Γιώργος Μαυρίδης-Ολοκληρώματα
DESCRIPTION
Ασκήσεις εφ' όλης της ύλης στα ολοκληρώματα, από το ανανεωμένο βιβλίο του Γιώργου Μαυρίδη για την Γ΄ Λυκείου. Οι ασκήσεις είναι μια φιλική προσφορά του φίλου και συναδέλφου, Γιώργου Μαυρίδη.TRANSCRIPT
9 786188 021495
ISBN: 978-618-80214-9-5
nȍȨȦ ȰȪȵȦȨ�ȩȨȮ�ȚȯȢȞȪȷȦȞȭȞ�ȭȷȪȚ� ȜȢȚ�ȦȚ ȣȚȭȚȦȨȗȬȞȭȞ�ȭȢȫ�țȚȬȢȣȖȫ�ȖȦȦȨȢȞȫ� ȣȚȢ�ȦȚ�ȞȥțȚȡȶȦȞȭȞ�ȬȭȨȦ�ȭȪȵȩȨ�ȬȣȖȱȠȫ� ȣȚȢ ȞȪȜȚȬȘȚȫ��ȡȚ�ȭȨȦ�ȣȞȪȝȘȬȞȭȞ� ȩȨȤȤȚȩȤȕȬȢȨ�ȚȪȜȵȭȞȪȚ.»
ȁȞȵȝȲȪȨȫ�Ȇ��ȃȚȟȚȦȭȟȗȫ
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ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ� ȞĮ� ĮʌȠIJİȜȑıȠȣȞ� ȝȑȡȠȢ� IJȠȣȜȐȤȚıIJȠȞ� IJȘȢ�ĮʌȐȞIJȘıȘȢ�ıIJȠȞ�ȚįİȫįȘ�țȩıȝȠ�IJȠȣ��ȥȑȝĮIJȠȢ�ȩʌȠȣ�IJĮ�ʌȐȞIJĮ�įȘȜȫȞȠȣȞ�țĮȚ�įȘȜȫ�ȞȠȞIJĮȚ�ȦȢ�ĮȜȘșȚȞȐ��
ĬĮ�ȝʌȠȡȠȪıĮȞ� IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�ĮʌȠIJİȜȑıȠȣȞ� IJȘȞ�țĮȪıȚȝȘ�ȪȜȘ�Ș�ȠʌȠȓĮ�șĮ�İȞİȡȖȠʌȠȚȠȪıİ�IJȠȞ�ȞĮȡțȦȝȑȞȠ�ȖȓȖĮȞIJĮ�IJȘȢ�ıȣȞİȓįȘıȘȢ��
ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ�� ȦȢ� ȖȜȫııĮ� IJȘȢ� ȜȠȖȚțȒȢ�� ȞĮ� ıȣȞįȡȐȝȠȣȞ�ĮʌȠijĮıȚıIJȚțȐ�IJȚȢ�ıIJȡȚȝȦȖȝȑȞİȢ�įȣȞȐȝİȚȢ�IJȘȢ�ıȪȞİıȘȢ��IJȘȢ�IJĮʌİȚȞȩIJȘIJĮȢ��IJȘȢ�ȖȞȫ�ıȘȢ�țĮȚ�IJȘȢ�ĮȜȒșİȚĮȢ��Įʌȩ�IJĮ�ȝĮȞȚĮıȝȑȞĮ�ıIJȓijȘ�IJȠȣ�©șȑȜȦª��IJȠȣ�©ijĮȓȞȠȝĮȚª��IJȠȣ�©ȑȤȦª��IJȠȣ�©ȖȡȒȖȠȡĮª��IJȠȣ�©țĮIJĮȞĮȜȫȞȦª��IJȠȣ�©İȞIJȣʌȦıȚĮțȠȪª��IJȠȣ�©İȖȫª�
ȅ�țĮșȑȞĮȢ�Įʌ¶�ȩıȠȣȢ�ȕȡȓıțȠȣȞ�ȗȦȠȖȩȞȠ�IJȠ�ȞĮ�ĮıȤȠȜȠȪȞIJĮȚ�ıİ�ȕȐșȠȢ�ȝİ�IJȘ�ȗȦȒ��șĮ�ȑįȚȞİ�IJȘ�įȚțȒ�IJȠȣ�ĮʌȐȞIJȘıȘ�
ǻȪıțȠȜĮ�ʌȐȞIJȦȢ�șĮ�ȣʌȒȡȤĮȞ�įȚĮijȦȞȓİȢ��ıIJȠ�ȩIJȚ�șĮ�ȒIJĮȞ��ĮįȪȞĮIJȠȞ��ĮțȩȝȘ�țĮȚ�ȖȚĮ�IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�İʌȚIJİȜȑıȠȣȞ�ȑȡȖȠ�ıİ�țȐʌȠȚȠ�Įʌȩ�IJĮ�ʌȡȠȘȖȠȪȝİȞĮ��ĮȞ�țȐIJȚ�IJĮ�ĮʌȑțȠʌIJİ�Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȡȠijȩ�IJȠȣȢ��Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȠȣ�ĮʌİȓȡȠȣ�
ǻİȞ�ȞȠȝȓȗȦ�ȩIJȚ�ȤȡİȚȐȗİIJĮȚ�Ș�ıȣȞįȡȠȝȒ�IJȘȢ�țȕĮȞIJȠȝȘȤĮȞȚțȒȢ�� IJȘȢ�İȟİȜȚțIJȚ�țȒȢ�ȕȚȠȜȠȖȓĮȢ�Ȓ�IJȘȢ�șİȦȡȓĮȢ�IJȠȣ�ȤȐȠȣȢ��ȫıIJİ�ȞĮ ȠȡȚıIJİȓ�ȦȢ ȜȠȖȚțȒ�ĮȞĮțȠȜȠȣșȓĮ���Ș�ıȪȞįİıȘ�IJȠȣ�ĮʌİȓȡȠȣ�IJȦȞ�įȣȞĮIJȫȞ�ʌİȡȚʌIJȫıİȦȞ��ȝİ�IJȘȞ�IJȣȤȠįȚȦțIJȚțȠȪ�IJȪʌȠȣ��İʌȚȜȠȖȒ�țȐʌȠȚȦȞ�Įʌȩ�ĮȣIJȑȢ�
Ȃ¶�ĮȣIJȒ� ĮțȡȚȕȫȢ� IJȘȞ� ʌİʌȠȓșȘıȘ� ĮʌȑijȣȖĮ� İʌȚȝİȜȫȢ�� țȐșİ� ȠȜȠțȜȘȡȦIJȚțȠȪ�IJȪʌȠȣ�ȠȝĮįȠʌȠȓȘıȘ�IJȦȞ�ĮıțȒıİȦȞ��ǹȣIJȩ�įȘȜĮįȒ�ʌȠȣ�IJȩıȠ�İȪıIJȠȤĮ�Ƞ�ĮȟȑȤĮıIJȠȢ�įȐıțĮȜȠȢ�ĬİȩįȦȡȠȢ�ȀĮȗĮȞIJȗȒȢ�ȤĮȡĮțIJȒȡȚȗİ�ȦȢ�©ʌİȡȚʌIJȦıȚȠȜȠȖȓĮª�
ǻİȞ�ȑȤİȚ�ȞȩȘȝĮ�ȞĮ�ĮȞĮȗȘIJȒıİȚ�țĮȞİȓȢ�IJȚȢ�ĮȚIJȓİȢ�ĮȣIJȒȢ�IJȘȢ�İȝȝȠȞȒȢ��İȐȞ�įȘ�ȜĮįȒ�ȠijİȓȜİIJĮȚ�ıİ� ȑȜȜİȚȝȝĮ�ȜȠȖȚțȒȢ�Ȓ� İȞIJȚȝȩIJȘIJĮȢ��ȆȐȞIJȦȢ�� ȝȠȣ� İȓȞĮȚ� ȚįȚĮȓIJİȡĮ�įȪıțȠȜȠ�ȞĮ�ĮʌȠijȪȖȦ�IJȠȞ�ʌİȚȡĮıȝȩ�țȐʌȠȚȦȞ�İȞȠȤȜȘIJȚțȫȞ�ĮȜȜȐ�ȝȐȜȜȠȞ�ʌȡĮȖȝĮIJȚ�țȫȞ� ıțȑȥİȦȞ��ȦȢ� ʌȡȠȢ� IJĮ� ĮʌȠIJİȜȑıȝĮIJĮ� ĮȣIJȫȞ� IJȦȞ� ĮȞIJȚȜȒȥİȦȞ� Ȓ� ȓıȦȢ� “ȠȡijĮ�ȞȫȞ” ʌȡĮțIJȚțȫȞ�
ȀȐșİ�ȥİȣįĮȓıșȘıȘ�țĮȚ�ȝȚĮ�ijȣȜĮțȒ���ȀȐʌȦȢ�ȑIJıȚ�țȚ�İįȫ�Ș�ȥİȣįĮȓıșȘıȘ�IJȘȢ�ıȣıIJȘȝĮIJȚțȒȢ�ȠȡȖȐȞȦıȘȢ�țĮȚ�ȝİȜȑIJȘȢ�ijȣȜĮțȓȗİȚ�IJȠ�įȡȠıİȡȩ�ʌȞİȪȝĮ�IJȦȞ�ȝĮșȘIJȫȞ��ıIJİȡȫȞIJĮȢ�Įʌ¶�ĮȣIJȠȪȢ�IJȘ�ȝĮșȘIJİȓĮ�IJȘȢ�İȟİȡİȪȞȘıȘȢ��IJȘȞ�ĮȞİȝİȜȚȐ�IJȘȢ�ʌİȡȚʌȜȐȞȘ�ıȘȢ�țĮȚ�IJȘ�ȤĮȡȐ�IJȘȢ�ĮȞĮțȐȜȣȥȘȢ��ȈIJİȡȫȞIJĮȢ�ijȣıȚțȐ�ıIJȘ�ıȣȞȑȤİȚĮ�ıIJȠ�ȕĮșȝȩ�ʌȠȣ�IJĮ�İʌȘȡİȐȗȠȣȞ��IJĮ�ȝĮșȘȝĮIJȚțȐ�ĮȜȜȐ�țȣȡȓȦȢ�IJȘȞ�țȠȚȞȦȞȓĮ��Įʌȩ�ȝȣĮȜȐ�İȟİȡİȣȞȘȝȑ�ȞĮ�țĮȚ�ȐȡĮ�İȟİȡİȣȞȘIJȚțȐ��ȝİ� IJİȜȚțȩ�ĮʌȠIJȑȜİıȝĮ�IJȘȞ�ȤȦȡȓȢ�ĮȞIJȓʌĮȜȠ�İʌȚțȡȐIJȘıȘ�IJȘȢ�ȝȓȝȘıȘȢ��IJȘȢ�ijĮȚįȡȩIJȘIJĮȢ�țĮȚ�IJȘȢ�İʌĮȞȐȜȘȥȘȢ��ȅȚ�ĮʌȠįİȓȟİȚȢ�ȩȜȦȞ�IJȦȞ�ʌȡȠȘ�ȖȠȪȝİȞȦȞ��ʌȡȠijĮȞȫȢ�įİȞ�ȤȡİȚȐȗİIJĮȚ�ȞĮ�ĮȞĮȗȘIJȘșȠȪȞ ıİ�ıțȠȞȚıȝȑȞĮ�ȝĮșȘȝĮIJȚțȐ�ȕȚȕȜȓĮ��ǹȞ�IJİȜȚțȐ�ijȠȕȓȗİȚ�IJȩıȠ�ʌȠȜȪ�ȝȚĮ�ȝĮIJȚȐ�ȝȑıĮ�ȝĮȢ��IJȠȣȜȐȤȚıIJȠȞ�IJȫȡĮ�İȓȞĮȚ�ĮȡțİIJȒ�ȝȚĮ�ȝĮIJȚȐ�įȓʌȜĮ�ȝĮȢ�
ĬİııĮȜȠȞȓțȘ��ǿȠȪȞȚȠȢ 2013 īȚȫȡȖȠȢ ȁ��ȂĮȣȡȓįȘȢ
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ȆȡȩȜȠȖȠȢ� ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��Ǻǯ�ȂȑȡȠȢ� ............................................................... 11� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 259 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 317 ȅȜȠțȜȘȡȦIJȚțȩȢ�ȁȠȖȚıȝȩȢ ........................................................................... 329 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 331 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 505 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 543 ǼʌĮȞĮȜȘʌIJȚțȐ�ĬȑȝĮIJĮ ................................................................................ 555 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 575 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 636
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� ȖȚĮ�țȐșİ� > @[ �� � ��
i) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�� � �f 2 0z .
ii) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ�� > @[ �� � ��
iii) ȃĮ�ȕȡİȓIJİ�IJȠȞ�IJȪʌȠ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�f . iv) ȃĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ�ȖȡĮ�
ijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ
� � � �� � > @2J [ I [ � [ �� � �
IJȠȞ�ȐȟȠȞĮ� x xc țĮȚ�IJȚȢ�İȣșİȓİȢ�� x 1 țĮȚ�� x 2.
ƭǞǔLj�
i) ȊʌȠșȑIJȠȣȝİ� �ĮʌĮȖȦȖȒ� ıİ� ȐIJȠʌȠ�� ȩIJȚ � �f 2 0. ĬȑIJȠȞIJĮȢ� ıIJȘ� įȠșİȓıĮ� ıȤȑıȘ� ȩʌȠȣ�x IJȠ���ȑȤȠȣȝİ
� � � � � �� �0
2
0
1 1f 0 f 2 f t dt 0 0 03 3
� � µ́¶
țĮȚ�ıȣȞİʌȫȢ � � � �f 0 f 2
ʌȠȣ��ȝİ�ȕȐıȘ�IJȘȞ�ȣʌȩșİıȘ��İȓȞĮȚ ȐIJȠʌȠ��ǼʌȠȝȑȞȦȢ � �f 2 0.z
ii) ĬȑIJȠȞIJĮȢ�ıIJȘ�įȠșİȓıĮ�ıȤȑıȘ�ȩʌȠȣ�[�IJȠ���ȑȤȠȣȝİ
� � � � � �� �2 2
0
1f 2 f 2 f t dt3
� µ́¶
Ȓ�ȚıȠįȪȞĮȝĮ
� � � �� �2 2
0
2 f 2 f t dt3
µ́¶
įȘȜĮįȒ������
� � � �� �2 2
0
3f 2 f t dt2
µ́¶
(1)
ȂǼĬȅǻȅȁȅīǿǹ ȆĮȡĮIJȘȡȠȪȝİ�ȩIJȚ
� �� �x 2
0f t dt 0tµ́
¶ ȖȚĮ�țȐșİ� > @x 0, 2�
ȅʌȩIJİ��ȜȩȖȦ�IJȘȢ�įȠșİȓıĮȢ�ȚıȩIJȘIJĮȢ
� � � � � �� �x 2
0
1f x f 2 f t dt3
� µ́¶
ȖȚĮ� ȞĮ� ĮʌȠįİȓȟȠȣȝİ� IJȠ� ȗȘIJȠȪȝİȞȠ�Įȡțİȓ�ȞĮ�ĮʌȠįİȓȟȠȣȝİ�ȩIJȚ � �f 2 0.!
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 489�
ǵȝȦȢ��Ș�ıȣȞİȤȒȢ�ıȣȞȐȡIJȘıȘ� � �� �2f t İȓȞĮȚ�ȝȘ�ĮȡȞȘIJȚțȒ�țĮȚ�ȩȤȚ�ʌĮȞIJȠȪ�ȝȘįȑȞ�ıIJȠ�
įȚȐıIJȘȝĮ� > @0, 2 ĮijȠȪ � �� �2f 2 0z . DZȡĮ�
� �� �2 2
0f t dt 0!µ́
¶
ȅʌȩIJİ��Įʌȩ�IJȘ�ıȤȑıȘ�(1) ʌȡȠțȪʌIJİȚ�ȩIJȚ � �f 2 0! . ǼʌȓıȘȢ�
� �� �x 2
0f t dt 0tµ́
¶ ȖȚĮ�țȐșİ�� > @x 0, 2� .
ǼʌȠȝȑȞȦȢ�
� � � � � �� �x 2
0
1f x f 2 f t dt 03
� !µ́¶
ȖȚĮ�țȐșİ�� > @x 0, 2� .
iii) Ǿ� ıȣȞȐȡIJȘıȘ� � �� �2f t İȓȞĮȚ� ıȣȞİȤȒȢ� țĮȚ� ıȣȞİʌȫȢ� Ș� ıȣȞȐȡIJȘıȘ� � �� �
x 2
0f t dtµ́
¶
İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȅʌȩIJİ��Įʌȩ�IJȘ�įȠșİȓıĮ�ıȤȑıȘ�ʌȡȠțȪʌIJİȚ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ�ȝİ�
� � � � � �� �
� � � �� � � �� �
� �� � > @
x 2
0
x 2 2
0
2
1f x f 2 f t dt3
1 f 2 f t dt 0 f x3
f x 0 ȖȚĮ țȐșİ [ �� � �
c§ ·c �¨ ¸© ¹
cc § ·§ · � �¨ ¸ ¨ ¸© ¹ © ¹
! �
µ́¶
µ́¶
ǻȘȜĮįȒ�
� �� �2
f x1
f xc
� � ȖȚĮ�țȐșİ�� > @x 0, 2�
� � � �1 x
f x
c§ · c� �¨ ¸¨ ¸© ¹
ȖȚĮ�țȐșİ�� > @x 0, 2�
� �1 x c
f x� � � ȖȚĮ�țȐșİ��� > @x 0, 2� ���ȝİ�� c�\
� � 1f xx c
� � �
ȖȚĮ�țȐșİ��� > @x 0, 2 .�
490 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ǵȝȦȢ�� Įʌȩ� IJȘ� įȠșİȓıĮ� ıȤȑıȘ�� șȑIJȠ�
ȞIJĮȢ�ȩʌȠȣ�x IJo ��ȑȤȠȣȝİ
� � � � � �� �
� � � �
� �
0 2
0
1f 0 f 2 f t dt31 1 1 1f 0 f 23 0 c 3 2 c
3 2 c c 6 3c c
2c 6 c 3.
�
� � �� � � �
� � � � � �
� �
µ́¶
ǼʌȠȝȑȞȦȢ�
� � 1f xx 3
� �
ȖȚĮ�țȐșİ��� > @x 0, 2 .�
iv) H ıȣȞȐȡIJȘıȘ�
� � � �� � > @2g t f t , t 0, 2 �
İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�ȝȘ�ĮȡȞȘIJȚțȒ��ȅʌȩIJİ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ İȓȞĮȚ�ȓıȠ�ȝİ
� � � �� �
� �� �� �
2 2 2
1 122 22
2 211 1
2 2
11
E g t dt f t dt
3 t1 1dt dt dtt 3 3 t 3 t
1 1 1 1 1 1dt 1 IJ�ȝ�3 t 3 t 3 2 3 1 2 2
c� �§ · ¨ ¸� �© ¹ � �
c§ · ª º � � ¨ ¸ « »� � � �© ¹ ¬ ¼
´ ´µ µ¶ ¶
´´ ´µµ µµ µ¶¶ ¶
´µ¶
Ȉȋȅȁǿȅ ǻİȞ�ȣʌȐȡȤİȚ�țȐʌȠȚĮ�ĮȡȤȚțȒ�ıȣȞșȒ�țȘ�ʌȠȣ�șĮ�ȕȠȘșȠȪıİ�ıIJȠȞ�ȣʌȠȜȠȖȚ�ıȝȩ�IJȘȢ�ıIJĮșİȡȐȢ�c. ǻȘȝȚȠȣȡȖȠȪȝİ�İȝİȓȢ� ȜȠȚʌȩȞ� ȝȓĮ� IJȑIJȠȚĮ� ıȣȞșȒțȘ�șȑIJȠȞIJĮȢ�ıIJȘ�įȠșİȓıĮ�ıȤȑıȘ�ȩʌȠȣ x IJȠ� �� țĮIJĮȜȒȖȠȞIJĮȢ� ȑIJıȚ� ıIJȘȞ� Țıȩ�IJȘIJĮ
� � � �1f 0 f 2 .3
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 491�
ƪƾǍǂ�104�
ǻȓȞȠȞIJĮȚ�ȠȚ�ıȣȞĮȡIJȒıİȚȢ� I � J � o\ \ ȠȚ�ȠʌȠȓİȢ�İȓȞĮȚ�ıȣȞİȤİȓȢ�țĮȚ�IJȑIJȠȚİȢ��ȫıIJİ
� � � �
x
0
tf x dt 1g t
�´µ¶
ȖȚĮ�țȐșİ�x�\
țĮȚ�
� � � �
x
0
tg x dt 1f t
�´µ¶
ȖȚĮ�țȐșİ� x�\ .
i) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ� x�\
țĮȚ � �g x 0! ȖȚĮ�țȐșİ� x�\ .
ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � � � �f x g x ȖȚĮ�țȐșİ� x�\ .
iii) ȃĮ�ȕȡİȓIJİ�IJȠȞ�IJȪʌȠ�IJȘȢ� ıȣȞȐȡIJȘıȘȢ�f. iv) ȃĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ�ȖȡĮ�
ijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ
� � � �
x
1
1K [ GW�f t
´µ¶
x�\
IJȠȣȢ�ȐȟȠȞİȢ� x xc �� y yc țĮȚ�IJȘȞ�İȣșİȓĮ� x 1. ƭǞǔLj� i) ǹʌȩ�IJȚȢ�įȠșİȓıİȢ�ıȤȑıİȚȢ�ʌȡȠțȪʌIJİȚ�ȩIJȚ
� �f x 0z ȖȚĮ�țȐșİ� x�\
țĮȚ�� � �g x 0z ȖȚĮ�țȐșİ�� x�\ .
ǻȚĮijȠȡİIJȚțȐ��ȝȓĮ�IJȠȣȜȐȤȚıIJȠȞ�Įʌȩ�IJȚȢ�ıȣȞĮȡIJȒıİȚȢ�� �t
g t țĮȚ�
� �t
f t įİȞ�șĮ�ȠȡȓȗȠ�
ȞIJĮȞ�ıIJȠ� \ țĮȚ�ıȣȞİʌȫȢ��ȝİ�ȕȐıȘ�IJȚȢ�įȠșİȓıİȢ�ıȤȑıİȚȢ��ȝȓĮ�IJȠȣȜȐȤȚıIJȠȞ�Įʌȩ�IJȚȢ�ıȣȞĮȡIJȒıİȚȢ��f țĮȚ��g įİȞ�șĮ�İȓȤİ�ʌİįȓȠ�ȠȡȚıȝȠȪ�IJȠ�\ ��ʌȡȐȖȝĮ�ȐIJȠʌȠ��ǼʌȓıȘȢ��ȠȚ�
492 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ıȣȞĮȡIJȒıİȚȢ�f țĮȚ� g İȓȞĮȚ�ıȣȞİȤİȓȢ�ıIJȠ�\ țĮȚ�ıȣȞİʌȫȢ�įȚĮIJȘȡȠȪȞ�ʌȡȩıȘȝȠ�ıIJȠ�.\ ȀĮȚ�İʌİȚįȒ
� � � �
0
0
tf 0 dt 1 0 1 1 0g t
� � !´µ¶
țĮȚ � � � �
0
0
tg 0 dt 1 0 1 1 0f t
� � !´µ¶
ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ� � �f x 0! ȖȚĮ�țȐșİ� x�\
țĮȚ � �g x 0! ȖȚĮ�țȐșİ� x�\ .
ii) ȅȚ�ıȣȞĮȡIJȒıİȚȢ � �t
g t țĮȚ�
� �t
f t İȓȞĮȚ�ıȣȞİȤİȓȢ�ıIJȠ� \ . ǼʌȠȝȑȞȦȢ��Įʌȩ�IJȚȢ�įȠșİȓ�
ıİȢ�ıȤȑıİȚȢ�ʌȡȠțȪʌIJİȚ�ȩIJȚ�ȠȚ�ıȣȞĮȡIJȒıİȚȢ� � �f x țĮȚ� � �g x İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝİȢ�ȝİ
� � � � � � � �
x
0
t xf x dt 1g t g x
c§ · cc � ¨ ¸¨ ¸© ¹
´µ¶
ȖȚĮ�țȐșİ�� x�\
țĮȚ
� � � � � � � �
x
0
t xg x dt 1f t f x
c§ · cc � ¨ ¸¨ ¸© ¹
´µ¶
ȖȚĮ�țȐșİ�� x�\ .
ǹʌȩ�IJȚȢ�ʌĮȡĮʌȐȞȦ�ȚıȩIJȘIJİȢ�ȑȤȠȣȝİ
� �� � � � � �
f x xf x f x g xc
ȖȚĮ�țȐșİ�� x�\
țĮȚ � �� � � � � �
g x xg x f x g xc
ȖȚĮ�țȐșİ�� x�\ .
ȅʌȩIJİ,
� �� �
� �� �
f x g xf x g xc c
ȖȚĮ�țȐșİ�� x�\ .
ǻȘȜĮįȒ�
� �� � � �� �ln f x ln g xc c ȖȚĮ�țȐșİ�� x�\ Ȓ�ȚıȠįȪȞĮȝĮ
� � � �ln f x ln g x c � ȖȚĮ�� x�\ ���ȩʌȠȣ� c�\ .
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 493�
ĬȑIJȠȞIJĮȢ�ıIJȘȞ�IJİȜİȣIJĮȓĮ�ȚıȩIJȘIJĮ�ȩʌȠȣ�x IJȠ�� ʌĮȓȡȞȠȣȝİ
� � � �ln f 0 ln g 0 c � ln1 ln1 c� � c 0.�
DZȡĮ�
� � � �ln f x ln g x ȖȚĮ�țȐșİ� x�\
țĮȚ�IJİȜȚțȐ
� � � �f x g x ȖȚĮ�țȐșİ� x�\ .
iii) ǹʌȩ�IJȠ�ʌȡȠȘȖȠȪȝİȞȠ�İȡȫIJȘȝĮ�ȑȤȠȣȝİ
� � � � � �x xf x
g x f xc ȖȚĮ�țȐșİ� x�\ .
ǻȘȜĮįȒ,
� � � �2f x f x 2xc ȖȚĮ�țȐșİ� x�\
� �� � � �2 2f x xc c� ȖȚĮ�țȐșİ� x�\
� �2 2f x x c� � ȖȚĮ�țȐșİ� x�\ ��ȩʌȠȣ� c�\ .
ĬȑIJȠȞIJĮȢ�ıIJȘȞ�IJİȜİȣIJĮȓĮ�ıȤȑıȘ�ȩʌȠȣ�x IJȠ���ʌĮȓȡȞȠȣȝİ
� �2 2f 0 0 c � � c 1.
ǼʌȠȝȑȞȦȢ�
� �2 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .
ȀĮȚ�İʌİȚįȒ
� �f x 0! ȖȚĮ�țȐșİ� x�\ ,
ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ
� � 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .
494 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
iv) ȆĮȡĮIJȘȡȠȪȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ � �1
f t İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ� .\ ǼʌȠȝȑȞȦȢ��Ș�ıȣȞȐȡ�
IJȘıȘ
� � � �
x
1
1h x dtf t
´µ¶
İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȐȡĮ�țĮȚ�ıȣȞİȤȒȢ���ıIJȠ�\ ȝİ
� � � � 2
1 1h x 0f x x 1
c !�
ȖȚĮ�țȐșİ� x�\ .
DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ�h İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ��ȀĮȚ�İʌİȚįȒ
� � � �
1
1
1h 1 dt 0f t
´µ¶
,
ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ
� � � � � �h x 0 h x h 1 x 1� � � � �
țĮȚ
� � � � � �h x 0 h x h 1 x 1.! � ! � !
ǼʌȠȝȑȞȦȢ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ
� � � � � � � �1 1 1
0 00E h x dx h x dx x h x dxc � �´ ´ ´µ µµ ¶ ¶¶
� � � � � � � � � �111
0 20 0
1xh x x h x dx 1h 1 0h 0 x dxx 1
c � � � � � �ª º¬ ¼�
´µ́ µ¶ ¶
� �1 1 1
2 2
2 000
2x0 0 dx x 1 dx x 12 x 1
c ª º � � � �¬ ¼�
´ ´µµ ¶¶
2 21 1 0 1 2 1 � � � � IJ�ȝ�
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 495�
ƪƾǍǂ�105�
DzıIJȦ�ıȣȞȐȡIJȘıȘ� f : o\ \ IJȑIJȠȚĮ��ȫıIJİ � � � �1f x f x
t 2x 12
1
e dt e et
� �´µµ¶
ȖȚĮ�țȐșİ��x�\ .
i) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ��x�\ .
ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�
� � 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .
iii) AȞ�F İȓȞĮȚ�Ș�ʌĮȡȐȖȠȣıĮ�IJȘȢ�f ıIJȠ� \ ȖȚĮ�IJȘȞ�ȠʌȠȓĮ�ȚıȤȪİȚ� � �) � �� ȞĮ�ȣʌȠȜȠȖȓıİIJİ� IJȠ� İȝȕĮįȩ�IJȠȣ� ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ� F& � IJȠȞ�ȐȟȠȞĮ�x xc țĮȚ�IJȚȢ�İȣșİȓİȢ� x 0 țĮȚ��x 1.
ƭǞǔLj��
i) Ǿ�ıȣȞȐȡIJȘıȘ�
1t
2et
ȑȤİȚ�ʌİįȓȠ�ȠȡȚıȝȠȪ�IJȠ�ıȪȞȠȜȠ� *\ țĮȚ�İȓȞĮȚ�ıȣȞİȤȒȢ�ı¶�ĮȣIJȩ�
ȦȢ� ĮʌȠIJȑȜİıȝĮ� ʌȡȐȟİȦȞ� ȝİIJĮȟȪ� ıȣȞİȤȫȞ� ıȣȞĮȡIJȒıİȦȞ�� ǼʌȠȝȑȞȦȢ�� Ș� ʌĮȡȐ�ıIJĮıȘ
� � 1f xt
21
e dtt
´µµ¶
ȠȡȓȗİIJĮȚ�ȖȚĮ�İțİȓȞĮ�IJĮ�x IJȑIJȠȚĮ��ȫıIJİ�IJĮ�ȩȡȚĮ�ȠȜȠțȜȒȡȦıȘȢ���țĮȚ� � �f x ȞĮ�ĮȞȒ�
țȠȣȞ�ıIJȠ�ȓįȚȠ�įȚȐıIJȘȝĮ�IJȠȣ� *.\ ǻȘȜĮįȒ��ȖȚĮ�İțİȓȞĮ�IJĮ�x IJȑIJȠȚĮ��ȫıIJİ�
� � � �f x 0, .� �f
ǵȝȦȢ��ıȪȝijȦȞĮ�ȝİ�IJȘ�įȠșİȓıĮ�ıȤȑıȘ��Ș�İȞ�ȜȩȖȦ�ʌĮȡȐıIJĮıȘ�ȠȡȓȗİIJĮȚ�ȖȚĮ�țȐșİ�x .�\ ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ
� �f x 0! ȖȚĮ�țȐșİ�� x .�\
496 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ii) Ǿ�įȠșİȓıĮ�ıȤȑıȘ�ȖȡȐijİIJĮȚ
� �� �2
f x f x1t x 1
21
1e dt e et
�§ ·� � � �¨ ¸© ¹
´µ¶
ȖȚĮ�țȐșİ�� x�\
� �� �2
f x f x1t x 1
1
1e dt e et
�c§ ·� � � �¨ ¸
© ¹
´µ¶
ȖȚĮ�țȐșİ�� x�\
� �� �2
f x f x1t x 1
1
e dt e e�c§ ·
� �¨ ¸© ¹
´µµ¶
ȖȚĮ�țȐșİ� x�\
� � � �2
f x f x1t x 1
1
e e e�ª º
� �« »¬ ¼
ȖȚĮ�țȐșİ�� x�\
� �� �2
f x1 1f x 1 x 1e e e e�� � � ȖȚĮ�țȐșİ�� x�\
� �� �2
f x1f x x 1e e �� ȖȚĮ�țȐșİ�� x�\
� �� �
2
f x1f x x 1
� �
ȖȚĮ�țȐșİ�� x�\
� �� �2 2f x x 1� � ȖȚĮ�țȐșİ�� x�\
ȀĮȚ�İʌİȚįȒ
� �f x 0! ȖȚĮ�țȐșİ�� x�\
ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ
� � 2f x x 1 � ȖȚĮ�țȐșİ�� x .�\
iii) Ǿ�ıȣȞȐȡIJȘıȘ�F İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȐȡĮ�țĮȚ�ıȣȞİȤȒȢ�ıIJȠ� .\ ǼʌȠȝȑȞȦȢ��IJȠ�ȗȘ�
IJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ� � �1
0E F x dx µ́
¶. ǼʌȓıȘȢ�
� � � � 2F x f x x 1 0c � ! ȖȚĮ�țȐșİ�� x .�\
DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ�F İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�țĮȚ�ıȣȞİʌȫȢ
� � � � � �F x F 1 0 ȖȚĮ țȐșİ [ ��� � �f .
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 497�
DzȤȠȣȝİ�ȜȠȚʌȠȞ
� �F x 0d ȖȚĮ�țȐșİ�� > @x 0,1� .
ȅʌȩIJİ�
� � � �� �1 1
0 0E F x dx F x dx �´ ´µ µ
¶ ¶
� � � � � � � � � �1 11
00 0x F x dx xF x x F x dxc c � � � �ª º¬ ¼
´ ´µ µ¶ ¶
� � � � � �1
01 F 1 0 F 0 xf x dx � � � � � µ́
¶
12
0x x 1dx �µ́
¶.
īȚĮ�IJȠȞ�ȣʌȠȜȠȖȚıȝȩ�IJȠȣ�IJİȜİȣIJĮȓȠȣ�ȠȜȠțȜȘȡȫȝĮIJȠȢ�șȑIJȠȣȝİ
2u x 1 � .
ȅʌȩIJİ 2 2u x 1 �
țĮȚ�ıȣȞİʌȫȢ
� � � �2 2u du x 1 dx 2udu 2xdx udu xdx.c c � � �
īȚĮ� x 0 ȑȤȠȣȝİ� u 1
īȚĮ� x 1 ȑȤȠȣȝİ� u 2.
ǼʌȠȝȑȞȦȢ�
1 1
2 2
0 0x x 1dx x 1 xdx� �´ ´µ µ
¶ ¶
2
1uu du µ́
¶
22
1u du µ́
¶
� �323 3
1
2u 13 3 3
ª º �« »¬ ¼
2 2 1.3�
ǻȘȜĮįȒ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ
2 2 1E IJ�ȝ�3�
498 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ���6�
DzıIJȦ�ıȣȞȐȡIJȘıȘ� f : o\ \ Ș�ȠʌȠȓĮ�İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�IJȑIJȠȚĮ��ȫıIJİ: z � � � �f 1 f 1� � z � �f x 0z ȖȚĮ�țȐșİ�� x *��
z � � � �x
1
f x1 2 f t dt
x� ³ ȖȚĮ�țȐșİ�� �x * .�
i) NĮ�ȕȡİȓIJİ�IJȘȞ�IJȚȝȒ� � �f 0 țĮȚ�IJȠ�ʌȡȩıȘȝȠ�IJȘȢ� f.
ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ� � � 2x 1f x xe � ȖȚĮ�țȐșİ� �x� . iii) NĮ�ȕȡİȓIJİ�IJĮ țȠȚȞȐ�ıȘȝİȓĮ�IJȘȢ� fC ȝİ�IJȘȞ�İȣșİȓĮ� y x. iv) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ� f ĮȞIJȚıIJȡȑijİIJĮȚ�țĮȚ� ȞĮ�ȣʌȠȜȠȖȓıİIJİ�
IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ıȤȘȝĮIJȓȗȠȣȞ�ȠȚ�ȖȡĮijȚțȑȢ�ʌĮȡĮıIJȐıİȚȢ�IJȦȞ�ıȣȞĮȡIJȒıİȦȞ�f țĮȚ�� 1f .�
ƭǞǔLj�
i) Aʌȩ�IJȘ�ıȤȑıȘ
� � � �x
1
f x1 2 f t dt
x� µ́
¶ ȖȚĮ�țȐșİ�� x *�� (1)
ʌĮȓȡȞȠȣȝİ
� � � �x
1f x x 2x f t dt � µ́
¶ ȖȚĮ�țȐșİ�� x *.�� (2)
Ǿ�ıȣȞȐȡIJȘıȘ� � �f t İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ� ,� ȠʌȩIJİ�Ș�ıȣȞȐȡIJȘıȘ� � �x
1f t dtµ́
¶ İȓȞĮȚ�
ʌĮȡĮȖȦȖȓıȚȝȘ� ȐȡĮ� țĮȚ� ıȣȞİȤȒȢ� ıIJȠ� ȓįȚȠ� įȚȐıIJȘȝĮ�� OʌȩIJİ�� Įʌȩ� IJȘ� ıȤȑıȘ� (2) ʌĮȓȡȞȠȣȝİ
� � � � � �x 0
x 0 x 0 x 0 x 0 1 1limf x lim x lim 2x lim f t dt 0 0 f t dt 0o o o o
� � � � ´ ´µ µ¶ ¶
țĮȚ�ȜȩȖȦ�IJȘȢ�ıȣȞȑȤİȚĮȢ�IJȘȢ�f � � � �x 0
f 0 limf x 0.o
H ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ıȣȞİȤȒȢ�
țĮȚ�įİȞ�ȝȘįİȞȓȗİIJĮȚ�ıİ�țĮȞȑȞĮ�ıȘȝİȓȠ�IJȠȣ� *.� ǼʌȠȝȑȞȦȢ��Ș�f įȚĮIJȘȡİȓ�ʌȡȩıȘ�ȝȠ�ıİ�țĮșȑȞĮ�Įʌȩ�IJĮ�įȚĮıIJȒȝĮIJĮ� � �, 0�f țĮȚ� � �0, .� f ǼʌȓıȘȢ, Įʌȩ�IJȘ�ıȤȑıȘ�(2) ȑȤȠȣȝİ
� � � �1
1f 1 1 2 f t dt 1 0 1 0 � � !µ́
¶
țĮȚ�ıȣȞİʌȫȢ � � � �f 1 f 1 1 0� � � � .
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 499�
ǹʌȩ�IJĮ�ʌĮȡĮʌȐȞȦ�ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ � �f x 0� ȖȚĮ�țȐșİ�� � �x , 0� �f
țĮȚ � �f x 0! ȖȚĮ�țȐșİ�� � �x 0, .� �f
ii) Aʌȩ�IJȘ�ıȤȑıȘ�(2) țĮȚ�ȝİ�ȕȐıȘ�IJȘȞ�ʌĮȡĮȖȦȖȚıȚȝȩIJȘIJĮ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ� � �x
1f t dtµ́
¶
ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ�ȝİ�
� � � � � � � � � � � �x x x
1 1 1f x x 2x f t dt 1 2x f t dt 2x f t dt
c c§ · § ·c cc � � �¨ ¸ ¨ ¸© ¹ © ¹
´ ´ ´µ µ µ¶ ¶ ¶
� � � �� � � � � �
x
1
f x1 2 f t dt 2xf x 2xf x
x � � �µ́
¶
1
� �1 2x f xx
§ · �¨ ¸© ¹
ȖȚĮ țȐșİ [ �� .
ǼʌȠȝȑȞȦȢ�� � �� � � �
f x 1 2x ȖȚĮ țȐșİ [ ĮijȠȪ I [ � ȖȚĮ țȐșİ [f x x
c � � z �� � �
Ȓ�ȚıȠįȪȞĮȝĮ
� �� � � �2ln f x ln x x ȖȚĮ țȐșİ [ � c c � �� (3)
z ȈIJȠ�įȚȐıIJȘȝĮ� � �, 0�f İȓȞĮȚ� x 0� țĮȚ�� � �f x 0� ��ȠʌȩIJİ�Ș�ıȤȑıȘ�(3) ȖȓȞİIJĮȚ
� �� �� � � �� �2ln f x ln x xc c� � � ȖȚĮ�țȐșİ�� x 0�
įȘȜĮįȒ � �� � � � 2
1ln f x ln x x c� � � � ȖȚĮ�țȐșİ�� x 0� ȝİ� 1c ��
īȚĮ� x 1 � ʌĮȓȡȞȠȣȝİ
� �� � 1 1 1ln f 1 ln1 1 c ln1 ln1 1 c c 1.� � � � � � � � �
ȅʌȩIJİ,
� �� � � � � �
� �� � � � � �
� �� � � � � �
� � � �
� � � �
2
2
x 1
2x 1
2x 1
2x 1
ln f x ln x x 1 ȖȚĮ țȐșİ [ ��
ln f x ln x ln e ȖȚĮ țȐșİ [ ��
ln f x ln xe ȖȚĮ țȐșİ [ ��
f x xe ȖȚĮ țȐșİ [ ��
f x xe ȖȚĮ țȐșİ [ �� �
�
�
�
�
� � � � � �f
� � � � � �f
� � � � �f
� � � � �f
� � �f
500 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
z ȈIJȠ�įȚȐıIJȘȝĮ� � �0, � f İȓȞĮȚ� x 0! țĮȚ� � �f x 0! ��ȠʌȩIJİ�Ș�ıȤȑıȘ�(3) ȖȓȞİIJĮȚ
� �� � � �2ln f x ln x x ȖȚĮ țȐșİ [ �cc � !
įȘȜĮįȒ � � 2
2ln f x ln x x c � � ȖȚĮ�țȐșİ�� x 0! ȝİ�� 2c .��
īȚĮ�� x 1 ʌĮȓȡȞȠȣȝİ � � 2 2 2ln f 1 ln1 1 c ln1 ln1 1 c c 1. � � � � � � �
ǼʌȠȝȑȞȦȢ�
� � � �
� � � �
� � � � � �
� � � �
2
2
2x 1
x 1
2x 1
ln f x ln x x 1 ȖȚĮ țȐșİ [ ��
ln f x ln x ln e ȖȚĮ țȐșİ [ ��
ln f x ln xe ȖȚĮ țȐșİ [ ��
f x xe ȖȚĮ țȐșİ [ �� �
�
�
�
� � � �f
� � � �f
� � �f
� � �f
z ȅ�ʌĮȡĮʌȐȞȦ�IJȪʌȠȢ�ȚıȤȪİȚ�țĮȚ�ȖȚĮ� x 0 ĮijȠȪ�įȓȞİȚ� � �f 0 0 .
DZȡĮ�
� � 2x 1f x xe � ȖȚĮ�țȐșİ�� x .�\
iii) Ǿ�İȟȓıȦıȘ� � �f x x ȚıȠįȪȞĮȝĮ�ȖȡȐijİIJĮȚ
� �2 2x 1 x 1
2x 1
2
xe x 0 x e 1 0
x 0 Ȓ H �x 0 Ȓ [ � �x 0 Ȓ [ � Ȓ [ ��
� �
�
� � �
�
� � � �
DZȡĮ��IJĮ�țȠȚȞȐ�ıȘȝİȓĮ�IJȘȢ� fC ȝİ�IJȘȞ�İȣșİȓĮ� y x İȓȞĮȚ�IJĮ�ıȘȝİȓĮ� � �A 1, 1� � ,
� �O 0, 0 țĮȚ�� � �B 1,1 .
iv) DzȤȠȣȝİ
� � � � � � � �2 2 2x 1 x 1 x 1 2
2 2x 1 2 x 1
f x xe x e x e x 1
e 2x e 0 ȖȚĮ țȐșİ [ �
� � �
� �
c ccc � � � � �
� ! ��
DZȡĮ�� Ș� ıȣȞȐȡIJȘıȘ� � f İȓȞĮȚ� ȖȞȘıȓȦȢ� ĮȪȟȠȣıĮ� țĮȚ� ıȣȞİʌȫȢ� 1 1� �� įȘȜĮįȒ� ĮȞIJȚ�ıIJȡȑȥȚȝȘ���ȁȩȖȦ�IJȘȢ�ıȣȝȝİIJȡȓĮȢ�IJȦȞ� fC țĮȚ� 1f
C � ȦȢ�ʌȡȠȢ�IJȘȞ�İȣșİȓĮ� y x Ș�
İȣșİȓĮ�ĮȣIJȒ�ȤȦȡȓȗİȚ�IJȠ�ȤȦȡȓȠ�ȍ��IJȠȣ�ȠʌȠȓȠȣ�IJȠ�İȝȕĮįȩ�ȗȘIJİȓIJĮȚ��ıİ�įȪȠ�ȤȦȡȓĮ� 1ȍ țĮȚ� 2ȍ ʌȠȣ�İȓȞĮȚ�ȓıĮ��ȐȡĮ�țĮȚ�ȚıİȝȕĮįȚțȐ��
ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 501�
ǹȞ� 1ȍ İȓȞĮȚ�IJȠ�ȤȦȡȓȠ�ʌȠȣ�ıȤȘȝĮIJȓȗȠȣȞ�Ș� fC țĮȚ�Ș��İȣșİȓĮ� y x , IJȩIJİ�ȑȤȠȣȝİ
� � � �1
1 1E ȍ I [ [ G[�
� �³
ȉȠ�ʌȡȩıȘȝȠ�IJȘȢ�įȚĮijȠȡȐȢ
� � � �2x 1f x x x e 1�� �
ıIJȠ�įȚȐıIJȘȝĮ� > @1, 1� ijĮȓȞİIJĮȚ�ıIJȠȞ�ʌĮȡĮțȐIJȦ�ʌȓȞĮțĮ��
ǼʌȠȝȑȞȦȢ�
� � � � � �0 1
11 0
E ȍ I [ [ G[ I [ [ G[�
� � �´ ´µ µ¶ ¶
� �� � � �� �
0 1
1 00 10 12 2x 1 x 1
1 001
f x x dx x f x dx
xe dx xdx xdx xe dx
�
� �
� �
� � �
� � �
´ ´µ µ¶ ¶
´ ´´ ´µ µµµ¶ ¶¶¶
2
0 10 12 2 2x 1 x 1
1 01 0
1 x x 1e e2 2 2 2
� �
� �
ª º ª ºª º ª º � � �« » « »« » « »¬ ¼ ¬ ¼¬ ¼ ¬ ¼
1 1 1 1 1 10 02e 2 2 2 2 2e
� � � � � � �
1 IJ�ȝ�e
DZȡĮ��IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ȍ�İȓȞĮȚ
� � � �12Ǽ ȍ �Ǽ ȍ IJ�ȝ�e
x 1� 0 1
x � � 2x 1e 1� � � �
� �f x x� � �
0
0 0
0 0 0
502 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ�107�
DzıIJȦ�ȝȚȖĮįȚțȩȢ�ĮȡȚșȝȩȢ�z IJȑIJȠȚȠȢ��ȫıIJİ
� �z 2 țĮȚ P ] � , t
i) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ƞ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�IJȘȢ�İȚțȩȞĮȢ�IJȠȣ�z ıIJȠ ȝȚȖĮįȚ�
țȩ�İʌȓʌİįȠ�İȓȞĮȚ�Ș�ȖȡĮijȚțȒ�ʌĮȡȐıIJĮıȘ�ȝȚĮȢ�ıȣȞȐȡIJȘıȘȢ�f ��ȆȠȚȠȢ�İȓ�
ȞĮȚ�Ƞ�IJȪʌȠȢ�IJȘȢ�f ;
ii) NĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�ȠȜȠțȜȒȡȦȝĮ� � �2
2f x dx .
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