenous function of order n in x and y. (Every term is of nth degree.)
f(x,y) = a0xn + a1xn-1y + a2xn-2y2 ………… an yn f(x,y) = xn ( )
Euler’s Theorem on Homogenous Function
If u be a homogenous function of order n in x and y then,
x
+y
= nu
x
+ 2xy
+ y
= n(n )u 1.4.3 Total Derivative
u= (x,y) ,x=φ(t), y=Ψ(t)
= . + . u =
x +
y
Monotonicity of a Function f(x)
1. f(x) is increasing function if for , ( ) ( ) Necessary and su icient condition, ’ (x) 2. f(x) is decreasing functionif for , , ( ) ( )
Necessary and sufficient condition, (x)
Note: is a monotonic unction on a domain ‘D’ then is one-one on D.
Maxima-Minima
a) Global b) Local
Rule for finding maxima & minima:
If maximum or minimum value of f(x) is to be found, let y = f(x)
Find dy/dx and equate it to zero and from this ind the values o x, say x is , β, …(called the critical points).
Find
at x = ,
If , y has a minimum value If ,y has a maximum value
If = , proceed urther and ind at x = .
If , y has neither maximum nor minimum value at x = But If
= , proceed further and find
at x = . If , y has minimum value
If , y has maximum value If = , proceed further
Note: Greatest / least value exists either at critical point or at the end point of interval.
Point of Inflexion
If at a point, the following conditions are met, then such point is called point of inflexion
i) = , ii) = 0 , iii)
Neither minima nor maxima exists Taylor Series:
(a h)= (a) h ’(a)
”(a) . . . Maclaurian Series:
(x) = ( ) x ’( )
( ) h
( ) Maxima &Minima (Two variables)
r = , s =
, t =
1. = 0, = solve these e uations. Let the solution be (a, b), (c, d)…
2. (i) if rt s and r maximum at (a, b) (ii) ifrt s and r minimum at (a, b)
(iii) ifrt s < 0 at (a, b), f(a,b) is not an extreme value i.e, f(a, b) is saddle point.
(iv) ifrt s > 0 at (a, b), It is doubtful, need further investigation.
Point of inflexion
1.4.4 Standard Integral Results
39. ∫ e [ (x) (x)]dx = e (x)
Integration by parts: ∫ u v dx = u. ∫ v dx ∫( ∫ v dx)dx
1.4.5 Rules for Definite Integral
1. ∫ (x)dx =∫ (x)dx+∫ (x)dx a<c<b
2. ∫ (x)dx =∫ (a b x)dx ∫ (x)dx =∫ (a x)dx
3. ∫ (x)dx =∫ / (x)dx+∫ / (a x)dx ∫ (x)dx = ∫ / (x)dx if f(a-x)=f(x)
= 0 if f(a-x)=-f(x)
4. ∫ (x)dx =2 ∫ (x)dxif f(-x) = f(x), even function = 0 if f(x) = -f(x), odd function Improper Integral
Those integrals for which limit is infinite or integrand is infinite in a x b in case of ∫ (x)dx, then it is called as improper integral.
1.4.6 Convergence:
∫ (x)dxis said to be convergent if the value of the integral is finite.
(i) (x) g(x) for all x and (ii) ∫ g(x)dx converges , then ∫ (x)dx also converges
(i) (x) g(x) for all x and (ii) ∫ g(x)dx diverges, then ∫ (x)dx also diverges
If lim ( )
( ) = c where c 0, then both integrals ∫ (x)dx and ∫ g(x)dx converge or both diverge.
∫ is converges when p and diverges when p
∫ e dx and ∫ e dx is converges for any constant p and diverges for p
The integral ∫ ( ) is convergent if and only ifp
The integral ∫ ( ) is convergent if and only ifp Selection of U & V I L A T E
E
Logarithmic Inverse circular (e.g.
tan
x)
Algebraic Trigonometric
Exponential
Note: Take that function as “u” which comes first in “ILATE”
1.4.7 Vector Calculus:
Scalar Point Function:
If corresponding to each point P of region R there is a corresponding scalar then (P) is said to be a scalar point function for the region R.
(P)= (x,y,z)
Vector Point Function:
If corresponding to each point P of region R, there corresponds a vector defined by F(P) then F is called a vector point function for region R.
F(P) = F(x,y,z) = f1(x,y,z) ̂ +f2(x,y,z)ĵ 3(x,y,z) ̂ Vector Differential Operator or Del Operator: = (
ĵ
̂
) Directional Derivative:
The directional derivative of f in a direction N⃗⃗ is the resolved part of in directionN⃗⃗ . . N⃗⃗ = | |cos
Where N⃗⃗ is a unit vector in a particular direction Direction cosine: l m n =
Where, l =cos , m=cos β , n=cos , 1.4.8 Gradient:
The vector function is defined as the gradient of the scalar point function f(x,y,z) and written as grad f.
grad f = = î ĵ + ̂
is vector function
If f(x,y,z)= 0 is any surface, then is a vector normal to the surface f and has a magnitude equal to rate of change of f along this normal.
Directional derivative of f(x,y,z) is maximum along and magnitude o this maximum is | |.
1.4.9 Divergence:
The divergence of a continuously differentiable vector point function F is denoted by div. F and is defined by the equation.
F = f + ĵ Ψ ̂
1.4.13 DEL Applied Twice to Point Functions:
1. div grad f = f=
f, g are scalar functions & F, G are vector functions 1. ( g) = + g
Also note:
1. (f/g)= (g f – f g)/g 2. ( . )’ = ’. . ’ 3. (F )’ = ’ G + F ’
4. (fg) = g f + 2 f. g + f g
1.4.15 Vector product
1. Dot product of A B with C is called scalar triplet product and denoted as [ABC]
Rule: For evaluating the scalar triplet product (i) Independent of position of dot and cross (ii) Dependent on the cyclic order of the vector [ABC] = A B. C = A. B C
= B C. A= B.C A = C A. B = C.A B A B. C = -(B A. C)
2. (A⃗⃗ B⃗⃗ ) C⃗ = (extreme adjacent) Outer
= (Outer. extreme) adjacent (Outer. adjacent) extreme
(A ⃗⃗⃗ B)⃗⃗⃗⃗ C⃗ = (C⃗ . A⃗⃗ )B⃗⃗ - (C⃗ . B⃗⃗ ) A⃗⃗
A⃗⃗ (B⃗⃗ C⃗ ) = (A⃗⃗ . C⃗ ) B⃗⃗ - (A⃗⃗ . B⃗⃗ )C⃗
(A⃗⃗ B⃗⃗ ) C⃗ A⃗⃗ (B⃗⃗ C⃗ )
1.4.16 Line Integral, Surface Integral & Volume Integral
Line integral = ∫ ( )d
( )= (x,y,z) ĵ (x,y,z) + ̂ Ψ(x,y,z) d = dx ĵ dy ̂ dz
∫ ( )d = ∫ ( dx dy Ψ dz )
Surface integral: ∫ ⃗ .ds⃗⃗⃗⃗ or∫ ⃗ . N⃗⃗ ds, Where N is unit outward normal to Surface.
Volume integral : ∫ dv
If F(R ) = f(x,y,z)î + (x,y,z)ĵ Ψ (x,y,z) ̂ and v = x y z , then
∫ dv = î∫ ∫ ∫ dxdydz ĵ ∫ ∫ ∫ dxdydz + ̂ ∫ ∫ ∫ Ψdxdydz 1.4.17 reen’s Theorem
If R be a closed region in the xy plane bounded by a simple closed curve c and if P and Q are continuous unctions o x and y having continuous derivative in , then according to reen’s theorem.
∮ (P dx dy)= ∫ ∫ ( x P
y) dxdy
1.4.18 Sto e’s theorem
If F be continuously differentiable vector function in R, then ∮ . dr = ∫ .N ds 1.4.19 Gauss divergence theorem
The normal surface integral of a vector point function F which is continuously differentiable over the boundary of a closed region is equal to the
∫ . N. ds = ∫ div dv