apply differential

31
บทที่ 3 การประยุกตอนุพันธ Calculus 1 1 (Differential) y = f (x) , x Δx f(x) Δx f(x) x d f (x) dy dx = Δx dx x d f (x) = dy = f(x) dx กกกก d c f (x) = c d f (x) dx n = nx n-1 dx บททีบททีบททีบทที่ 3 3 3 3 การประยุกตอนุพันธ การประยุกตอนุพันธ การประยุกตอนุพันธ การประยุกตอนุพันธ 1. 2. 3. 4. 5.

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Apply Differential

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Page 1: Apply Differential

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����

Page 2: Apply Differential

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

Page 3: Apply Differential

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

Page 4: Apply Differential

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

Page 5: Apply Differential

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

Page 6: Apply Differential

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

Page 7: Apply Differential

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+�

Page 8: Apply Differential

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

Page 9: Apply Differential

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

Page 10: Apply Differential

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

Page 11: Apply Differential

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

Page 12: Apply Differential

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

Page 13: Apply Differential

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

Page 14: Apply Differential

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

Page 15: Apply Differential

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

Page 16: Apply Differential

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

Page 17: Apply Differential

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

Page 18: Apply Differential

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(��

Page 19: Apply Differential

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

Page 20: Apply Differential

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

Page 21: Apply Differential

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

Page 22: Apply Differential

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

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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

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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

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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

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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

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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

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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

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Calculus 1

29

GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG

������ �� 4 6(����#1��1�4������ก�(7(�������ก@97)�.��/��+ 10,800 ��(����� 6+-7(���

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GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG

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บทที่บทที่บทที่บทที่ 3333 การปรการปรการปรการประยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธะยกุตอนพุนัธ

Calculus 1

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GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG

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NNOOTTEE GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG