Applications of the derivative
      Tangent, normal, subtangent and subnormal
         Property of the parabola
         Property of power functions
         Property of the exponential function
Tangent, normal, subtangent and subnormal
A segment of a tangent to a curve lying between the tangency point (the point at which a tangent is drawn to a curve) and the intercept of the tangent with the x-axis is called the length of the tangent.
A segment of the x-axis lying between the x-coordinate of the tangency point and the intercept of the tangent with the axis is called the sub tangent.
Therefore, the sub tangent is the projection of the segment of the tangent onto the x-axis.
A segment of a line normal to a tangent lying between the tangency point and the intercept of the normal with the x-axis is called the length of the normal.
A segment of the x-axis lying between the x-coordinate of the tangency point and the intercept of the normal with the axis is called the subnormal.
Therefore, the subnormal is the projection of the segment of the normal onto the x-axis.
In the figure below denoted are, 
the length of the tangent  tl = PT,                      the subtangent  st = TX
the length of the normal  nl = PN        and         the subnormal  sn = NX.
In the right triangle PTX,    
In the right triangle PNX,  
Therefore, the subtangent, and the subnormal
   
   
The length of the tangent, and the length of the normal
   
   
 
At every point of the parabola y2 = 2px the subnormal have the same value p.
By differentiating both sides of the equation of the parabola with respect to x we get
2yy' = 2p   or    yy' = p = sn.
We use this property to construct the normal and the tangent at a point of the parabola. 
Thus, the normal line passes through the given point P(x, y) of the parabola and the point x + p, lying on the x-axis.
Further, since  
therefore, the vertex of the parabola  y2 = 2p  bisects the sub tangent.
Property of power functions
The sub tangent of the power function  y = axn,  where n is a positive integer, 
Thus for example, the tangent to the parabola  y = ax2  at P (x, y) bisects the abscissa of P that is, st = x/2, as shows the figure below.
 
Property of the exponential function
The sub tangent of the exponential function  y = erxr R and r ¹ 0, since
Thus for example, at every point of the exponential function  y = ex the sub tangent  st = 1, as shows the figure above.
Calculus contents D
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