12.5 Taylor formulas with remainders 6376 Little things like the formula(1) f(1) = f(0) (1 - 0) +LEO)(1 - 0)2
f(n)(0) f(n+I)(1*
- ... + -Fl
n( (1 -0)n+(n+1)1(1 -0)n
and the capacitors that appear in electrical networks have surprising applications.
As we shall see, simple applications of (1) give Taylor formulas for functions of
"several variables," "several" meaning more than one. Extensions to functions
of more variables being easily made, we suppose that G is a function of two vari-
ables x and y. We suppose that (xo,yo) and (x,y) are interior points of some
convex region, a circular disk, for example, over which C is continuous and has
all of the continuous partial derivatives we want to use. Supposing finally that
0<t<_1,let
(2) f(t) = G(xo + t(x - x0), yo + t(y - yo)).Then f(1) = G(x,y) and f (0) = G(xo,yo) and we can start production of the Taylor
formulas. Differentiating (2) with the aid of the chain rule of Theorem 11.23
gives(3) f(t) = G=(xo + t(x- xo), yo+ t(y - Yo))(x - xo)
+G,(xo + t(x - xo), yo + t(Y - yo))(Y - Yo).We can use (1) with n = 0 and obtain the primitive but neverthelessuseful
Taylor formula(4) G(x,y) = G(xo,yo) +Gz(x*,Y*)(x - xo) + G,,(x*,Y*) (Y - Yo),where(5) x* = x0 + t*(x - xo), Y* = yo + t*(Y - YO).To prepare for more elaborate Taylor formulas, we put t =0 in (3) to obtain(6) f'(0) = G.(xo,Yo)(x - xo) +Gv(xo,Yo)(Y - yo)
and differentiate (3) with the aid of the chain rule to obtain(7) f"(t) = G::(xo + t(x - x0), yo +t(y- yo))(x -x0)2
+ G=v(xo + t(x - xo), yo + t(y - yo))(x - xo)(Y -yo)
+Gxv(xo + t(x - xo), Yo + t(Y - Yo))(Y - Yo)(x -xo)
+Gvv(xo + t(x - xo), yo + t(y - yo))(y -yo)2.We now have the material required to use (1) with n = I. It is easy to continue
the procedure, but the expressions we write will become moreand more pon-
derous unless we introduce simplifying notation. Webegin by abbreviating
(3) to the form(8) f' (t) = C (x - xo)ax+ (Y _ Yo)yJ Gand observing that, on account of the equality of themixed partial derivatives,