Page 98 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 89

4.82. A particle travels with constant velocities v 1 and v 2 in mediums I and II, P
Medium I
respectively (see adjoining Fig. 4-11). Show that in order to go from point 1
P to point Q in the least time, it must follow path PAQ where A is such velocity 1
that
A
Medium II
ðsin 1 Þ=ðsin 2 Þ¼ v 1 =v 2 2
velocity 2
Note: This is Snell’s Law; a fundamental law of optics first discovered Q
experimentally and then derived mathematically.
Fig. 4-11
4.83. A variable is called an infinitesimal if it has zero as a limit. Given two
infinitesimals and ,wesay that is an infinitesimal of higher order (or the same order)if lim = ¼ 0(or
2
lim = ¼ l 6¼ 0). Prove that as x ! 0, (a) sin 2x and ð1 cos 3xÞ are infinitesimals of the same order,
3
3
(b) ðx sin xÞ is an infinitesimal of higher order than fx lnð1 þ xÞ 1 þ cos xg.
2
x sin 1=x
4.84. Why can we not use L’Hospital’s rule to prove that lim ¼ 0 (see Problem 3.91, Chap. 3)?
x!0 sin x
2
3 n
4.85. Can we use L’Hospital’s rule to evaluate the limit of the sequence u n ¼ n e , n ¼ 1; 2; 3; ... ? Explain.
4.86 (1) Determine decimal approximations with at least three places of accuracy for each of the following
p ffiffiffi p ffiffiffi 1=3
irrational numbers. (a) 2; ðbÞ 5; ðcÞ 7
3
2
(2) The cubic equation x 3x þ x 4 ¼ 0 has a root between 3 and 4. Use Newton’s Method to
determine it to at least three places of accuracy.
3
2
4.87. Using successive applications of Newton’s method obtain the positive root of (a) x 2x 2x 7 ¼ 0,
(b)5 sin x ¼ 4x to 3 decimal places.
Ans. (a)3.268, (b)1.131
k
k
k
4.88. If D denotes the operator d=dx so that Dy dy=dx while D y d y=dx ,prove Leibnitz’s formula
n
n
2
n
r
D ðuvÞ¼ ðD uÞv þ n C 1 ðD n 1 uÞðDvÞþ n C 2 ðD n 2 uÞðD vÞþ þ n C r ðD n r uÞðD vÞþ þ uD v
n
where n C r ¼ð Þ are the binomial coefficients (see Problem 1.95, Chapter 1).
r
d n
2
2
4.89. Prove that ðx sin xÞ¼ fx nðn 1Þg sinðx þ n =2Þ 2nx cosðx þ n =2Þ.
dx n
4.90. If f ðx 0 Þ¼ f ðx 0 Þ¼ ¼ f ð2nÞ ðx 0 Þ¼ 0 but f ð2nþ1Þ ðx 0 Þ 6¼ 0, discuss the behavior of f ðxÞ in the neighborhood
0
00
of x ¼ x 0 . The point x 0 in such case is often called a point of inflection. This is a generalization of the
previously discussed case corresponding to n ¼ 1.
4.91. Let f ðxÞ be twice differentiable in ða; bÞ and suppose that f ðaÞ¼ f ðbÞ¼ 0. Prove that there exists at least
0
0
4
one point in ða; bÞ such that j f ð Þj A f f ðbÞ f ðaÞg. Give a physical interpretation involving
00
2
ðb aÞ
velocity and acceration of a particle.