641.sagews

x=var('x')

s=var('s')

delta_x=var('delta_x')

p=3.14159

f(x,s) = 1/p*(s/(x^2+s^2))

f_prime(x,s)=f(x,s).diff(x)

f_2prime(x,s)=f_prime.diff(x)

f_3prime(x,s)=f_2prime.diff(x)

f_4prime(x,s)=f_3prime.diff(x)

f_5prime(x,s)=f_4prime.diff(x)

f_6prime(x,s)=f_5prime.diff(x)

f_7prime(x,s)=f_6prime.diff(x)

f_8prime(x,s)=f_7prime.diff(x)

p=plot([f(x,1),f_prime(x,1)],-10,10)

p.show(xmin=-5,xmax=5,ymin=-0.5,ymax=0.5)

delta_f2(x,delta_x,s)=delta_x*f_prime(x,s)+delta_x^2/2*f_2prime(x,s)+delta_x^3/factorial(3)*f_3prime(x,s)+delta_x^4/factorial(4)*f_4prime(x,s)

delta_delta_f2(x,delta_x,s)=delta_f2(x+delta_x,delta_x,s)-delta_f2(x,delta_x,s)

p=plot(f(x,1),-10,10,rgbcolor='red')

p += plot(f_prime(x,1),-10,10,rgbcolor='red')

p += plot(delta_f2(x,1,1)/1,-10,10,rgbcolor='green')

p += plot(delta_f2(x,.1,1)/.1,-10,10,rgbcolor='blue')

p += plot(delta_f2(x,.01,1)/.01,-10,10,rgbcolor='black')

p += plot(delta_delta_f2(x,.01,1)/(.01)^2,-10,10,rgbcolor='black')

p.show(xmin = -3, xmax = 3,ymin = -0.5, ymax = 0.5)

delta_f2(x,delta_x,s)=delta_x*f_prime(x,s)#+delta_x^2/2*f_2prime(x,s)+delta_x^3/factorial(3)*f_3prime(x,s)+delta_x^4/factorial(4)*f_4prime(x,s)+delta_x^5/factorial(5)*f_5prime(x,s)+delta_x^6/factorial(6)*f_6prime(x,s)+delta_x^7/factorial(7)*f_7prime(x,s)

delta_f2_2(x,delta_x,s)=delta_x*f_prime(x,s)+delta_x^2/2*f_2prime(x,s)#+delta_x^3/factorial(3)*f_3prime(x,s)+delta_x^4/factorial(4)*f_4prime(x,s)+delta_x^5/factorial(5)*f_5prime(x,s)+delta_x^6/factorial(6)*f_6prime(x,s)+delta_x^7/factorial(7)*f_7prime(x,s)

delta_f2_3(x,delta_x,s)=delta_x*f_prime(x,s)+delta_x^2/2*f_2prime(x,s)+delta_x^3/factorial(3)*f_3prime(x,s)#+delta_x^4/factorial(4)*f_4prime(x,s)+delta_x^5/factorial(5)*f_5prime(x,s)+delta_x^6/factorial(6)*f_6prime(x,s)+delta_x^7/factorial(7)*f_7prime(x,s)

delta_f2_4(x,delta_x,s)=delta_x*f_prime(x,s)+delta_x^2/2*f_2prime(x,s)+delta_x^3/factorial(3)*f_3prime(x,s)+delta_x^4/factorial(4)*f_4prime(x,s)#+delta_x^5/factorial(5)*f_5prime(x,s)+delta_x^6/factorial(6)*f_6prime(x,s)+delta_x^7/factorial(7)*f_7prime(x,s)

delta_f2_5(x,delta_x,s)=delta_x*f_prime(x,s)+delta_x^2/2*f_2prime(x,s)+delta_x^3/factorial(3)*f_3prime(x,s)+delta_x^4/factorial(4)*f_4prime(x,s)+delta_x^5/factorial(5)*f_5prime(x,s)#+delta_x^6/factorial(6)*f_6prime(x,s)+delta_x^7/factorial(7)*f_7prime(x,s)

p=plot(f(x+0.8,1)-f(x,1),-10,10)

p += plot(delta_f2(x,.8,1),-10,10,rgbcolor='red')

p += plot(delta_f2_2(x,.8,1),-10,10,rgbcolor='green')

p += plot(delta_f2_3(x,.8,1),-10,10,rgbcolor='black')

p += plot(delta_f2_4(x,.8,1),-10,10,rgbcolor='yellow')

p += plot(delta_f2_5(x,.8,1),-10,10,rgbcolor='purple')

p.show(xmin=-3,xmax=3,ymin=-0.2,ymax=0.2)