Dear readers of this group, I wrote a simple sagelet which describes
approximation of a function in two variabels using differential. The
sagelet is published at http://www.sagenb.org/home/pub/298/
I hope, somebody finds it useful in her/his courses.
Any suggestions are welcomed.
Robert Marik
---------------
code:
x,y=var('x y')
html('<h2>Explaining approximation of a function in two \
variables by differential</h2>')
html('Points x0 and y0 are values where the exact value of the
function \
is known. Deltax and Deltay are displacements of the new point. Exact
value \
and approximation by differential at shifted point are compared.')
@interact
def _(func=input_box('sqrt(x^3+y^3)',label="f(x,y)=",type=str), x0=1,
y0=2, \
deltax=slider(-1,1,0.01,0.2),\
deltay=slider(-1,1,0.01,-0.4), xmin=0, xmax=2, ymin=0, ymax=3):
f=sage_eval('lambda x,y: ' + func)
derx(x,y)=diff(f(x,y),x)
dery(x,y)=diff(f(x,y),y)
tangent(x,y)=f(x0,y0)+derx(x0,y0)*(x-x0)+dery(x0,y0)*(y-y0)
A=plot3d(f(x,y),(x,xmin,xmax),(y,ymin,ymax),opacity=0.5)
B=plot3d(tangent(x,y),(x,xmin,xmax),
(y,ymin,ymax),color='red',opacity=0.5)
C=point3d((x0,y0,f(x0,y0)),rgbcolor='blue',size=9)
CC=point3d((x0+deltax,y0+deltay,f
(x0+deltax,y0+deltay)),rgbcolor='blue',size=9)
D=point3d((x0+deltax,y0+deltay,tangent
(x0+deltax,y0+deltay)),rgbcolor='red',size=9)
exact_value_ori=f(x0,y0).n(digits=10)
exact_value=f(x0+deltax,y0+deltay)
approx_value=tangent(x0+deltax,y0+deltay).n(digits=10)
abs_error=(abs(exact_value-approx_value))
html(r'Function $ f(x,y)=%s \approx %s $ '%(latex(f(x,y)),latex
(tangent(x,y))))
html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori)))
html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay))))
html(r'Value of the function in shifted point is $%s$'%f
(x0+deltax,y0+deltay))
html(r'Value on the tangent plane in shifted point is $%s$'%latex
(approx_value))
html(r'Error is $%s$'%latex(abs_error))
show(A+B+C+CC+D)
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