(^640) Series
takes extreme values. 4ns.: The critical points where the first-order partial
derivatives both vanish are (0,0) and (a,a). At (0,0), G has neither a maximum
nor a minimum. At (a,a), G has a minimum if a > 0 and a maximum if a < 0
Moreover, G(a,a) = -a3.
10 Assuming that
G(x,y) = ao + bix + b,y + c1x2 + C2xy+ C3y2
- dix3+ d2x2y+ d3xy' + d4x3 + e1x4 +
over some neighborhood of the origin, and that the series can be differentiated
term rise with respect to x and y as often as desired, determine enough coefficients
to verify three or four terms in the expansion
G(x,y) = G(0,0) + [Gx(0,0)x + Gv(0,0)y]
+2![Gxx(0,0)x2 + 2Gxv(0,0)xy + Gvv(0,0)Y21
1 - -! [Gxx:(0,0)x3 + 3G.xv(0,0)x'y + 3Gxvv(0,0)xy2 + Gvvv(0,0)Y3] + ...
11 Suggest two or more ways to obtain the power series in x and y which
converges to ex+v and use each method to obtain a few of the terms.
12.6 Euler-Maclaurin summation formulasf As we shall see, we
need only one very simple idea to obtain some remarkably useful and
important formulas involving the Bernoulli functions and numbers.
While the index reveals locations of more information about Bernoulli
functions and numbers, we start with the facts that
(12.611) Bo(x) = 1
(12.612) Bn(x) = (n =1,2,3, )
(12.613) foI dx = 0 (n = 1,2,3,
(12.614) Bn(x + 1) = (n =0,1,2,
over the interval - < x < co, except that (12.612) fails to hold when
n is 1 or 2 and x is an integer. The function Bj(x) is the saw-tooth func-
tion having the graph shown in Figure 12.62. The Bernoulli numbers
Figure 12.62
f This section can be omitted. It is not claimed that the section is easy. It is not
claimed that the material can be thoroughly digested ina day or a week or a year. It is,
however, claimed that students of calculus shouldsee a substantial and useful application
of calculus. Even though we build only more modest structures in examinations in this
course, we should see at least one cathedral.