Index

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE.

TANGENT PLANE AND NORMAL LINE TO A SURFACE

TAYLOR’s EXPANSION FOR FUNCTION OF TWO VARIABLES.

LAGRANGE’s METHOD OF UNDETERMINED MULTIPLES.

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

The Function f(x,y) is maximum at (x,y) if for all small positive or negative values of h and k; we have

f(x+h , y+k) – f(x,y) < 0

Similarly f(x,y) is minimum at (x,y) if for all small positive or negative values of h and k, we have

f(x+h , y+k) – f(x,y) > 0

Thus ,from the defination of maximum of f(x,y) at (x,y) we note that f(x+h , y+k) – f(x,y) preserves the same sign for a maximum it is negative and for a minimum it is positive

Working rule to find maximum and minimum values of a function f(x,y)

(1) find ∂f/∂x and ∂f/∂y(2) a necessary condition for maximum or

minimum value is ∂f/∂x=0 , ∂f/∂y=0

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

solve simultaneous equations ∂f/∂x=0 , ∂f/∂y=0

Let (a₁,b₁) , (a₂,b₂) … be the solutions of these equations.

Find ∂²f/∂x²=r , ∂²f/∂x ∂y=s , ∂²f/∂y²=t

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

(3) a sufficient condition for maximum or minimum value is rt-s²>0.

(4 a ) if r>0 or t>0 at one or more points then those are the points of minima.

(4 b) if r<0 or t<0 at one or more points then those points are the points of maxima.

(5) if rt-s²<0 ,then there are no maximum or minimum at these points. Such points are called saddle points.

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

(6) if rt-s²=0 nothing can be said about the maxima or minima .it requires further investigation.

(7) if r=0 nothing can be said about the maximum or minima . It requires further investigation.

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

Example discuss the maxima and minima of xy+27(1/x + 1/y)

∂f/∂x=y-(27/x²) , ∂f/∂y=x-(27/y²)For max. or min ,values we have ∂f/∂x=0 ,

∂f/∂y=0.y-(27/x²)=0…(1)x-(27/y²)=0…(2)Giving x=y=3

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

∂²f/∂x²=r =54/x³ ∂²f/∂x ∂y=s=1 , ∂²f/∂y²=t=27/y³r(3,3)=3s(3,3)=1t(3,3)=3rt-s²=9-1=8>o , since r,t are both >0We get minimum value at x=y=3 which is 27.

MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLE

TANGENT PLANE AND NORMAL LINE

LET THE EQUATION OF THE SURFACE BE f(x,y,z)=0

The equation of the tangent plane at P(x₁,y₁,z₁) to the surface is

(x-x₁)(∂f/∂x)p + (y-y₁)(∂f/∂y)p +(z-z₁)(∂f/∂z)p=0

And the equations of the normal to the surface at P(x₁,y₁,z₁) which is a line through P are:

x-x₁/ (∂f/∂x)p = y-y₁/(∂f/∂y)p = z-z₁/(∂f/∂z)p

Example find the equations of the tangent plane and

normal to the surface z=x²+y² at the point (1,-1,2).

∂f/∂x=-2x ∂f/∂y=-2y ∂f/∂z=1

TANGENT PLANE AND NORMAL LINE

At (1,-1,2), ∂f/∂x=-2 ∂f/∂y=2 ∂f/∂z=1Therefore equation of the tangent plane at (1,-1,2) is (x-1)(-2)+(y+1)(2)+(z-2)(1)=0Or -2x+2+2y+2+z-2=0Or 2x-2y-z=2Equations of the normal arex-1/-2 = y+1/2 = z-2/1

TANGENT PLANE AND NORMAL LINE

TAYLOR’S & MACLAURIN’S SERIES

(Taylor’s series):-If f(x) is an infinitely differentiable function of x which can be expanded as a convergent power series in (x-a), then

f(x)= f(a)+ (x-a)f’(a)/1! +(x-a)2 f’’(x)/2!+ (x-a)3 f’’’(x)/3!+……….+(x-a)n fn(a)/n!+……….

Where a is constant.

By putting x-a=h; that is, x=a+h in equation, we get

f(a+h)= f(a)+ h f’(a)/1!+ h2 f’’(a)/2!+ h3f’’’(a)/3!+….+hn fn(a)/n!+……

By putting a=0 in above equation, we have

f(x)=f(0)+ x f’(0)/1! +x2f’’(0)/2! +……+ xn fn(0)/n!+……

TAYLOR’S & MACLAURIN’S SERIES

Statement of Maclaurin’s Series:-If f(x) is an infinitely differentiable function of x which can be expanded as a convergent power series in x ,then

f(x)=f(0)+ x f’(0)/1! +x2f’’(0)/2! +……+ xn

fn(0)/n!+……

TAYLOR’S & MACLAURIN’S SERIES

Expansions of some standard function

The following are some expansion of standard treated as standard expansions,obtained with the help of maclaurin’s series. These standard expansions are useful in obtaining the expansion of oher functions.

1.Expansion of e x , e –x ,cosh(x) ,sinh(x) Let f(x)=e x, then

f’(x)=f’’(x)=f’’’(x)=….=ex

Also, f’(0)= f’’(0)= f’’’(0)=….=1

By substituting the values of f(0), f’(0), f’’’(0),…. In Maclaurin’s series ,We get

e x =1+ x+ x2/2!+ x3 /3!+…….

Expansions of some standard function

Replacing x by –x, yields e –x= 1- x+x2/2! – x3/3!+……..Adding e x and e –x , we obtain Cosh(x)=( e x +e –x)/2 =1+ x2/2! + x4 /4!+……Again, by subtracting e x and e –x ,we obtain Sinh(x)=(e x- e –x)/2 =x+ x3/3!+ x5/5!+……Note The expansions of e –x, cosh(x), sinh(x) can

also be obtained by using Maclaurin’s series.

Expansions of some standard function

Example:-1. Expansion of tan x Let y= f(x)= tanx, Y1=sec2x = 1+tan2x=1+y2,

Y2=2yy1,

Y3=2y12+2yy2,

…… ……… ……. .……..

Expansions of some standard function

Y(0)=0, Y1(0)=1+y2(0) =1,

Y2(0)=2y(0)y1(0)=2(0)(1)=1,

Y3(0)=2y 12(0)+ 2y(0)y2(0)

=2(1)2+2(0)(0)=2, ……… ……….. ……… ………..

Expansions of some standard function

By putting these values of y(0),y1(0),y2(0),….. in Maclaurin’s series,

Y(x)=y(0)+ xy1(0)+ x2y2(0)/2!+ x3y3(0)/3!+…….

we get, tan x=x+x3/3+ x52/15+…….

Expansions of some standard function

LINEARIZATION

(Linearization):-Linearization means to replace given function of two variables. This can be achieved by tangent plane approximation.

The linearization of a function f(x , y) at a point (x0,y0) where f is differentiable is the function

L(x , y)=f(x0,y0)+ fx(x0,y0)(x –x0)+fy(x0,y0)(y-y0)

Which is the equation of the tangent plane to the graph of a function f of two variables at the point (x0,y0, f(x0,y0).

The approximation f(x , y)= L(x ,y ) is called the Standard Linear

approximation or Linear approximation or the tangent plane approximation of f at (x0,y0).

LINEARIZATION

EXAMPLE:- Find the linearization of f(x ,y) =X2 –xy+

(1/2) y2+ 3 at the point (3, 2).

Let us first evaluate f, f x and f y at (3 ,2).

f(3 ,2)=(3)2 -(3)(2)+(1/2)(2)2+3=8. F x =2x- y=> (f x)(3, 2)= 2(3)-2 =4.

F y=-x+ y=> (f y) (3, 2) =-3 +2=-1.

LINEARIZATION

Using differentiable function, the required linearization is

L(x ,y)=f(3 ,2)+ f x (3 ,2)(x -3) + f y (3,2) (y-

2) =8+ (4) (x -3)+ (-1)(y-2) = 4x- y -2.

LINEARIZATION

To find extreme values of a function we consider a

function of three variables with one restriction.

Lagrange’s Method of Undetermined Multipliers

REFERENCE

CALCULUSDr.K.R.KachotMahajan publishing

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