James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

782 Chapter 11 Infinite Sequences and Series
velocity of a wave tends to be independent of the length of
the wave.)
(c) Use the Alternating Series Estimation Theorem to show
that if L . 10d, then the estimate v 2 < td is accurate to
within 0.014tL.
d
36. A uniformly charged disk has radius R and surface charge density
as in the figure. The electric potential V at a point P at a
distance d along the perpendicular central axis of the disk is
V − 2k e(sd 2 1 R 2 2 d)
where k e is a constant (called Coulomb’s constant). Show that
L
38. The period of a pendulum with length L that makes a maximum
angle 0 with the vertical is
T − 4Î L t
y y2
0
dx
s1 2 k 2 sin 2 x
where k − sins 1 2 0d and t is the acceleration due to gravity.
(In Exercise 7.7.42 we approximated this integral using
Simpson’s Rule.)
(a) Expand the integrand as a binomial series and use the
result of Exercise 7.1.50 to show that
T − 2Î L t
F1 1 12
2 2 k 2 1 12 3 2
2 2 4 2 k 4 1 12 3 2 5 2
2 2 4 2 6 2 k 6 1 ∙ ∙ ∙G
If 0 is not too large, the approximation T < 2sLyt ,
obtained by using only the first term in the series, is often
used. A better approximation is obtained by using two
terms:
V < keR2
d
for large d
T < 2Î L t s1 1 1 4 k 2 d
R
d
P
(b) Notice that all the terms in the series after the first one have
coefficients that are at most 1 4 . Use this fact to compare this
series with a geometric series and show that
2Î L t s1 1 1 4 k 2 d < T < 2Î L 4 2 3k 2
t 4 2 4k 2
37. If a surveyor measures differences in elevation when making
plans for a highway across a desert, corrections must be made
for the curvature of the earth.
(a) If R is the radius of the earth and L is the length of the
highway, show that the correction is
C − R secsLyRd 2 R
(b) Use a Taylor polynomial to show that
C < L 2
2R 1 5L 4
24R 3
(c) Compare the corrections given by the formulas in parts (a)
and (b) for a highway that is 100 km long. (Take the radius
of the earth to be 6370 km.)
(c) Use the inequalities in part (b) to estimate the period of
a pendulum with L − 1 meter and 0 − 10°. How does
it compare with the estimate T < 2sLyt ? What if
0 − 42°?
39. In Section 4.8 we considered Newton’s method for approximating
a root r of the equation f sxd − 0, and from an initial
approximation x 1 we obtained successive approximations
x 2, x 3, . . . , where
x n11 − x n 2 f sxnd
f 9sx nd
Use Taylor’s Inequality with n − 1, a − x n, and x − r to show
that if f 0sxd exists on an interval I containing r, x n, and x n11,
and | f 0sxd | < M, | f 9sxd | > K for all x [ I, then
L
C
| xn11 2 r | < M 2K | xn 2 r | 2
R
R
[This means that if x n is accurate to d decimal places, then x n11
is accurate to about 2d decimal places. More precisely, if the
error at stage n is at most 10 2m , then the error at stage n 1 1 is
at most sMy2Kd10 22m .]
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