Tuesday, 25 March 2014

Regions of the Curve




Now let we will discuss about Regions of the Curve. 

-----  Solve the equation for  y.

If  y   is imaginary when   x   lies between   a  and   b,

then the curve does not lie in the region bounded by 

x=a   and x=b

-----  Find asymptotes parallel to axis and the curve will not go beyond its asymptotes.

-----  Sometimes it is possible to detect values of 

x   and   y    for which two sides of the equation assume opposite signs.

The curve does not exist for such values.

Increase or Decrease of the Curve:-

-----  Solve the equation for  y  or  x   whichever is found convenient.

now see the behaviour of  y   or  x   for different values of  

x  or  y  giving particular attention to those values for which 

y  or  x  tends to infinity or zero.

If there is symmetry about axis of   x   or   y   i.e. in opposite quadrant, only positive values need be considered. The other branches are drawn by symmetry.

-----  Find   (dy/dx)  and points where tangents are parallel to axis. 

There are maxima and minima (which already discussed in my previous post in this blog) of the curve and here are the ordinates cease to increase or decrease.    



Thursday, 20 March 2014

Cardioide

 


Let we will discuss about Cardioide

r=a(1+cos theta)

-----  If we put  -theta  for  theta  in the equation of the curve we find that;

r=a{1+cos (-theta)}=a(1+cos theta)

i.e. the equation of the curve does not change. Therefore the given curve is symmetrical about the initial line.

-----  Now    r=0  when;

1+cos theta = 0 

i.e.  cos theta = -1 

therefore    theta = pi 

Hence the curve passes through the origin and the equation of the tangent at the pole is

theta = pi  i.e. the initial line.

-----  Now we plot some of the points on the curve.

When  theta=0,  then  r=2a;

when  theta=pi/3,  then  r=3a/2

when  theta=pi/2,  then  r=a

when  theta=pi,  then  r=0.

-----  From the given equation

r=a(1+cos theta),  when we have

(dr/d theta)=-a sin theta 

That is, when the value of  theta  increases from

0  to  pi,  then the value of   r  decreases and as been stated earlier decreases from

2a   to  0

Again, since the given curve is symmetrical about the initial line, therefore when the value of    theta   increases from    pi  to    2pi,

Then the value of   r    increases from   0  to  2a.



Tuesday, 18 March 2014

Cycloid and Catenary


This is a Cycloid Pendulum


If there are equation

x=a(theta - sine theta)  

and,  y=a(1-cos theta)

Also   OX   be a fixed straight line called  x-axis 

And a circle of radius  a  roll, without sliding, along this line,
Then the cycloid is the curve traced out by a point   on the circumstances of the circle. Also if  C  be the center of circle. The point moves  to  P  such that

OM=arcPM

Take  angle PCM = theta   and the point  P(x,y), then ;

 x=ON=OM-MN=OM-PK

= a theta - a sin theta
(because  OM=arcPM=a theta)

=a(theta-sin theta)   and;

y=PN=KM=CM-CK

=a-a.cos theta

=a(1-cos theta)

It is clear that in one complete revolution of the circle but point   P  describes the curve  ODA  when  y=0 i.e.   theta=0 or 2pi. If the motion is continued, we get an infinity number of such curves.

This fixed line is called the base and the highest point from the fixed line is called the vertex or cusp.

When the curve is inverted the equation become;

x=a(theta+sin theta)

y=a(1-cos theta)

Centenary is such a curve in in which a uniform chain hangs freely under gravity.  If the curve be measured from   A  to any point and  arc AP=s . Tangent at  P  makes an angle  sai   with  x-axis  . Then;

s=a.tan sai,  where   a=constant .



Saturday, 15 March 2014

Spiral




Spirals are of three kinds :-

1)  Equiangular Spiral or Logarithmic Spiral :-

Its equation is  r=ae to the power (theta cot alfa)

As   theta  ranges from  -infinity to  infinity.

r  ranges from   theta  to  infinity.

The curve has a characteristic property that the   tangent  makes a constant angle with the radius vector

i.e.  phay=alfa (constant)

origin is not on the curve.

2)  Spiral or Archimedes :-

The equation is  r=a theta

This is a curve described by a point which moves along a straight line with constant velocity while the line rotates with constant angular velocity about a fixed point in it.

Thus;  r=ut

and  theta= omega.t

Hence;  a=u/omega  

3)  Reciprocal Spiral.

r=a/theta.

Clearly,  y=r.sine theta

              =(r.sine theta.theta)/theta

            =(a.sine theta)/theta

therefore;  y=Lt theta-->0  (a.sine theta)/theta=a

therefore;   y=a   is any asymptote .



Wednesday, 12 March 2014

Tracing of Curves in Polar Co-ordinates

 


1)  If    theta   be replaced by   -theta  and the equation remains unaltered, the curve is symmetrical about the initial line .

2)  If only even powers of   r    occur in the equation, the curve is symmetrical about the pole or origin .

3)  The curve is symmetrical about the line

theta=pi/2

If the equation remains unaltered when

theta   is changed into    pi-theta    or when

theta   is changed into    -theta 

and   r   into  -r.

d)  The curve is symmetrical about the line

theta=pi/4

if the equation of the curves remains unaltered when

theta   is changed into   (pi/2)-theta 

------  If the curve passes through the pole, The value of

theta   for which   r   is    zero   gives the tangent at the pole.

------  In most popular equations only periodic functions occur and so

value of    theta    from  0  to    2pi 

need alone be consider.



Sunday, 9 March 2014

Folium of Descartes

 



Folium of Descartes :-

cube of  x  + cube of   y = 3axy

It is symmetrical about the line

y=x

and meets in at the point

(3a/2, 3a/2)

It passes through the origin and

x=0    y=0  

are the tangents there so that the origin is a node on the curve.

It meets the co-ordinate axes at the origin only.

x+y+a=0    is its only asymptote.

x,  y   can not both be negative .

So, that no part of the curve lies in the third quadrant. 



Polar Equations Of Curves

 


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

r=a(1+cos theta)

is symmetrical about the initial line , for ;

r=a(1+cos theta)=a[1+cos(-theta)]

It may be noted that ;

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

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



Wednesday, 5 March 2014

Limit of f(x) at x=a

 


A function   f(x)  is said to tend to the limit   l   at   x=a  

if given   element>0  there exists a number  delta>0


such that ;  /f(x)-l/< that element 


whenever ; /x-a/<delta 


In simple words ,   f(x)  is said to tends to the


limit l  at  x=a 


if   f(x)  tends to  as  x  approaches  a  


through values of  x    greater then  


as well as   values of  smaller than  a.


The limit l  of  f(x)  at  x=a  denoted by


lim x-->a  f(x)=l


It is clear from above that


f(x)  tends to the limit  at  x=a  implies .


lim x-->a+0  f(x) = lim x-->a-0  f(x) 

=lim x-->a  f(x) = l

provided both the right-hand and left-hand limits exists for

x=a  .

If the domain of definition of   f(x)  is

[a, b] , then existence of

lim x-->a+0  f(x)   does not arise .

Similarly the question of existence

 lim x-->b+0  f(x)   also does not arise .



Limits from the Left

 


A function   f(x)   is said to tend to the limit   l'   as      tends to    alfa    from the left

if given element is > 0

there exists a number   delta>0 

such that;    /f(x)-l'/<of an element

whenever ;   a-delta<x<a  .

This number   l'  is called the left  hand limit of 

 f(x)  at   x=a   and it is denoted by ;

lim x-->a  f(x)=l'  or  lim x-->a-0 f(x)=l'

or ;  lim h-->0  f(a-h)=l'

This limit is also written as   f(a-0)

In simple words ,  f(x)  is said to tend to the limit   l'  from the left if    f(x)  tends to   l'  as   approaches   a   through values of   x   smaller then   a .

Working Rule for finding the limit from the left at

x=a 

A)  Put   a-h  for   x  in  f(x)   to get   f(a-h)

B)  Make   h-->0  in  f(a-h) .



Monday, 3 March 2014

Limit of a Function

 


A function   f(x)  is said to tend to the limit   l   as   x  tends to   a   from the right

if given element is >0  there exists a number

delta>0  , such that ;

/f(x)-l/< of that element 

whenever , a<x<a+delta .

This number   l  is called the right hand limit of  f(x)  at

x=a and it is denoted by

lim x-->a  f(x)=l  or   lim x-->a=0  f(x)=l

or ,  lim h-->0  f(a+h)=l

This limit is also written as   f(a+0)

In simple words ,   f(x)  is said tend to the limit   l  from the right if  f(x)  tends to   l   as   approaches   through value of   x   greater then  a .

working rule for finding the limit from the right at  x=a.

a)  --  Put   a+h  for  in  f(x)  to get   f(a+h)

b)  --    h-->0  in   f(a+h).



Use of Comparison Test

 


The test is applicable for series of positive numbers only .

For given series    summation of un   of positive terms,

a series   summation of vn   of positive terms

is to be selected whose convergence is known and

lim n--> infinty un/vn   is non zero finite number .

The efficiency is choosing
summation of vn   requires practice .

However if ,

un=P(x)/Q(x),  then choose 

vn=term containing highest power of x in P(x)/term containing highest power of x in Q(x) 

which will be finally of the form

1/n to power p .

Thus summation of   1/n to power p   will act as primary or auxiliary series for testing the convergence of a series by comparison test .



Direct Consequences of Convergence of Series

 

Followings are the Direct Consequences of  the Definition of Convergence of Series :-

1)  -----  The alteration , addition or omission of a finite number of terms of a series has no effect on its being Convergent , Divergent or Oscillatory .

2)  -----  Multiplication of all the terms of a series by a fixed non zero number has no effect on its vbeing convergent , divergent or oscillatory

i.e.   summation of ku is convergent, divergent or oscillatory 

according as , summation of  u is convergent, divergent or oscillatory .


It is important here that the sum of infinity of a convergent series is essentially a limit and is not a sum in the sense described in the defination of addition . We are therefore not justified in assuming that the sum of an infinite series is unaltered by changing the order of the terms or by the introduction or removal of brackets .


In fact, changes of this kind may alter the sum or they may transform a convergent series into one which diverges or oscillates .

For example , the series 

(1-1)+(1-1)+(1-1)+...................... is convergent and its sum is zero , but the series 

1-1+1-1+.............. which is obtained by removing the brackets of the original series , oscillates .



Sequence of Partial Sums of a Series

 


Let;   u1+u2+ ..............+un+.......

be an infinite series . 

If  Sn denotes the sums of first   n   terms of the series ,

So that ;

Sn=  u1+u2+ ..............+un+,   then ; the sequence   {Sn}   is called the sequence of partial sums of the given series .

Convergent Series :-   If    {Sn}  tends to a finite and infinite limit  S,  then  S  is defined to be the sums to infinity of the series and the series is said to be convergent to the sum  S. 

Thus    is defined as

lim n-->infinity  Sn=S

or,  Sn-->S,  as  n-->infinity 

Divergent Series :-  If  {Sn}  tends to +infinity  or  -infinity

as  n tends to  infinity , then the series is said to be divergent .

in other words , the series is divergent if having given any positive number delta whatsoever , we can find a finite number   m  such that   Sn>delta  ,
when   n is less and equal to m .

Oscillatory Series :-  If  {Sn} tends to no definite limit whether  finite or infinite as   n tends to infinity   , then the series is aid to be Oscillate. We say that the series oscillates finitely or infinitely according as  Sn  oscillates between finite limits or between positive and negative infinity .


Our Latest Post

How to Evaluate an Integral

In this video you will see how to evaluate an Integral. This video shows an example, by this example you will see about how to evaluate an ...

Popular Post