Behaviour

of

Function f(x)=(1+ecosx) /√(1+e2+2ecosx) where e is[0,1]

1⋜e⋝0  
    x in °  
         
    x/2 in °  
    cos(x/2)  
    y1=f(x1)=1+e*cosx  
    f(x1)max =1+e  at x=0° and x=360°  
    f(x1)min=  1-e  at x=180°  
    f(x1) =1  at x=90° which is independent of e    
    Gradient of f(x1)=f'(x1)= e*sinx  
    e*cosx  
    Max. height of the curve =2e [it spreads symmetrically on both sides of f(x1)=1]  
    Area enclosed between the curve and f(x1)=1+e ; I=2* 0∫180° x*f(x1)dx=[x2/2 +e(xsinx+cosx)]from 0 to 180

= π2 ; Normalized Area=(1/π2)*I =1

  
         
         
    y2=f(x2)=√(1+e2+2e*cosx)  
    1/y2=1/f(x2)=1/√(1+e2+2e*cosx)  
    1  -  1/y2   =  
    esinx/ (1+e2+2e*cosx)3/2  
    f(x2)max =1+e  at x=0° and x=360°  
    f(x2)min= 1-e  at x=180°  
    f(x2)  at x=90 is √(1+e2)  
    Gradient of f(x2)=f'(x2)=esinx/f(x2)=f'(x1)/f(x2)  
    Max. height of the curve =2e [it does not spread symmetrically either on both sides of f(x2)=1 or f(x2)=√(1+e2) ]  
    Area enclosed between the curve and f(x1)=1+e ; I=2* 0∫180° x*f(x2)dx=??from 0 to 180    
         
   

   
    y1/y2=f(x) =f(x1) / f(x2)=(1+e*cosx) /√(1+e2+2e*cosx)=sinα

f(x) is extremum when f' (x)=0 or e(e+cosx)

  
    f(x=90) =1/√(1+e2)  
    x=cos-1(-e)  
    f(x) minimum at cosx=-e is √( 1-e2)=(sinα)min  
    intersection of f(x) with f(x)=1 at e(e+2cosx)=0  => x°=cos-1(-e/2)  
    180° - cos-1(-e/2)  
    (α)min  
         
    g(x)=y1-y2=(1+e*cosx) - √(1+e2+2e*cosx)  
    g(x) =1-√(1+e2) at x=90 degree  
    g(x)=(1 - e/2) - √(1+e2-e) at x=120 degree  
    |g(x)|max is at cosx=-e/2 and its value is (1 - e2/2) -1 =-e2/2

The absolute value of g(x) starts with zero at 0 degree , reaches maximum of e2/2 at x= cos-1(-e/2) and then again falls off to 0 at 180 degree. Range of g(x)=[0,-1/2 ] for e=[0,1] & x=[90°,120°]

  
    g(x) =0 when e=0 irrespective of x value

g(x)=0 when x=0 irrespective of value of e

    
    g(x) =2cos(x/2)(cos x/2 - 1) when e=1  
         
    gradient of g(x)= g'(x)=esinx[1- 1/√(1+e2+2e*cosx)]=esinx[1 - 1/f(x2)] =f'(x1)[1 - 1/f(x2)]  
    g'(x) =0 when x=0  OR e(e+2cosx) =0 or, cosx=-e/2 or x°=cos-1(-e/2)    
    g"(x) =ecosx[1- 1/√(1+e2+2e*cosx)] +esinx * esinx/ (1+e2+2e*cosx)3/2    
    g"(x)=ecosx[1 - 1/f(x2)] + esinx*f'(x1)/f(x2)3  
    g"(x)min =e2/(1+e) which is > 0  when x=0 whch means g(x) has minimum value at x=0  
    g"(x)min =e2(1-e2/4)which is >  0  
    gradient of g(x) between x=0 & x=cos-1(-e/2) assuming g(x) to be a st. line: tanφ1 =-e2 / 2*x0r  
    gradient of g(x) between x=cos-1(-e/2) &x=180 assuming g(x) to be a st. line: tanφ2 =-e2 / [2*(π-x0r)]  
   

   
    OE=  
    ED=  
    CE=  
    OC=  
    CD=