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บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
1
���������� � (Differential) ���ก�� ����!"#$��%�&�'�% %�#(������������ �
����� ��� y = f (x) �� ����ก������������������ ��!�, x #�$ ∆x �� �%&��'�%()�*+ , �(�-ก
f′ (x) • ∆x '��.����)���� ��!�/�� f(x) ��� x �/�-�#��+�'- d f (x) 1(2� dy
����� ��� dx = ∆x �(�-ก dx '�� .����)���� ��!�/�� x 34���� ���/%&��'�%()�*+ ,
%�ก�)-��%$5+�'�� d f (x) = dy = f′ (x) dx
ก�(1�.����)���� ��!�/�����ก�����&�5+�6+-*��78�(�����+�-'ก�9ก�(1���� ��!� ���� d c f (x) = c d f (x) dxn = nxn-1 dx
บทที ่บทที ่บทที ่บทที ่3 3 3 3 การประยกุตอนพุนัธการประยกุตอนพุนัธการประยกุตอนพุนัธการประยกุตอนพุนัธ
)���ก����*#�+,
1. .����)���� ��!� 2. .'���(@'.'���(�� 3. ���(�7�� ��!� 4. .'��1��-/����� ��!�*�����(/�.C)� 5. .��78�7�+#�$.����&�7�+/�����ก����
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
2
������ �� 1 %�1�.����)���� ��!�/�����ก���� y = 2x3 - 4x2 - 5x + 7 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
������ �� 2 %�1�.����)���� ��!�/�� ƒ(X) = ln xsin1xsin1
−
+
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
3
ก�� ����!"#$��%�&
%�ก�)-��5+�'�� dy = ƒ′(x) dx ���*1� dx �� �.�������-�#���/�� x #�$ dx �� �.�������-�#���/�� y ��%1�.��6+-�($��C%�ก��������� ��!�+����'�-������5���L
������ �� 1 %�1�.�� 3
2
)5.28( 6+-*����������� ��!� GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 2 #���6�1$ก��#���1�4��/-�-��'��2��5+�(�9.'��(���6+-���(�N��� )��/4L�%�ก
12.5 3�. �� � 12.65 3�. %��($��C.�� 2L����/��#���6�1$ก������ )��/4L�%�ก�+)�
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
4
������ �� 3 ��ก&�5(%�กก�(/�-7)�.����)+1�4���-8�*�(8�/�����ก���� P = 242x - 3x2 + 45 6+-*1� x �� �(�.�7)�.�� %�1�7�'�� )��/��ก&�5(����)L����� )��(�.�/�-/4L� %�ก�)L��$ 40 9�� �� � 43 9��
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 4 1�96�1$(8��8ก9�Nก� '�+.'��ก'���R�-*�#���$+���-�'+����$ 1 ���
��.'��1�� 0.25 �)L' %��($��C�()���(6�1$���*��*�ก�(�&�1�9 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
5
S1
S2
C B A
lim
�'�%��-'�'�%���� (Velocity and Acceleration)
)%�(C�ก�(�.�2�������� �#�'�(�/��'���� ���(8����5���L %�ก(8�ก&�1�+*1� '�����.�2������%�ก A 5� B �� �($-$��� s1 1��'- #�$*���'�� t1 1��'- (�'��) '�����.�2������%�ก A 5� B �� �($-$��� s2 1��'- #�$*���'�� t2 1��'- (�'��)
+����L� '�����.�2������%�ก B 5� C �� �($-$��� s2 X s2 = ∆s
#�$*���'��%�ก B 5� C ����ก�9 t2 X t2 = ∆t ���ก&�1�+*1�'�����.�2�������� �#�'�(�5����7�ก�(/�����ก���� S = f(t) #��' %$5+�
s + ∆s = f(t + ∆t)
∆s = f(t + ∆t) - s
= f(t + ∆t) - f(t) 6+-��� s .2� ($-$������'�+%�ก%�+�()�����5�-��%�+7�+���-/��'������ �#�'�(� t .2� �'�������9%�ก%�+�()�����/��ก�(�.�2������5�-��%�+7�+���-/��'���� %�ก��L���L�1�+���5+� )%�(C��� 7���(�1�.'���(@'�Z���-/��'��������.�2������%�ก B 5� C 5+� +����L VBC =
= ts∆
∆
= t
)t(f)tt(f∆
−∆+ (1)
%�ก (1) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(@'�Z���- (VBC) %$ก��-�� �.'���(@'*�/C$*+/C$1�4�� (V) +����L
∴ v = t
)t(f)tt(f∆
−∆+
($-$������'�����.�2������%�ก B 5� C
�'�����'����*���.�2������%�ก B 5� C
∆t 0
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
6
v1 v2
∆t 0 lim
���5�*1� )%�(C�(8����5���L t1 t2 B C %�ก(8� ��2���'�� t1 '�����.�2������+�'-.'���(@' v1 ��2���'�� t2 '�����.�2������+�'-.'���(@' v2
∴ .'���(@'%�ก%�+ B 5� C �����-�5� v2 - v1 = ∆v
#�$%�ก B 5� C *���'��5� t2 - t1 = ∆t
%�ก (2) ; v = f′(t)
∴ v + ∆v = f′(t + ∆t )
∆v = f′(t + ∆t ) - v
= f′(t + ∆t ) - f′(t) %�ก��L�1�+���5+� )%�(C��� 7���(�1�.���(���Z���- (aBC) /��'�����.�2������ %�ก B 5� C 5+�+����L aBC =
= tv∆
∆
= t
)t(f)tt(f∆
′−∆+′
%�ก (3) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(���Z���- (aBC) %$ก��-5��� �.'���(@'*�/C$*+/C$1�4�� (a) +����L
∴ a = t
)t(f)tt(f∆
′−∆+′
= f″(t)
.'���(@'��������-�%�ก B 5� C
�'�����*��5�%�ก B 5� C
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
7
���� ���'�����)L�1�4���.�2�������� �#�'�7���(�5����7�ก�(/�����ก���� s = f(t) 6+-��� s �� �($-$������'�����.�2������5� #�$ t �� ��'�����'����*��*�ก�(�.�2������#��'
.'���(@'/��'������2���'�� t *+ , .2�
a = f′(t) .'���(��/��'������2���'�� t *+ , .2�
a = v′ = f′(t) ������ �� 1 '�����)L�1�4���.�2�������� �#�'�(� 6+-��.'��7�� ��!�($1'���($-$��� s ก�9
�'�� t +��7�ก�( s = 6t2 - 2t3 %�1�($-$��� .'���(@' #�$.'���(�� /��ก�(�.�2������/��'������L ��2���'�� t = 1 ')����
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 2 '�����)L�1�4���.�2��������ก%�ก%�+ A �� �#�'�(� +�'-.'���(@'.����
4 ���(/')���� �������ก 2 ')���� '������ก�)L�1�4��ก@�.�2��������ก%�ก%�+ A 5�
*��)N����+�-'ก��+�'-7�ก�(ก�(�.�2������ s = 5t + 2t23
6+-��� s �� �
($-$��� #�$ t �� ��'�� %�1�'��'�����)L�1�����.'���(@'#�$.'���(������5( ��2���.�2������5����'�����)L�#(ก
'�����.�2������%�ก A 5� B �� �($-$��� s1 1��'- #�$*���'�� t1 1��'- (�'��) '�����.�2������%�ก A 5� B �� �($-$��� s2 1��'- #�$*���'�� t2 1��'- (�'��)
+����L� '�����.�2������%�ก B 5� C �� �($-$��� s2 X s2 = ∆s
#�$*���'��%�ก B 5� C ����ก�9 t2 X t2 = ∆t ���ก&�1�+*1�'�����.�2�������� �#�'�(�5����7�ก�(/�����ก���� S = f(t) #��' %$5+�
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
8
lim
v1 v2
s + ∆s = f(t + ∆t)
∆s = f(t + ∆t) - s
= f(t + ∆t) - f(t) 6+-��� s .2� ($-$������'�+%�ก%�+�()�����5�-��%�+7�+���-/��'������ �#�'�(� t .2� �'�������9%�ก%�+�()�����/��ก�(�.�2������5�-��%�+7�+���-/��'���� %�ก��L���L�1�+���5+� )%�(C��� 7���(�1�.'���(@'�Z���-/��'��������.�2������%�ก B 5� C 5+� +����L VBC =
= ts∆
∆
= t
)t(f)tt(f∆
−∆+ (1)
%�ก (1) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(@'�Z���- (VBC) %$ก��-�� �.'���(@'*�/C$*+/C$1�4�� (V) +����L
∴ v = t
)t(f)tt(f∆
−∆+
���5�*1� )%�(C�(8����5���L t1 t2 B C %�ก(8� ��2���'�� t1 '�����.�2������+�'-.'���(@' v1 ��2���'�� t2 '�����.�2������+�'-.'���(@' v2
∴ .'���(@'%�ก%�+ B 5� C �����-�5� v2 - v1 = ∆v
#�$%�ก B 5� C *���'��5� t2 - t1 = ∆t
%�ก (2) ; v = f′(t)
∴ v + ∆v = f′(t + ∆t )
∆v = f′(t + ∆t ) - v
($-$������'�����.�2������%�ก B 5� C
�'�����'����*���.�2������%�ก B 5� C
∆t 0
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
9
∆t 0 lim
= f′(t + ∆t ) - f′(t) %�ก��L�1�+���5+� )%�(C��� 7���(�1�.���(���Z���- (aBC) /��'�����.�2������ %�ก B 5� C 5+�+����L aBC =
= tv∆
∆
= t
)t(f)tt(f∆
′−∆+′
%�ก (3) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(���Z���- (aBC) %$ก��-5��� �.'���(@'*�/C$*+/C$1�4�� (a) +����L
∴ a = t
)t(f)tt(f∆
′−∆+′
= f″(t)
���� ���'�����)L�1�4���.�2�������� �#�'�7���(�5����7�ก�(/�����ก���� s = f(t) 6+-��� s �� �($-$������'�����.�2������5� #�$ t �� ��'�����'����*��*�ก�(�.�2������#��'
.'���(@'/��'������2���'�� t *+ , .2�
a = f′(t) .'���(��/��'������2���'�� t *+ , .2�
a = v′ = f′(t) ������ �� 1 '�����)L�1�4���.�2�������� �#�'�(� 6+-��.'��7�� ��!�($1'���($-$��� s ก�9
�'�� t +��7�ก�( s = 6t2 - 2t3 %�1�($-$��� .'���(@' #�$.'���(�� /��ก�(�.�2������/��'������L ��2���'�� t = 1 ')����
'�����.�2������%�ก A 5� B �� �($-$��� s1 1��'- #�$*���'�� t1 1��'- (�'��)
.'���(@'��������-�%�ก B 5� C
�'�����*��5�%�ก B 5� C
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
10
lim
v1 v2
'�����.�2������%�ก A 5� B �� �($-$��� s2 1��'- #�$*���'�� t2 1��'- (�'��)
+����L� '�����.�2������%�ก B 5� C �� �($-$��� s2 X s2 = ∆s
#�$*���'��%�ก B 5� C ����ก�9 t2 X t2 = ∆t ���ก&�1�+*1�'�����.�2�������� �#�'�(�5����7�ก�(/�����ก���� S = f(t) #��' %$5+�
s + ∆s = f(t + ∆t)
∆s = f(t + ∆t) - s
= f(t + ∆t) - f(t) 6+-��� s .2� ($-$������'�+%�ก%�+�()�����5�-��%�+7�+���-/��'������ �#�'�(� t .2� �'�������9%�ก%�+�()�����/��ก�(�.�2������5�-��%�+7�+���-/��'���� %�ก��L���L�1�+���5+� )%�(C��� 7���(�1�.'���(@'�Z���-/��'��������.�2������%�ก B 5� C 5+� +����L VBC =
= ts∆
∆
= t
)t(f)tt(f∆
−∆+ (1)
%�ก (1) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(@'�Z���- (VBC) %$ก��-�� �.'���(@'*�/C$*+/C$1�4�� (V) +����L
∴ v = t
)t(f)tt(f∆
−∆+
���5�*1� )%�(C�(8����5���L t1 t2 B C %�ก(8� ��2���'�� t1 '�����.�2������+�'-.'���(@' v1 ��2���'�� t2 '�����.�2������+�'-.'���(@' v2
∴ .'���(@'%�ก%�+ B 5� C �����-�5� v2 - v1 = ∆v
($-$������'�����.�2������%�ก B 5� C
�'�����'����*���.�2������%�ก B 5� C
∆t 0
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
11
∆t 0 lim
#�$%�ก B 5� C *���'��5� t2 - t1 = ∆t
%�ก (2) ; v = f′(t)
∴ v + ∆v = f′(t + ∆t )
∆v = f′(t + ∆t ) - v
= f′(t + ∆t ) - f′(t) %�ก��L�1�+���5+� )%�(C��� 7���(�1�.���(���Z���- (aBC) /��'�����.�2������ %�ก B 5� C 5+�+����L aBC =
= tv∆
∆
= t
)t(f)tt(f∆
′−∆+′
%�ก (3) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(���Z���- (aBC) %$ก��-5��� �.'���(@'*�/C$*+/C$1�4�� (a) +����L
∴ a = t
)t(f)tt(f∆
′−∆+′
= f″(t)
���� ���'�����)L�1�4���.�2�������� �#�'�7���(�5����7�ก�(/�����ก���� s = f(t) 6+-��� s �� �($-$������'�����.�2������5� #�$ t �� ��'�����'����*��*�ก�(�.�2������#��'
.'���(@'/��'������2���'�� t *+ , .2�
a = f′(t) .'���(��/��'������2���'�� t *+ , .2�
a = v′ = f′(t)
.'���(@'��������-�%�ก B 5� C
�'�����*��5�%�ก B 5� C
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
12
lim
������ �� 1 '�����)L�1�4���.�2�������� �#�'�(� 6+-��.'��7�� ��!�($1'���($-$��� s ก�9 �'�� t +��7�ก�( s = 6t2 - 2t3 %�1�($-$��� .'���(@' #�$.'���(�� /��ก�(�.�2������/��'������L ��2���'�� t = 1 ')����
'�����.�2������%�ก A 5� B �� �($-$��� s1 1��'- #�$*���'�� t1 1��'- (�'��) '�����.�2������%�ก A 5� B �� �($-$��� s2 1��'- #�$*���'�� t2 1��'- (�'��)
+����L� '�����.�2������%�ก B 5� C �� �($-$��� s2 X s2 = ∆s
#�$*���'��%�ก B 5� C ����ก�9 t2 X t2 = ∆t ���ก&�1�+*1�'�����.�2�������� �#�'�(�5����7�ก�(/�����ก���� S = f(t) #��' %$5+�
s + ∆s = f(t + ∆t)
∆s = f(t + ∆t) - s
= f(t + ∆t) - f(t) 6+-��� s .2� ($-$������'�+%�ก%�+�()�����5�-��%�+7�+���-/��'������ �#�'�(� t .2� �'�������9%�ก%�+�()�����/��ก�(�.�2������5�-��%�+7�+���-/��'���� %�ก��L���L�1�+���5+� )%�(C��� 7���(�1�.'���(@'�Z���-/��'��������.�2������%�ก B 5� C 5+� +����L VBC =
= ts∆
∆
= t
)t(f)tt(f∆
−∆+ (1)
%�ก (1) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(@'�Z���- (VBC) %$ก��-�� �.'���(@'*�/C$*+/C$1�4�� (V) +����L
∴ v = t
)t(f)tt(f∆
−∆+
($-$������'�����.�2������%�ก B 5� C
�'�����'����*���.�2������%�ก B 5� C
∆t 0
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
13
v1 v2
∆t 0 lim
���5�*1� )%�(C�(8����5���L t1 t2 B C %�ก(8� ��2���'�� t1 '�����.�2������+�'-.'���(@' v1 ��2���'�� t2 '�����.�2������+�'-.'���(@' v2
∴ .'���(@'%�ก%�+ B 5� C �����-�5� v2 - v1 = ∆v
#�$%�ก B 5� C *���'��5� t2 - t1 = ∆t
%�ก (2) ; v = f′(t)
∴ v + ∆v = f′(t + ∆t )
∆v = f′(t + ∆t ) - v
= f′(t + ∆t ) - f′(t) %�ก��L�1�+���5+� )%�(C��� 7���(�1�.���(���Z���- (aBC) /��'�����.�2������ %�ก B 5� C 5+�+����L aBC =
= tv∆
∆
= t
)t(f)tt(f∆
′−∆+′
%�ก (3) ���*1��'�����'����*��*�ก�(�.�2��������.�����-��ก (∆t 0) .'���(���Z���- (aBC) %$ก��-5��� �.'���(@'*�/C$*+/C$1�4�� (a) +����L
∴ a = t
)t(f)tt(f∆
′−∆+′
= f″(t)
.'���(@'��������-�%�ก B 5� C
�'�����*��5�%�ก B 5� C
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
14
���� ���'�����)L�1�4���.�2�������� �#�'�7���(�5����7�ก�(/�����ก���� s = f(t) 6+-��� s �� �($-$������'�����.�2������5� #�$ t �� ��'�����'����*��*�ก�(�.�2������#��'
.'���(@'/��'������2���'�� t *+ , .2�
a = f′(t) .'���(��/��'������2���'�� t *+ , .2�
a = v′ = f′(t) ������ �� 1 '�����)L�1�4���.�2�������� �#�'�(� 6+-��.'��7�� ��!�($1'���($-$��� s ก�9
�'�� t +��7�ก�( s = 6t2 - 2t3 %�1�($-$��� .'���(@' #�$.'���(�� /��ก�(�.�2������/��'������L ��2���'�� t = 1 ')����
������ �� 3 6-�ก���1)�/4L�5�*���ก�N���#�'+)��+�'-.'���(@' 96 ���/')���� 6+-ก���
1)��.�2������5����7�ก�( h = 96t + 16t2 6+-��� h �� �.'��78� (���) /��ก���1)�%�ก 2L�+)� t �� ��'�� (')����) ��9%�ก/C$���6-�ก���1)�/4L�5� %�1� 1. .'���(@'#�$.'���(��/��ก���1)�/4L�5� #�$*1� )%�(C�+�'-'��ก���1)�ก&����
�.�2������/4L�1(2��� 2. ก���1)���-�-8�*���ก�N�� ��'���������5( 3. ก���1)�/4L�5�78�7�+ก�����
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
15
∆t 0 ∆t 0
∆t 0 ∆t 0
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
��4��)�%��5 � (Related Rate)
���(�7�� ��!�%$����กbC$�����+�-'ก��ก�9ก�(1�.'���(@'/��'���� .2� .'���(@'�� �ก�(��(�-9���-9($-$���ก�9�'��#�����(�7�� ��!�%$�� �ก�(��(�-9���-9�()��C*+ , ก@5+�ก�9�'������
� ������� ��ก�����
*1� 2L���� A �����-�#���5�����'�� t ���.'��7�� ��!�/�� ���ก���� A = f(t) ���(�ก�(�����-�#���/��+/�� 2L���� ��2���'�� t *+ , 1�5+�+����L
lim tA∆
∆= lim
t)t(f)tt(f
∆
−∆+ =
dtdA
� ����������ก�����
*1��()���( V �����-�#����()��C5�����'�� t ���.'��7�� ��!�/�����ก���� V = f(t) ���(�ก�(�����-�#����()���( ��2���'�� t *+ , 1�5+�+����L
lim tV∆
∆ = lim
t)t(f)tt(f
∆
−∆+ =
dtdV
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
16
������ �� 1 ����L&�ก������1�4�� (�N��/������ )��/4L�+�'-���(� 0.01 ���( / ���� .'�� -�'/������+��+�'-���(� 0.02 ���( / ���� %�1�'�� 2L��)'/�����%$�����-�#���5�+�'-���(�����5( *�/C$��������.'��-�' 6 ���( #�$(�N�� 0.2 ���(
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 2 ��c+�L&��/����� %$5+��L&� V �)�( R�-*��'�� t ')���� .'��7�� ��!�($1'���
V ก�9 t ก&�1�+7�ก�( V = 2t - 1t− %�1�'����2����c+�L&�.(9 10 ')���� %$���L&�*����ก���)�( #�$*�/C$��L�L&�51��/�����+�'-���(��(@'����5(
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
17
• °
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
������ �� 3 %�ก(8� ก #�$ / -2�1���ก�� 80 ���( ก #�$ / �()����ก�+)� (���ก�� 5�����)N�8กN( 6+- ก �+)�+�'-.'���(@' 25 ���(/���� #�$ / �+)�+�'-.'���(@' 20 ���( / ���� %�1����(�ก�(�����-�#���($-$1���($1'��� ก ก�9 / ��2����L�.8��+)�5�5+� 2 ����
ก / GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
18
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
�'�% %�#(����� �75����(��&�4
����� .'�����/��ก(�����%�+ P .2�.'�����/���7��7����7ก(�����%�+ P ��2�� P �� �%�+*+ , 9�ก(��
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
19
h 0
%�ก(8� 1.3 *1� y = f(x) �� ����ก����/��ก(��*+ , %�+ P(x1, y1) #�$ Q(x2, y2) �� � 2 %�+*+ , �������ก��9�ก(��
.'�����/�� PQ = 12
12
xxyy
−
−
= 11
12
xx)x(f)x(f
−
− GG.(1)
*1� x2 - x1 = h 1(2� x2 = x1 + h
#��*� (1) ; .'�����/�� PQ = h
)x(f)hx(f 11 −+
�����2��� Q �/��*ก�� P ���#�'ก(�� %$�&�*1�($-$ h ��.���/��*ก�� O �7����+ก(�� PQ %$1����/��1��7��7����7 ก(�� L �&�*1�.'�����/���7����+ก(�� PQ ��.���/��*ก��.'�����/���7��7����7ก(�� L
��L�.2� .'�����/�� L = lim h
)x(f)hx(f 11 −+ = f′(x1)
���� .'�����/�����ก���� f(x) ���%�+ (x1, y1) 1(2�.'�����/���7��7����7ก(�����%�+ (x1, y1)
1�5+�%�ก ��� ��!����+�9 1 /�� f ��� x1 1(2� f′(x1) ������� ������ �� 1 %�1�.'�����/��ก(�� f(x) = 2 - 3x2 ���%�+ (-2, -10) GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
20
������ �� 2 %�1�.'�����/��ก(�� f(x) = x1
x−
���%�+ (2, -2)
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
������ �� 3 %�1�.'�����/��ก(�� y = x31
5−
���%�+ (2, -1)
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
21
�� ����!��ก��"
��2��.&��'C1�.'�����/���7��7����7ก(��#��' �������ก�(1�7�ก�(/���7��7����7ก(�� %$1�5+�%�ก7�ก�(���5���L
y - y1 = m(x - x1) = f′ (x1)(x-x1) ��2�� (x1, y1) .2� %�+7����7ก(�� m .2� .'�����/���7��7����7ก(�����%�+ (x1, y1) ������ �� 4 %�1�7�ก�(/���7��7����7ก(�� f (x) = 2x3 - 5x + 10 ���%�+ (0, 1) GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
������ �� 5 %�1�7�ก�(/���7��7����7ก(�� y = 2x1x − ���%�+ (0, 0) GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
22
������ �� 6 ก&�1�+*1� y = xx3
x 2
3
−− %�1�%�+7����7#�$�7��7����7ก(�������
.'��������%�+7����7����ก�9 2 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
���)+�)"������489�)"(��:;�ก���
��� ��!�/�����ก����*�����(/�.C)� .2� .'�����/���7��7����7ก(��/�����ก���� ���.'��������%�+*+%�+1�4���� �N8�-� 1��-.'��'��%�+��L���%%$�� �%�+78�7�+ 1(2�%�+��&�7�+6+- )%�(C�%�ก�)-�� #�$�gbh����5���L �)-�� 1 ������ก���� f ��.'�������2���*���'����ก&�1�+*1� #�$�� x0 �-8�*���'����
ก&�1�+*1�#��'%$�(�-ก f ��.��78�7�+7�� ��!����%�+ x = x0 ��2���� q > 0
34���&�*1� f(x) ≤ f(x0) ��ก.��/�� x *���'� (x0 - q, x0 + q) ∩ Df �)-�� 2 ������ก���� f ��.'�������2���*���'����ก&�1�+*1� #�$�� x0 �-8�*���'����
ก&�1�+*1�#��'%$�(�-ก'�� f ��.����&�7�+7�� ��!����%�+ x = x0 ��2���� q > 0 34��
�&�*1� f(x) ≥ f(x0) ��ก.��/�� x ����-8�*���'� (x0 - q, x + q0) ∩ Df
�)-�� 3 ������ก���� f �����2��� #�$�� x0 ∈ Df %$�(�-ก'�� f ��.��78�7�+7��98(C�
���%�+ x0 ��� f(x0) ≥ f(x) ��ก.��/�� x ����-8�*� Df
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
23
�)-�� 4 ������ก���� f �����2��� #�$�� x0 ∈ Df %$�(�-ก'�� f ��.����&�7�+7��98(C�
���%�+ x0 ��� f(x0) ≤ f(x) ��ก.��/�� x ����-8�*� Df �gbh�9� ��� f(x) 1�.��5+�7&�1(�9��ก.��/�� x *� [a , b] #�$���ก������.��78�7�+
1(2�.����&�7�+7�� ��!����%�+ x = c 6+-��� a < c < b #�$��� ��!�/�����ก����
1�.��5+���� x = c #��' %$5+� f′ (c) = 0 %��ก&�ก��%�'���(���)%����*���)����� +, -)�&. �����+,��)��%�/��
��2��ก&�1�+���ก���� y = f(x) �(�%$1�.��78�7�+7�� ��!� 1(2�.����&�7�+7�� ��!�/�����ก����5+����/�L�������5���L
1. 1�.�� dxdy
2. ก&�1�+*1� dxdy
= 0 #�$1�.�� x 7���)'�� x = x0
3. �('%7�9+8'���� �.��78�7�+7�� ��!� 1(2�.����&�7�+7�� ��!� �&�5+�+����L
��2�� x = x0 34���&�*1� dxdy
= 0
���#�� x < x0 , #�$ x > x0
3.1 ��� x < x0 �&�*1� dxdy
> 0
#�$ x > x0 �&�*1� dxdy
< 0
∴ f ��.��78�7�+7�� ��!����%�+ x = x0 y0 �� �.��78�7�+7�� ��!� #�$%�+ (x0 , y0) �� �%�+78�7�+7�� ��!�
3.2 ��� x < x0 �&�*1� dxdy
< 0
#�$ x > x0 �&�*1� dxdy
> 0
∴ f ��.����&�7�+7�� ��!����%�+ x = x0 y0 �� �.����&�7�+7�� ��!� #�$%�+ (x0 , y0) �� �%�+��&�7�+7�� ��!�
3.3 ��� x < x0 �&�*1� dxdy
< 0
#�$ x > x0 �&�*1� dxdy
< 0
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
24
∴ f ��%�+�����-��'�����%�+ x = x0 #�$%�+ (x0 , y0) �� �%�+�����-��'��
3.4 ��� x < x0 �&�*1� dxdy
> 0
#�$ x > x0 �&�*1� dxdy
> 0
∴ f ��%�+�����-��'�����%�+ x = x0 #�$%�+ (x0 , y0) �� �%�+�����-��'�� ������ �� 1 %�1�.��78�7�+1(2�.����&�7�+7�� ��!�/�����ก���� y = 4x - x2 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 2 %�1�.��78�7�+1(2�%�+��&�7�+7�� ��!�/�����ก���� y = 2x3 - 24x + 5 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
25
%��ก&�ก��%�'���(���)����� +, %��'����*���)����� +, -)�&. �����+,��)�� ���� ��2��ก&�1�+���ก���� y = f(x) �(�%$1�.��78�7�+7�� ��!� 1(2� .����&�7�+7�� ��!�/�����ก����5+����/�L�������5���L
1. 1�.�� f′ (x)
2. ก&�1�+ f′ (x) = 0 #��' 1�.�� x 7���)'�� x = x0 3. �('%7�9+8'���� �.��78�7�+7�� ��!� 1(2�.��78�7�+7�� ��!� �&�5+�+����L
3.1 %�+ (x0 , y0) �� �%�+78�7�+7�� ��!� ��2�� f″ (xo) < 0
3.2 %�+ (x0 , y0) �� �%�+��&�7�+7�� ��!� ��2�� f″ (xo) > 0
3.3 ��� f″ (x0) = 0 5��7���(��+7�9')!���L5+� ����*���+7�96+-*�� ��� ��!����+�9���1�4��
������ �� 3 %�1�.��78�7�+/�����ก���� f (x) = -x2 + 3x + 5 9���'� [-1 , 3] GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
26
������ �� 4 %�1�.��78�7�+1(2�.����&�7�+7�� ��!�/�����ก���� y = x3 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 5 %�1�.��78�7�+7�� ��!�/�� f (x) = 3x4 + 4x3 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
27
ก����0��ก�,ก��%�'���(���)1�0'����*���)
/�L����/��ก�(#ก���q1�6%�-��($-�ก�� ก�(1�.��78�7�+ #�$.����&�7�+ ��+�����5���L /�L���� 1 *1��/�-�(8��($ก�9 ����� �5�5+� /�L���� 2 ก&�1�+��'��กb(*1�ก�9�()��C���� , �����*���q1� /�L���� 3 ��2�ก�()��C�������ก�(1�.��78�78+ 1(2�.����&�7�+ #��'7(������ก����/��
�()��C+��ก���'*1��-8�*� %��/���()��C�2�� , /�L���� 4 ������ก�������7(���/4L���1��-��'#�( �&�*1���2�ก��'#�(�+��-'6+-��N�-
��2���5/%�ก��q1� /�L���� 5 #��'*��')!�1�.��78�7�+1(2���&�7�+.���������ก�( ������ �� 1 %�1�%&��'�7��%&��'�34��('�ก������ก�9 30 #�$��9'ก/��ก&����7��/��
%&��'�1�4��ก�97)9����/����ก%&��'�1�4�� ��.�����-���7�+ GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
28
������ �� 2 ���#���ก($+�b(8�7���1���-����Z�กก'��� 10 �)L' -�' 16 �)L' �������ก�(�&� ก����5����r��c+ 6+-ก�(��+���ก($+�b��ก�� �(8�7���1���-�%���(�7 #��' �9/4L�5��� �ก���� %�1�/��+/��(8�7���1���-�%���(�7�����+�)L� 34���&�*1�ก����*9��L���()���(��ก���7�+
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ������ �� 3 ��'���8�1�4�����'+1���-�' 600 '� �������/4�����(�L'�����6+-#9���������ก
�� � 2 #�������ก�� #�$*1�(�L'+���*�/���ก�9+�����ก+���*++���1�4�� %�1�'��%$����(�L'�-���5(%4�%$5+� 2L����('� 2 #�����ก���7�+
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
29
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
������ �� 4 6(����#1��1�4������ก�(7(�������ก@97)�.��/��+ 10,800 ��(����� 6+-7(���
ก&�# �� �-� 3 +��� (+���1����� ��4ก6(����) .��*��%��-*�ก�(7(���ก&�# �+���/��� (�.� 200 9�������� #�$.��*��%��-*�ก�(7(���ก&�# �+���1���(�.� 300 9�������� %�1�'��%$����7(���ก&�# ��-���5( %4�%$�&�*1��7�-.��*��%��-���-���7�+
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ
Calculus 1
30
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
บทท่ี 3 การประยุกตอนุพันธ
Calculus 1
31
NNOOTTEE GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
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