Tuesday, 21 January 2014

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

tan45degree=1.

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



No comments:

Post a Comment

Our Latest Post

Introduction of Circle

All of my lessons and teaching videos are in English and most of them are for students of Logistics Management. But many of mys students an...

Popular Post