Showing posts from January, 2014

Some Important Differential Terms

Logarithmic Differentiation :- 

In order to differentiate a function of the form     v power on u    , where    u, v   are both variables , it is necessary to take its logarithm and then differentiate . This process which is known as Logarithmic Differentiation is also useful when the function to be differentiated is the product of a number of factors .

Preliminary Transformation :- 

In some cases , a preliminary transformation of the function to be differentiated facilitates the process differentiation a good deal .

Differentiation "ab into" :- 

To differentiate "ab into" or from first principles means that the process of differentiation is to be performed without making any use of the theorems on the differentiations of sums , products functions and functions etc. nor is any use to be made of the differential co-officiants of standard  forms  .

The   n th  derivative of the product of the powers of  sines and cosines :-

In order to find out   n th derivative of such…

Polar Equation of Curves

Any explicit or implicit relation between    r    and    theta      will give a curve determined by the points whose co-ordinates satisfies that relation .

Thus the equations ;

r=f(theta)   or   F(r , theta)=0

determine curves .

The co-ordinates of two points symmetrically situated about the initial line are of the form   (r,Theta)    and    (r,-theta)   so that their vertical angles differ in sign only .

Hence a curve will be symmetrical about the initial line if on changing    theta    to     -theta    its equation does not change . For instance the curve


is symmetrical about the initial line , for


It may be noted that

r=a    represents a circle with its center at pole and radius   a ;  and

theta=b    represents a line through the pole obtained by revolving the initial line through the angle   b   .

A few important curves will not be traced . To trace polar curves , we generally consider the variations in     r    as    theta 

Approximate Value

Approximate Value :-

Let    y=f(x)

Now ,   Lt delta  x--->0 (delta  y)/(delta  x)=dy/dx ,

therefore from the definition of the limit

as    delta  x   approaches   0  ,

(delta  y)/(delta  x)  approaches  dy/dx

therefore ,  (delta  y)/(delta  x)=dy/dx

[approx . when   delta  x   is small]

or,  (delta  y)=dy/dx  (delta  x)

Thus ,  (delta  y)=dy/dx  (delta  x)


Important Result  Essential For Problems :-

If   x   and   y   are functions of time ,  t  then

(dx/dt)=change of  x  in unit time 

         =rate of change of    x    .

and   (dy/dt)=change in   y  in unit time 

                  =rate of change of    y  .

Also , (dy/dx)=(dy/dt)/(dx/dt)

Existence of Tangent

Let   y=f(x)    be any function and

dy/dx   exists at     x=a    

Then at the corresponding point    [a , f(a)]    of the function tangent to the curve exists and if the tangent

makes an angle    theta     with the positive direction of    x-axis    we have

tan(theta)=dy/dx    at     x=a .

tan(theta) = gradient of the tangent to the curve

  y=f(x)     at    x=a 

therefore ; if    f(x)   is differentiable at    x=a    then tangent to the curve at

x=a    must exist and it must be unique .

In any graph of    /x/   tangent is not unique at    x=0   .

i.e. ; tangent at point    x=0   when     x-->0    from left is not same as the tangent

x=0    when   x-->0   from right .

when   x-->0   from left , gradient of the tangent is

tan135degree=-1    and when   x-->0  from right , gradient of the tangent is


Hence tangent is not unique at    x=0    and consequently    /x/  is not differentiable at     x=0    .

Formulas For Differential Co-officiant

Before find differential co-officiant of any mathematical term it is important to remember these formulas . These formulas are very helpful to find differential co-officiant .

1)          (d/dx)(c)=0

                   where    c    is a constant 

2)          (d/dx)(u+v)=(du/dx)+(dv/dx)

3)          (d/dx)(u-v)=(du/dx)-(dv/dx)

4)          (d/dx)(u.v)=(vdu/dx)+(udv/dx)

5)          (d/dx)(u/v)=[(vdu/dx)+(udv/dx)]/v.v

6)          (d/dx)(cu)=cdu/dx

                    where    c      is a constant .

7)          dy/dx=(dy/du)(du/dx)

8)          dy/dx=1/(dx/dy)

9)          dy/dx= (dy/du)(du/dv)(dv/dx)  

Angle Between Curves

Here is some Working Rule on Angle between Curves ;

1)  If Equations of two curves are given . Find the value of    dy/dx     fom the equation of two curves .

2)  If the value of     dy/dx    obtained from the two equations are equal then , angle between the curve is    0 degree    i.e. they touch each other at each point  i.e. the two curves are same .

     If product of the values of     dy/dx     for two curves is    -1,   then the two curves cut each other orthogonally (perpendicularly) .

3)  If given condition (2) is not true , find the co-ordinates of the points of intersection of the two curves by solving the equation of the two curves .

  Then find the values of     dy/dx    from the equation of two curves at one point of intersection .

 This will give gradient of the tangent to the two curves
i.e.    m1     and    m2    .

Then find the angle     A       between the curves by the formula

   tanA= +and- [m1 -m2]/1 + m1m2    .

Do this fo every point of intersection .

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Differentiation as rate measurer

dy/dx  as rate measurer  ;

dy/dx=(dy/dt)/(dx/dt)   , if any two of

dy/dx  ,  dy/dt  and  dx/dt   are

given the value of the third can be determined

    For working out problems there will be two variables    and   y   .   From given condition  a relation between    x    and    y    can be found and differentiating this expression we can find

dy/dx   ,

Either     dy/dt    or    dx/dt    is given

Thus if   dy/dt   is given we can find    dx/dt     and vice versa  .

If there is only one variable and its rate of change is given , then we can find the value of the variable in term of time    t    by integrating this expression .

We already know that ;

dy/dx=(dy/dt)/(dx/dt)=rate of change of (y)/rate of change of (x)

thus differential coefficient of   y    with respect to    x     is equal to the ratio of the rate of change of    y     and rate of change of      x      . 

Exact Equation

The differential equation     Mdx+Ndy=0    ,

where both    and   N   are functions of     and   y     is said to be exact when there is a function     u     of     x, y    such that 

Mdx+Ndy=du ,

i.e. ,  when   Mdx+Ndy   becomes a partial differential .

Now , we know from Differential Calculus that ;

Mdx+Ndy   should be a perfect differential if ;

DM/Dy=DN/Dx  ,   Hence the condition that

Mdx+Ndy=0  should be an exact differential equation is


The method of solving and exact equation of the type

Mdx+Ndy=0   .

First integrate the term in   Mdx   as if    were constant then integrate the terms in   Ndy   considering    x   as constant and rejecting the terms already obtained equate the sum of these integrals to a constant .

Solutions of Differential Equation

A solution of a differential equation with arbitrary constant equal in number to the order if differential equation  is called General Solution .

Other solutions obtained by  giving particular values to the arbitrary constant in the general solution are called Particular Solutions .

Also we know that the general solution of a function contains an arbitrary constant . Therefore the solution of differential equations , resulting as they do from the operations of integration , must contain arbitrary constants, equal in number to the number of time the integration is involved in obtaining the solution which is equal to the order of differential equation .

Thus we see that the most general solution of a differential equation of the   nth   order must contain       and only    n    independent arbitrary constants .

An ordinary differential equation  is a differential equation in which the unknown function is a function of a single independent variable. In the simplest form, the unknown f…

Functions in R

A rule of correspondence   f   is called a real value function if it associates each member    of the set    to an uniquely determined member    y    (called image of   x ) of   B  , a subset of    . is denoted by

    f: A--> B   or    y=f(x) ,

indicating that the real number     corresponds to 

    is element of    A

                                             under the rule      f      .

The set  A  is called domain of the function and

set   f(A)={ y/ y=f(x) ;   x   is element of  A}

i.e. the set of images of  A  is called the range of   f  .

Also   x   is called the independent variable and    y   is called a dependent variable .

Features of a function    y=f(x)   in    R   are as follows ;

1)   y    is uniquely determined for every value of   x   from the domain . The value of     corresponding to the value    a    of    x    is denoted by     f(a)  .

2)   For two or more values of    x ,   y    can have the same value .

3)   The image i.e. the value of  

Laws Of Motion

Laws of Motion discovered by Newton , They are as follows ;

Law 1  :-     Every body  perseveres in its state of rest or of uniform motion in a straight line unless it will compelled to change that state by impressed force .

Law 2  :-     The rate of change of momentum is proportional to the impressed force and take place in the direction of the straight line in which the force acts .

Law 3  :-     To every section there is always an equal and opposite reaction or the mutual action of any two bodies are always equal and oppositely directed .

The first law implies :

1)-----  That the internal force in a body do not play a part in the state of rest or of uniform motion .

2)-----  The definition of an external force as that which tends to change the inertia of a body on which it acts .

The second law enables us to measure force . If a mass    m     moves with a velocity    v     then its momentum is   mv     .

According to second law ;

F  (an external force)   varies as     (d/dt) (mv) 


Properties of Continuous Function

Theorem :- 

A continuous function which has opposite signs at two points vanishes at least once between these points , that is if   f(x)   be continuous in the closed interval    [a,b]   and    f(a)    and    f(b)   have opposite signs , then there is at least one value of   x   between    and   b   for which    f(x)=0  .  

Proof :-

For the sake of definiteness , let us suppose that ;

F(a)<0   and    f(b)>0   .

Science   f(x)  is continuous and   f(a)<0  , therefore   f(x)   will be negative in the neighborhood of   a   .
Again Since    f(x)   is  continuous and     f(b)>0   therefore   f(x)   will be positive in the neighborhood of   b   .
The set of values of   x   between    and   b   which make   f(x)    positive is bounded below by  a  and hence possesses an exact lower bound    k  .

Hence ;   a<k<b 

In this way we find that   f(x)   is positive in the interval
k<x<b   and is negative or zero in the interval
a   is less then and equal to   x<k  .


Intervals in R

At the time of analyze  the Real Number Set   R   it is essential to associates subsets of     with real numbers . This purpose is served by taking subsets of    in a spacial way as follows .

Open and Closed Intervals :-

An Open Interval of real number is  a subset of   defined as

 {x/  a<x<b}  i.e.  it is the set of all real numbers that are greater then   a   but less then    b    . This open interval is denoted by
   (a ,b)   or  ]a, b[    .

Thus ;  (a, b) = {x/  a<x<b , x   is element of  R }

and ;   y   is element of (a, b) => a<y<b .

A closed interval of real number is subset of    defined as

{ x/  a  is less then an equal to   x   is less then and equal to  b }

i.e.  it is the set of all real numbers that are equal to or greater then   but equal to or less then    b   . This closed interval is denoted by    [a , b] . 

[a, b]= {x/  a  is less then and equal to   x   is less then and equal to  b  ,

Differentiability Theorem

Theorem :-  A function   f    is differentiable  at     x=a     if and only if there exists a number   l    .
                   such that ;


                   Where     n    denotes a quantity which tends to    0     as   h-->0     .

Proof :-     Let    f     be differentiable at    x=a    . Then there exists a number   l    
                Such that ;

                lim x-->a  [f(x)-f(a)]/[x-a]=l

               putting ;     x=a+h ,

  lim h-->0  [f(a-h)-f(a)]/h=l

           or;    lim h-->0 [{f(a+h)-f(a)/h}-l]=0

              therefore   [{f(a+h)-f(a)}/h]-l     is equal to    n  

            where    n-->0    as     h-->0    

          Therefore f(a+h)-f(a)=lh+hn 

          where    n-->0    as    h-->0     .

          Thus it is the necessary condition .

As the argument is reversible , the condition is also sufficient .

Polar Co-ordinates

Beside the Cartesian , there are other systems also for  representing points and curve analytically . Polar system which is one of them .

---------In this system we started with a fixed line    OX   called the Initial Line and a fixed point on it , called the Pole .

---------If   P   be any given point, the distance   OP=r   is called the Radius Vector and    Angle XOP=theta    the Vectorial Angle . The two together are referred to as the Polar Co-ordinates of     .

Unrestricted Variation of Polar Co-ordinates :-

If we concerned with  assigning polar co-ordinates to only individual points in the plane , then it would clearly be enough to consider the radius vector to have positive values only and the vectorial angle    theta     to lie between    0 and 2pi   .

Transformation of Co-ordinates :-

Take the initial line   OX    of the polar system as the positive direction of   X-axis and the Pole   O    as origin for the Cartesian system .The positive direction of  Y-axis is to be such …