The Chemistry Maths Book, Second Edition

(Grace) #1

638 Solutions to exercises



  1. 11 − 13 x 1 + 13 x


2

1 − 1 x


3


  1. 11 + 112 x 1 + 154 x


2

1 + 1108 x


3

1 + 181 x


4


  1. 11 − 120 x 1 + 1160 x


2

1 − 1640 x


3

1 + 11280 x


4

1 − 11024 x


5


  1. 811 − 1216 x 1 + 1216 x


2

1 − 196 x


3

1 + 116 x


4


  1. 7291 + 11458 x 1 + 11215 x


2

1 + 1540 x


3

1 + 1135 x


4

1



  • 118 x


5

1 + 1 x


6


  1. (i)


(ii)(a


4

1 + 1 b


4

1 + 1 c


4

) 1 + 1 4(a


3

b 1 + 1 a


3

c 1 + 1 b


3

a 1 + 1 b


3

c 1



  • 1 c


3

a 1 + 1 c


3

b) 1 + 1 6(a


2

b


2

1 + 1 a


2

c


2

1 + 1 b


2

c


2

) 1



  • 1 12(a


2

bc 1 + 1 b


2

ca 1 + 1 c


2

ab)



  1. (i)


(ii)(a


3

1 + 1 b


3

1 + 1 c


3

1 + 1 d


3

) 1 + 1 3(a


2

b 1 + 1 a


2

c 1 + 1 a


2

d 1



  • 1 b


2

a 1 + 1 b


2

c 1 + 1 b


2

d 1 + 1 c


2

a 1 + 1 c


2

b 1 + 1 c


2

d 1



  • 1 d


2

a 1 + 1 d


2

b 1 + 1 d


2

c) 1 + 1 6(abc 1 + 1 abd 1



  • 1 acd 1 + 1 bcd)




  1. 10211 37.






Section 7.4



  1. (i) 11 + 13 x 1 + 19 x


2

1 + 127 x


3

1 + 181 x


4

1 + 1243 x


5

1



  • 1729 x


6

1 +1-


(ii)|x| 1 < 13



  1. (i) 11 − 15 x


2

1 + 125 x


4

1 − 1125 x


6

1 +1-


(ii)



  1. (i)


(ii)|x| 1 < 12




  1. (i)0.000001000001000001 -






Section 7.5


46.diverges 47.converges


48.converges for all a


49.no conclusion (see Exercise 50)


50.diverges ifa 1 ≤ 11 , converges ifa 1 > 11


51.diverges


qe


T

v

v

=−










1


1

θ

1


2 4 8 16 32 64 128


2345 6

−+ − + − + −


xx x x x x






||x< 15


n


nnn


2

432

12


2651 ++−








3


4


23


21 2







++


n


()( )nn


3


1110


6


!


!!!!


=


3


3000


1


3


2100


3


!


!!!!


,


!


!!!!


==,


4


211


12


!


!!!


=


4


400


1


4


310


4


4


220


6


!


!!!


,


!


!!!


,


!


!!!


== =,


Section 7.6




  1. 4 53. 1 54. 1




  2. 1 56. 0 57.






  3. 11 − 1 x




2

1 + 1 x


4

1 − 1 x


6

1 + 1 x


8

1































  1. (i)


(ii)



  1. (i) (ii) 01 < 1 x 1 < 12

  2. (i) (ii)all x

  3. (i) (ii)all x

  4. (i) (ii) 01 < 1 x 1 ≤ 14


Section 7.7



  1. (i)


(ii)2.08008214


(iii)



  1. 1 74. 122 75.− 2 76. 122


2 0800832 9 2 0800840


3

.<<.


2


12 288


5


20736


5


248832


23 4

+− + −


xx x x


ln 2


12


2


1











=


n

n

n


x



n

nn

x


n


=


−−


0

2

1


2



()( )


()!


π 2


e


x


n


n

n

2

0

2


=



!



()


n

n

x


=



0

1



()


Bb


ab


RT


a


RT


2

2

2

22

2


=− +


BBb


a


RT


01

==−, 1 ,


BBb


a


RT


Bb


01 2

2

==−,= 1 ,


→→


1


2


0


0

2

mcvv as


Tm


cc


=+






























1


2


1


3


4


5


8


0

2

24

v


vv














x


xxx x


++++


357 9

2 6 24 120


13


9


2


9


2


27


8


23 4

−+ − +x


xx x


−− − 2 − −


8


3


4


32


5


32


3


2

34

x


x


xx


2


4


3


4


15


8


315


4


2835


2

6101418

x


xx x x


−+ − +


1


3 9 27 81 243


23 4

−+ − +


xx x x


1


2


3


8


5


16


35


128


23 4

++ + +


xx x x


1


39


5


81


10


243


23 4

+− + −


xx x x


3

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