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 June 12th, 2012, 10:30 AM #1 Senior Member   Joined: Apr 2012 Posts: 106 Thanks: 0 Taylor polynomial with two variables Dear all, I need to approximate the following term: $(1,02)^3(0,97)^2$ with a Taylor polynomial of the first degree. How do we do that?
 June 12th, 2012, 11:58 AM #2 Global Moderator   Joined: May 2007 Posts: 5,819 Thanks: 405 Re: Taylor polynomial with two variables Expand each factor using binomial {(1+.02)^3 and (1-.03)^2} multiply out and keep the first non-zero term after 1.
 June 12th, 2012, 08:15 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,131 Thanks: 437 Math Focus: Calculus/ODEs Re: Taylor polynomial with two variables Let $z=f$$x,y$$=x^3y^2$ The differentials of the independent variables are: $dx=\Delta x,\,dy=\Delta y$ The total differential of the function is: $dz=f_x$$x,y$$\,dx+f_y$$x,y$$\,dy$ A Taylor polynomial of the first degree is linear, so we may use: $\Delta z\approx dz$ $f$$x+\Delta x,y+\Delta y$$-f$$x,y$$\approx f_x$$x,y$$\,dx+f_y$$x,y$$\,dy$ $f$$x+\Delta x,y+\Delta y$$-f$$x,y$$\approx $$3x^2y^2$$dx+$$2x^3y$$dy$ Let: $x=1,\,y=1,\,\Delta x=0.02,\,\Delta y=-0.03$ $f$$1.02,0.97$$-f$$1,1$$\approx $$3(1)^2(1)^2$$(0.02)+$$2(1)^3(1)$$(-0.03)$ $f$$1.02,0.97$$\approx 1+0.06-0.06$ $(1.02)^3(0.97)^2\approx1$
 June 13th, 2012, 01:05 AM #4 Senior Member   Joined: Apr 2012 Posts: 106 Thanks: 0 Re: Taylor polynomial with two variables Thanks a lot!

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