1000 Solved Problems in Modern Physics

(Tina Meador) #1

1.3 Solutions 49


h=−
f(v)
f′(v)

;f(v)=f(− 2 .1)

f′(v)=

dy
dx

|v;f′(v)= 10. 23

h=−

0. 039

10. 23

=− 0. 0038

To a first approximation the root is− 2. 1 − 0 .0038123 or− 2 .1038123. As a
second approximation, assume the root to be
a=− 2. 1038123 +h,
Putv 1 =− 2. 1038123
h 1 =−f(v 1 )/f′(v 1 )=− 0. 000814 / 6. 967 =− 0. 0001168
The second approximation, therefore, givesa=− 2 .1039291.
The third and higher approximations can be made in this fashion. The first
approximation will be usually good enough in practice.

1.34 y(x)=x


(^2) exp(−x (^2) )(1)
Turning points are determined from the location of maxima and minima. Dif-
ferentiating (1) and setting dy/dx= 0
dy/dx= 2 x(1−x^2 )exp(−x^2 )= 0
x= 0 ,+ 1 ,−1. These are the turning points.
We can now find whether the turning points are maxima or minima.
dy
dx
=2(x−x^3 )e−x
2
y
′′
=2(2x^4 − 5 x^2 +1)e−x
2
Forx= 0 ,d
(^2) y
dx^2 =+^2 →minimum
Forx=+ 1 ,,d
(^2) y
dx^2 =−^4 e
− (^1) →maximum
Forx=− 1 ,,d
(^2) y
dx^2 =−^4 e
− (^1) →maximum
y(x)=x^2 e−x
2
is an even function becausey(−x)=+y(x)


1.3.6 Series............................................


1.35 The given series isx−


x^2
22

+

x^3
32


x^4
42

+··· (A)

The series formed by the coefficients is

1 −

1

22

+

1

32


1

42

+··· (B)

lim
n=∞

(

an+ 1
an

)

=lim
n=∞

[


n^2
(n+1)^2

]

=



Apply L’Hospital rule.
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