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Prévia do material em texto

0 D W H P i W L F D
A u t o r i a : C a r l o s H e n r i
q u e D i a s
Te m a 0 8
A p l i c a ç ã o d a D e r i v a d a n o E s t u d o d a s F u n ç õ e s d a s Á r e a s E c o n ô m i c a s e 
A d m i n i s t r a t i v a s
7 H P D � � �
A p l i c a ç ã o d a D e r i v a d a n o E s t u d o d a s F u n ç õ e s d a s Á r e a s E c o n ô m i c a s e A d m i n i s t r a t i v a s
A u t o r i a : C a r l o s H e n r i q u e D i a s
C o m o c i t a r e s s e d o c u m e n t o :
D I A S , C a r l o s H e n r i q u e . M a t e m á t i c a : A p l i c a ç ã o d a D e r i v a d a n o E s t u d o d a s F u n ç õ e s d a s Á r e a s E c o n ô m i c a s e A d m i n i s t r a t i v a s . C a d e r n o d e 
A t i v i d a d e s . A n h n a g u e r a P u b l i c a ç õ e s : V a l i n h o s , 2 0 1 4 .
Í n d i c e
‹ � � � � � � $ Q K D Q J X H U D � ( G X F D F L R Q D O � � 3 U R L E L G D � D � U H S U R G X o m R � ¿ Q D O � R X � S D U F L D O � S R U � T X D O T X H U � P H L R � G H � L P S U H V V m R � � H P � I R U P D � L G r Q W L F D � � U H V X P L G D � R X � P R G L ¿ F D G D � H P � O t Q J X D �
p o r t u g u e s a o u q u a l q u e r o u t r o i d i o m a .
P á g . 1 3
P á g . 1 4 P á g . 1 4
P á g . 1 3
P á g . 1 0P á g . 9
A C O M PA N H E N A W E B
P á g . 3
C O N V I T E À L E I T U R A
P á g . 4
P O R D E N T R O D O T E M A
�
C o n t e ú d o 
N e s t a a u l a , v o c ê e s t u d a r á :
• O c o n c e i t o d e c u s t o m a r g i n a l .
• A e s t i m a t i v a d o c u s t o m a r g i n a l a p a r t i r d o u s o d a d e r i v a d a .
• O s c o n c e i t o s d e r e c e i t a e l u c r o m a r g i n a l .
• A e s t i m a t i v a d a r e c e i t a e d o l u c r o m a r g i n a l a p a r t i r d o u s o d a d e r i v a d a .
• 2 � V L J Q L ¿ F D G R � G D � I X Q o m R � P D U J L Q D O � H P � X P � S U R E O H P D � S U i W L F R �
• 2 � S R Q W R � G H � P i [ L P R � R X � P t Q L P R � G H � X P D � I X Q o m R � T X D G U i W L F D � D � S D U W L U � G R � F i O F X O R � G D V � G H U L Y D G D V �
H a b i l i d a d e s 
$ R � ¿ Q D O � � Y R F r � G H Y H U i � V H U � F D S D ] � G H � U H V S R Q G H U � D V � V H J X L Q W H V � T X H V W } H V �
• 2 � T X H � V m R � F X V W R � � U H F H L W D � H � O X F U R � P D U J L Q D O � H � F R P R � G H W H U P L Q i � O R V "
• & R P R � H V W L P D U � R � Y D O R U � P D U J L Q D O � X W L O L ] D Q G R � D � G H U L Y D G D "
• & R P R � L Q W H U S U H W D U � R � Y D O R U � P D U J L Q D O "
• & R P R � H Q F R Q W U D U � R � Y D O R U � T X H � P D [ L P L ] D � R X � P L Q L P L ] D � X P D � I X Q o m R � T X D G U i W L F D � X W L O L ] D Q G R � D � G H U L Y D G D "
C ú
C O N V I T E À L E I T U R A
�
A p l i c a ç ã o d a D e r i v a d a n o E s t u d o d a s F u n ç õ e s d a s Á r e a s E c o n ô m i c a s e A d m i n i s t r a t i v a s
I n t r o d u ç ã o
( P � D O J X P D V � V L W X D o } H V � � X P � H P S U H V i U L R � S R G H � W H U � G H � W R P D U � D � G H F L V m R � G H � D X P H Q W D U � R X � Q m R � R � Q t Y H O � G H � S U R G X o m R � � 3 D U D �
t a n t o , u m a a n á l i s e d o l u c r o m a r g i n a l � � W D P E p P � F X V W R � P D U J L Q D O � H � U H F H L W D � P D U J L Q D O � � S R G H � P R V W U D U � D � H V V H � H P S U H V i U L R � V H �
S U R G X ] L U � P D L V � H � Y H Q G H U � P D L V � V L J Q L ¿ F D � O X F U D U � P D L V � � 1 D V � i U H D V � G H � D G P L Q L V W U D o m R � � F R Q W D E L O L G D G H � H � H F R Q R P L D � � X W L O L ] D � V H � R �
F R Q F H L W R � G H � I X Q o m R � P D U J L Q D O � S D U D � D Y D O L D U � R � H I H L W R � F D X V D G R � H P � X P D � I X Q o m R � � T X H � S R G H � V H U � F X V W R � � U H F H L W D � R X � O X F U R � � S R U � X P D �
S H T X H Q D � Y D U L D o m R � Q D � T X D Q W L G D G H � Y H Q G L G D � R X � S U R G X ] L G D � � 8 P D � I H U U D P H Q W D � S U i W L F D � S D U D � H V V H � F i O F X O R � p � D � G H U L Y D G D �
E x e m p l o P r á t i c o – O C u s t o M a r g i n a l
( P � X P D � L Q G ~ V W U L D � G H � F D O o D G R V � G H � O X [ R � � Q D � S U R G X o m R � G H � [ � X Q L G D G H V � G H � F H U W R � W L S R � G H � V D S D W R � � R � F X V W R � & � H P � U H D L V � I R L �
H V W X G D G R � H � H V W D E H O H F L G R � F R P R � & � [ � � � � � � [
3 
– 1 5 x
�
� � � � � [ � � � �
$ � S D U W L U � G H V V D V � L Q I R U P D o } H V � p � S R V V t Y H O � G H W H U P L Q D U � R � F X V W R � � S R U � H [ H P S O R � � S D U D � D � S U R G X o m R � G H � � � � V D S D W R V �
& � � � � � � � ˜ � �
3
– 1 5 ˜ � �
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� � � � ˜ � � � � Ÿ & � � � � � � � � � � � � � � � � � � � � � � � � �
Ÿ C ( 3 4 ) = 7 5 2 3 , 8 0 r e a i s
7 D P E p P � p � S R V V t Y H O � G H W H U P L Q D U � R � F X V W R � G H � I D E U L F D o m R � G H � � � � V D S D W R V �
& � � � � � � � ˜ 3 5
3
– 1 5 ˜ 3 5
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3 5 + 3 Ÿ & � � � � � � � � � � � � � � � � � � � � � �
Ÿ C ( 3 5 ) = 7 7 0 3 r e a i s
( Q W U H W D Q W R � � V H � D � H P S U H V D � G H V H M D � V D E H U � D S H Q D V � R � F X V W R � S D U D � S U R G X ] L U � R � W U L J p V L P R � T X L Q W R � V D S D W R � � � �
o
� � � G H Y H � V H � V X E W U D L U � G R �
F X V W R � G H � S U R G X o m R � G H � � � � V D S D W R V � R � F X V W R � G H � S U R G X o m R � G H � � � � V D S D W R V � � R X � V H M D �
& � � � � � ± � & � � � � � � � � � � � � � � � � � � � � 1 7 9 , 2 0 r e a i s .
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P O R D E N T R O D O T E M A
�
3 R G H � V H � W D P E p P � L Q W H U S U H W D U � H V W H � U H V X O W D G R � F R P R � � Q R � Q t Y H O � G H � S U R G X o m R � G H � � � � X Q L G D G H V � G H � V D S D W R V � � R � F X V W R � D G L F L R Q D O �
S D U D � D � S U R G X o m R � G H � P D L V � X P D � X Q L G D G H � p � G H � 5 � � � � � � � � � � $ � H V V H � W L S R � G H � F X V W R � G i � V H � R � Q R P H � G H � c u s t o m a r g i n a l .
O c u s t o m a r g i n a l , q u a n d o a p r o d u ç ã o é d e 3 4 u n i d a d e s d e s a p a t o s , é d e R $ 1 7 9 , 2 0 p o r u n i d a d e .
3 D U D � R � F i O F X O R � G R � F X V W R � P D U J L Q D O � � D O J X Q V � H F R Q R P L V W D V � � S H O D � S U D W L F L G D G H � H � U D S L G H ] � � S U H I H U H P � X W L O L ] D U � D � G H U L Y D G D � � D S H V D U �
G H � D � G H U L Y D G D � V H U � D S H Q D V � X P D � D S U R [ L P D o m R � S D U D � H V V H � F D V R � � , V V R � D F R Q W H F H � S R U T X H � R � F i O F X O R � G D � G H U L Y D G D � S U H V V X S } H � R �
n ú m e r o d e s a p a t o s c o m o u m a v a r i á v e l c o n t í n u a � � $ V V L P � � D � G H U L Y D G D � G D � I X Q o m R � & � [ � � Q R � S R Q W R � [ � � � � � p �
C
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C ' � [ � 15 0 03 0 x0 , 6 x
2
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2
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3 D U D � [ � � � � & ' � � � � � � � ˜ � �
�
� � � ˜ � � � � � � Ÿ � I ' � � � � � � � � � � � � � � � � � � � � � Ÿ Ÿ C ' ( 3 4 ) = 1 7 3 , 6 0 r e a i s p o r u n i d a d e .
3 R U W D Q W R � � S R U � P H L R � G R � F i O F X O R � G D � G H U L Y D G D � � R � F X V W R � P D U J L Q D O � � T X D Q G R � D � S U R G X o m R � p � G H � � � � X Q L G D G H V � � p � & ' � � � � � � 5 � � � � � � � � �
S R U � X Q L G D G H � � 2 � Y D O R U � G R � F X V W R � P D U J L Q D O � S R U � P H L R � G D � G H U L Y D G D � p � X P D � E R D � D S U R [ L P D o m R � S D U D � R � F X V W R � P D U J L Q D O � U H D O �� T X H � �
F R Q I R U P H � R E W L G R � D Q W H U L R U P H Q W H � � p � G H � 5 � � � � � � � � � � 2 � T X H � V L J Q L ¿ F D � T X H � R � X V R � G D � G H U L Y D G D � H P � D O J X P D V � V L W X D o } H V � S R G H �
V L P S O L ¿ F D U � R � F i O F X O R � G H V W H � F X V W R � � 1 R V � S U y [ L P R V � F i O F X O R V � G D � I X Q o m R � P D U J L Q D O � V H U i � X W L O L ] D G D � D S H Q D V � D � G H U L Y D G D �
E x e m p l o P r á t i c o – A R e c e i t a M a r g i n a l
$ � H P S U H V D � G H � F D O o D G R V � G H � O X [ R � � D S y V � H V W X G R V � � D Y D O L D � T X H � D � I X Q o m R � U H F H L W D � � S D U D � D W p � F H U W R � Q t Y H O � G H � Y H Q G D V � � p �
G D G D � S H O D � I X Q o m R � 5 � [ � � � � � [
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G H P D Q G D � [ � p � S � � � � [
� �
± � � � [ � � � � � � � $ O p P � G L V V R � � Y D O H � D � U H O D o m R � 5 � � S ˜ [ � � H P � T X H � D � U H F H L W D � p � L J X D O � D � S U H o R � S � P X O W L S O L F D G R �
S H O D � T X D Q W L G D G H � Y H Q G L G D � [ �
( V V D � H P S U H V D � S R G H � H V W L P D U � D � Y D U L D o m R � G D � U H F H L W D � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p � G H � � � � V D S D W R V � � R X � V H M D � � D � H P S U H V D � S R G H �
d e t e r m i n a r a r e c e i t a p a r a a v e n d a d o 3 5
o
� V D S D W R � � R � T X H � V L J Q L ¿ F D � H Q F R Q W U D U � D � r e c e i t a m a r g i n a l � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V �
p � G H � � � � V D S D W R V � � 3 D U D � L V V R � � p � Q H F H V V i U L R � R E W H U � D � I X Q o m R � U H F H L W D � P D U J L Q D O � � ' H U L Y D Q G R � D � I X Q o m R �
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2
- 3 2 x + 6 0 0 .
P O R D E N T R O D O T E M A
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2 � Y D O R U � G D � U H F H L W D � P D U J L Q D O � T X D Q G R � [ � � � � � X Q L G D G H V �
5 ' � � � � � � � ˜ � �
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� � � ˜ � � � � � � Ÿ 5 ' � � � � � � � � � � � � � � � � � � � � � Ÿ 
Ÿ R ' ( 3 4 ) = 2 0 5 , 6 r e a i s p o r u n i d a d e .
3 R U W D Q W R � � D � U H F H L W D � P D U J L Q D O � � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p � G H � � � � V D S D W R V � � p � G H � 5 � � � � � � � � � , V W R � V L J Q L ¿ F D � T X H � � Q D � Y H Q G D � G R �
3 5
o
� V D S D W R � � D � H P S U H V D � W H U i � X P � D X P H Q W R � G H � 5 � � � � � � � � Q D � U H F H L W D �
E x e m p l o P r á t i c o – O L u c r o M a r g i n a l
& R Q V L G H U D Q G R � T X D Q W L G D G H � Y H Q G L G D � L J X D O � j � T X D Q W L G D G H � S U R G X ] L G D � � D � I X Q o m R � O X F U R � p � G D G D � S H O D � G L I H U H Q o D � H Q W U H � D �
I X Q o m R � U H F H L W D � H � D � I X Q o m R � F X V W R �
L = R - C
/ � � � � � [
3 
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3 
– 1 5 x
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3 
+ 1 5 x
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L = - x
2
 + 1 0 0 x - 3
( V V D � H P S U H V D � S R G H � H V W L P D U � D � Y D U L D o m R � G R � O X F U R � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p � L J X D O � D � � � � V D S D W R V � � R X � V H M D � � D � H P S U H V D � S R G H �
d e t e r m i n a r o l u c r o p a r a a v e n d a d o 3 5
o
� V D S D W R � � R � T X H � V L J Q L ¿ F D � H Q F R Q W U D U � R � l u c r o m a r g i n a l � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p �
L J X D O � D � � � � V D S D W R V � � 3 D U D � L V V R � � L Q F L D O P H Q W H � � p � Q H F H V V i U L R � R E W H U � D � I X Q o m R � O X F U R � P D U J L Q D O � � ' H U L Y D Q G R � D � I X Q o m R � O X F U R �
L ' � [ � 
� � �x� � �x� �3�� � � [x�
x d
d
��
�˜�˜ �
Ÿ L ' ( x ) = - 2 x + 1 0 0 .
2 � Y D O R U � G R � O X F U R � P D U J L Q D O � T X D Q G R � [ � � � � � X Q L G D G H V � � / ' � � � � � � � ˜ � � � � � � � � 3 2 r e a i s p o r u n i d a d e .
3 R U W D Q W R � � R � O X F U R � P D U J L Q D O � � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p � G H � � � � V D S D W R V � � p � G H � 5 � � � � � � � � S R U � X Q L G D G H � � , V V R � V L J Q L ¿ F D � T X H � � Q D �
v e n d a d o 3 5
o
� V D S D W R � � D � H P S U H V D � W H U i � X P � D X P H Q W R � Q R � O X F U R � G H � 5 � � � � � � � �
& R Q V L G H U H � D � V L W X D o m R � H P � T X H � D � H P S U H V D � G H V H M D � G H W H U P L Q D U � R � O X F U R � P D U J L Q D O � T X D Q G R � R � Q t Y H O � G H � Y H Q G D V � p � L J X D O � D � � � �
X Q L G D G H V � � $ V V L P � � S D U D � [ � � � � � X Q L G D G H V �
L ' � � � � - 2 ˜ � � + 1 0 0 = - 4 r e a i s p o r u n i d a d e .
P O R D E N T R O D O T E M A
�
( V V H � Y D O R U � Q H J D W L Y R � L Q G L F D � T X H � � Q D � Y H Q G D � G R � � �
o
� V D S D W R � � K D Y H U i � X P D � U H G X o m R � G H � 5 � � � � � � � Q R � O X F U R � G D � H P S U H V D � � 2 E V H U Y H �
T X H � � Q H V W H � F D V R � � D � U H G X o m R � G R � O X F U R � Q m R � V L J Q L ¿ F D � T X H � D � H P S U H V D � W H U i � S U H M X t ] R � D R � Y H Q G H U � � � � X Q L G D G H V � G H � V D S D W R V � � P D V �
V L P � X P � O X F U R � P H Q R U � H P � 5 � � � � � � �
E x e m p l o P r á t i c o – O L u c r o M á x i m o
2 � S R Q W R � G H � P i [ L P R � G H � X P D � I X Q o m R � T X D G U i W L F D � S R G H � V H U � G H W H U P L Q D G R � S H O R � F i O F X O R � G D � G H U L Y D G D � � 3 D U W H � V H � G R � S U L Q F t S L R �
G H � T X H � � Q R � Y p U W L F H � G D � S D U i E R O D � � R Q G H � R F R U U H � R � P i [ L P R � R X � P t Q L P R � G D � I X Q o m R � � � D � U H W D � t a n g e n t e � W H P � L Q F O L Q D o m R � L J X D O � D � ] H U R � �
R X � V H M D � � H V W i � Q D � K R U L ] R Q W D O � �
$ V V L P � � F R P R � D � G H U L Y D G D � I R U Q H F H � D � L Q F O L Q D o m R � G D � U H W D � W D Q J H Q W H � � E D V W D � G H U L Y D U � D � I X Q o m R � T X D G U i W L F D � H � L J X D O D U � R � U H V X O W D G R � D �
] H U R � S D U D � H Q F R Q W U D U � R � S R Q W R � H P � T X H � D � U H W D � W D Q J H Q W H � H V W i � Q D � K R U L ] R Q W D O � � R X � V H M D � � S D U D � H Q F R Q W U D U � R � S R Q W R � G H � P t Q L P R � R X � G H �
P i [ L P R � � 2 E V H U Y H � D � ) L J X U D � � � � �
+ +
F i g u r a 8 . 1 � 3 R Q W R � G H � P i [ L P R � R X � G H � P t Q L P R � G D � I X Q o m R �
& R P R � R � F i O F X O R � G D � G H U L Y D G D � G D � I X Q o m R � O X F U R � M i � I R L � G H W H U P L Q D G R � D Q W H U L R U P H Q W H � � W H P � V H � T X H �
L ' ( x ) = - 2 x + 1 0 0 .
P O R D E N T R O D O T E M A
�
9 D O H � O H P E U D U � T X H � R � U H V X O W D G R � G D � G H U L Y D G D � G H � X P D � I X Q o m R � W D P E p P � U H S U H V H Q W D � D � L Q F O L Q D o m R � m � G D � U H W D � W D Q J H Q W H � j � I X Q o m R �
H P � X P � S R Q W R � [ � T X D O T X H U � � R X � V H M D �
m
t a n g e n t e
= - 2 x + 1 0 0 .
$ J R U D � � S D U D � R E W H U � R � S R Q W R � G H � P i [ L P R � G D � I X Q o m R � � L J X D O D � V H � D � L Q F O L Q D o m R � D � ] H U R � � R X � V H M D � � G H W H U P L Q D � V H � R � Y D O R U � G H � [ � H P � T X H �
D � U H W D � H V W i � Q D � K R U L ] R Q W D O �
m
t a n g e n t e
 � Ÿ � � [ � � � � � � � �
˜
 x = 5 0 u n i d a d e s
2 � O X F U R � P i [ L P R �R F R U U H � T X D Q G R � [ � � � � � X Q L G D G H V � F R P � X P � Y D O R U � F R U U H V S R Q G H Q W H � D � �
/ � � � � � � � � �
�
� � � � � ˜ � � � � Ÿ � / � � � � � � � � �
�
� � � � � ˜ � � � � � � 2 4 9 7 , 0 0 r e a i s .
3 R U W D Q W R � � D � S D U W L U � G D � I X Q o m R � O X F U R � / � � � [
�
� � � � � [ � � � � S D U D � D � H P S U H V D � P D [ L P L ] D U � R � O X F U R � � H O D � G H Y H � S U R G X ] L U � H � Y H Q G H U � � � �
V D S D W R V � J H U D Q G R � X P � O X F U R � G H � 5 � � � � � � � � � � �
( V W H � S U R F H V V R � p � H T X L Y D O H Q W H � D � H Q F R Q W U D U � R � Y p U W L F H � G D � S D U i E R O D � � F R P R � D S U H Q G L G R � Q R � 7 H P D � � � � 3 D U D � R X W U R V � W L S R V � G H �
I X Q o } H V � S R O L Q R P L D L V � G H � J U D X � P D L R U � T X H � G R L V � � D � G H U L Y D G D � W R U Q D � V H � X P D � H [ F H O H Q W H � I H U U D P H Q W D � Q D � E X V F D � G R V � P t Q L P R V � H �
m á x i m o s l o c a i s .
P O R D E N T R O D O T E M A
�
L e i a o t e x t o M a t e m á t i c a a p l i c a d a à A d m i n i s t r a ç ã o .
• 7 H [ W R � E H P � F R P S O H W R � V R E U H � D � I X Q o m R � P D U J L Q D O � � 2 � D U T X L Y R � G R � s i t e � P R V W U D � D � U H V R O X o m R � G H � G L Y H U V R V �
e x e r c í c i o s .
L i n k : � K W W S � � � Z Z Z � H E D K � F R P � E U � F R Q W H Q W � $ % $ $ $ $ X � � $ + � P D W H P D W L F D � D S O L F D G D � D � D G P L Q L V W U D F D R ! . A c e s s o 
H P � � � � P D L � � � � � � �
/ H L D � D � G H ¿ Q L o m R � G H � & X V W R � P D U J L Q D O �
• & R Q W p P � D � G H ¿ Q L o m R � I R U P D O � V R E U H � D � I X Q o m R � P D U J L Q D O � F R P � D � D S O L F D o m R � G H V V H � F R Q F H L W R � H P �
e c o n o m i a .
L i n k : � K W W S � � � S W � Z L N L S H G L D � R U J � Z L N L � & X V W R B P D U J L Q D O ! � � $ F H V V R � H P � � � � P D L � � � � � � �
A c e s s e o s i t e d a B i b l i o t e c a V i r t u a l d a A n h a n g u e r a .
• 1 R � F D P S R � S D U D � S H V T X L V D � � G L J L W H � d e r i v a d a � � $ S D U H F H U m R � Y i U L R V � e - b o o k s � T X H � F R Q W r P � R V � F R Q F H L W R V �
V R E U H � G H U L Y D G D V � � F R P � D S O L F D o m R � Q D � G H W H U P L Q D o m R � G D V � I X Q o } H V � P D U J L Q D L V � �
L i n k : � K W W S � � � Z Z Z � D Q K D Q J X H U D � F R P � E L E O L R W H F D V � E L E O L R W H F D � Y L U W X D O � F X U V R � H D G � D G P L Q L V W U D F D R ! � � $ F H V V R � H P � � � � P D L � � � � � � �
F u n ç õ e s M a r g i n a i s A p l i c a d a s à A d m i n i s t r a ç ã o .
L i n k : � K W W S � � � Z Z Z � \ R X W X E H � F R P � Z D W F K " Y 1 7 T J = F B P � E � ! � � $ F H V V R � H P � � � � P D L � � � � � � �
7 H P S R � � � � � � �
( V W H � Y t G H R � P R V W U D � D � U H V R O X o m R � G H � G L Y H U V R V � H [ H U F t F L R V � U H O D F L R Q D G R V � D � I X Q o } H V � P D U J L Q D L V � D S O L F D G D V � j � D G P L Q L V �
W U D o m R � H � H F R Q R P L D �
á à
A C O M PA N H E N A W E B
� �
I n s t r u ç õ e s :
$ J R U D � � F K H J R X � D � V X D � Y H ] � G H � H [ H U F L W D U � V H X � D S U H Q G L ] D G R � � $ � V H J X L U � � Y R F r � H Q F R Q W U D U i � D O J X P D V � T X H V W } H V � G H � P ~ O W L S O D �
H V F R O K D � H � G L V V H U W D W L Y D V � � / H L D � F X L G D G R V D P H Q W H � R V � H Q X Q F L D G R V � H � D W H Q W H � V H � S D U D � R � T X H � H V W i � V H Q G R � S H G L G R �
A G O R A É A S U A V E Z
Q u e s t ã o 1
& R Q V L G H U H � D � V L W X D o m R � H P � T X H � X P D � H P S U H V D � J D V W D � & � [ � � � m x + b � � S D U D � S U R G X ] L U � X P D � T X D Q W L G D G H � [ � G H � X P � S U R G X W R � � H P � T X H � m e b 
V m R � F R Q V W D Q W H V � � 8 W L O L ] D Q G R � D � G H U L Y D G D � � G H W H U P L Q H � D � I X Q o m R � P D U J L Q D O � G R � F X V W R � S D U D � X P � Q t Y H O � G H � S U R G X o m R � [ � � ( [ S O L T X H � R � U H V X O W D G R �
R E W L G R � F R P H Q W D Q G R � R � F R P S R U W D P H Q W R � G D � I X Q o m R � P D U J L Q D O � S D U D � T X D O T X H U � Y D O R U � G H � [ �
A t e n ç ã o : A s Q u e s t õ e s d e 2 a 5 d e v e m s e r r e s p o n d i d a s c o m b a s e n o e n u n c i a d o a s e g u i r :
( P � X P D � L Q G ~ V W U L D � � Q D � S U R G X o m R � G H � [ � X Q L G D G H V � G H � F H U W R � W L S R � G H � S H o D � P H W i O L F D � � R � F X V W R � & � H P � U H D L V � I R L � H V W X G D G R � H � H V W D E H O H F L G R � F R P R �
& � [ � � � � � � [
3
� ± � � [
�
� � � � [ � � � �
Q u e s t ã o 2
2 � F X V W R � G D � S U R G X o m R � S D U D � D � I D E U L F D o m R � G D � � �
a
� S H o D � I R L � G H � �
a ) � 5 � � � � � � � � �
b ) � 5 � � � � � � � � �
c ) � 5 � � � � � � � �
d ) � 5 � � � � � � � �
e ) � 5 � � � � � � � �
� �
A G O R A É A S U A V E Z
Q u e s t ã o 3
$ � S D U W L U � G R � F i O F X O R � G D � G H U L Y D G D � � D � I X Q o m R � F X V W R � P D U J L Q D O � G H � & � [ � � p �
a ) C ' � [ � � � [
�
� � [ � � � �
b ) C ' � [ � � � [
�
� � [ � � � �
c ) C ' � [ � � � � � [
�
� � [ � � � �
d ) C ' � [ � � � � � [
�
� � [ � � � �
e ) C ' � [ � � � � � [
�
� � [ � � � �
Q u e s t ã o 4
8 W L O L ] D Q G R � D � I X Q o m R � F X V W R � P D U J L Q D O � R E W L G D � Q D � 4 X H V W m R � � � � G H W H U P L Q H � R � F X V W R � P D U J L Q D O � T X D Q G R � R � Q t Y H O � G H � S U R G X o m R � p � [ � � � � S H o D V �
a ) � 5 � � � � � � � �
b ) � 5 � � � � � � � �
c ) � 5 � � � � � � � �
d ) � 5 � � � � � � � �
e ) � 5 � � � � � � �
� �
A G O R A É A S U A V E Z
Q u e s t ã o 5
6 H P � X W L O L ] D U � D � G H U L Y D G D � � T X D O � R � F X V W R � P D U J L Q D O � T X D Q G R � R � Q t Y H O � G H � S U R G X o m R � p � [ � � � � � S H o D V �
a ) 5 � � � � � � � �
b ) 5 � � � � � � � �
c ) 5 � � � � � � � �
d ) 5 � � � � � � � �
e ) 5 � � � � � � � �
A t e n ç ã o : A s Q u e s t õ e s d e 6 a 1 0 d e v e m s e r r e s p o n d i d a s c o m b a s e n o e n u n c i a d o a s e g u i r :
( P � X P D � L Q G ~ V W U L D � W r [ W L O � � D � U H F H L W D � Q D � Y H Q G D � G H � F H U W R � W L S R � G H � W R D O K D � G H � P H V D � p � G D G D � S R U � 5 � [ � � � � � � � � � [
�
� � � [ � � H P � T X H � � � < x < � � � � � � �
2 � F X V W R � S D U D � D � S U R G X o m R � G H V V D V � W R D O K D V � p � G D G R � S R U � & � [ � � � � [ � � � � � � � $ � Y D U L i Y H O � [ � U H S U H V H Q W D � D V � T X D Q W L G D G H V � G H � W R O K D V � S U R G X ] L G D V �
e v e n d i d a s .
Q u e s t ã o 6
' H W H U P L Q H � �
D � � $ � I X Q o m R � O X F U R �
E � � $ � I X Q o m R � O X F U R � P D U J L Q D O � � 3 D U D � H V W H � F i O F X O R � � X W L O L ] H � D � G H U L Y D G D �
Q u e s t ã o 7
2 E W H Q K D � R � O X F U R � P D U J L Q D O � Q R V � Q t Y H L V � G H � S U R G X o m R � [ � � � � � � H � [ � � � � � � � X Q L G D G H V � � , Q W H U S U H W H � R V � U H V X O W D G R V �
� �
Q u e s t ã o 8
' H W H U P L Q H � D � U H F H L W D � H � R � F X V W R � P D U J L Q D O � T X D Q G R � [ � � � � � � � X Q L G D G H V � � & R P S D U H � R � U H V X O W D G R � F R P � R � R E W L G R � Q D � 4 X H V W m R � � �
Q u e s t ã o 9
8 W L O L ] D Q G R � D � G H U L Y D G D � � G H W H U P L Q H � D � T X D Q W L G D G H � [ � T X H � I R U Q H F H � D � U H F H L W D � P i [ L P D � � ' H W H U P L Q H � W D P E p P � D � U H F H L W D � P i [ L P D �Q u e s t ã o 1 0
8 W L O L ] D Q G R � D � G H U L Y D G D � � G H W H U P L Q H � D � T X D Q W L G D G H � [ � T X H � I R U Q H F H � R � O X F U R � P i [ L P R � � ' H W H U P L Q H � W D P E p P � R � O X F U R � P i [ L P R �
A G O R A É A S U A V E Z
1 H V W H � W H P D � � Y R F r � D S U H Q G H X � R � F R Q F H L W R � V R E U H � I X Q o m R � P D U J L Q D O � � L Q F O X L Q G R � R � H V W X G R � V R E U H � F X V W R � � U H F H L W D � H � O X F U R �
P D U J L Q D O � � $ O p P � G L V V R � � Y R F r � D S U H Q G H X � T X H � p � S R V V t Y H O � H V W L P D U � D � I X Q o m R � P D U J L Q D O � S R U � P H L R � G R � X V R � G D � G H U L Y D G D � � ( V W X G R X �
W D P E p P � S U R E O H P D V � D S O L F D G R V � D R � X V R � G D � I X Q o m R � P D U J L Q D O � � 3 R U � ¿ P � � H V W X G R X � X P D � L P S R U W D Q W H � D S O L F D o m R � G D � G H U L Y D G D � � T X H � p �
D � R E W H Q o m R � G H � S R Q W R V � G H � P i [ L P R V � H � P t Q L P R V � �
8 W L O L ] D Q G R � D � G H U L Y D G D � � G H W H U P L Q H � D � T X D Q W L G D G H � [ � T X H � I R U Q H F H � R � O X F U R � P i [ L P R � � ' H W H U P L Q H � W D P E p P � R � O X F U R � P i [ L P R �
1 r G L E I L O L O L G G E L O
F I N A L I Z A N D O
0 8 5 2 / 2 � � $ I U k Q L R � & D U O R V � � % 2 1 ( 7 7 2 � � * L i F R P R � � M a t e m á t i c a A p l i c a d a a A d m i n i s t r a ç ã o , E c o n o m i a e C o n t a b i l i d a d e � � � � � H G � � 6 m R � 3 D X O R � �
& H Q J D J H � / H D U Q L Q J � � � � � � �
T A N , S o o T a n g . M a t e m á t i c a A p l i c a d a à A d m i n i s t r a ç ã o e E c o n o m i a � � � � � H G � � 6 m R � 3 D X O R � � 3 L R Q H L U D � � � � � � �
D � R E W H Q o m R � G H � S R Q W R V � G H � P i [ L P R V � H � P t Q L P R V �
R E F E R Ê N C I A S
� �
C o n t í n u a : � X P D � Y D U L i Y H O � F R Q W t Q X D � p � D T X H O D � T X H � S R G H � D V V X P L U � T X D O T X H U � Y D O R U � Q X P p U L F R �
I n s t a n t â n e o : � T X H � D F R Q W H F H � H P � D S H Q D V � X P � L Q V W D Q W H � H V S H F t ¿ F R �
M a r g i n a l : � X P D � S U R G X o m R � P D U J L Q D O � p � D � T X D Q W L G D G H � H [ W U D � S U R G X ] L G D � S R U � X P D � X Q L G D G H � G H � S U R G X o m R �
T a n g e n t e : � X P D � U H W D � W D Q J H Q W H � W R F D � X P D � F X U Y D � V H P � F R U W i � O D � � F R P S D U W L O K D Q G R � X P � ~ Q L F R � S R Q W R � � F R Q K H F L G R � F R P R � S R Q W R � G H �
t a n g ê n c i a .
C t í L i O W t p O G L O O p L
G L O S S Á R I O
t a n g ê n c i a .
G A B A R I T O
Q u e s t ã o 1
R e s p o s t a : � ' H U L Y D Q G R � D � I X Q o m R � & � [ � � m x + b :
C
'
� [ � � � � � � � � �mE
x d
d
x
x d
d
mEx m
x d
d
� � � 
Ÿ
 C
'
( x ) = m
$ � I X Q o m R � P D U J L Q D O � R E W L G D � p � F R Q V W D Q W H � � H � D � Y D U L i Y H O � [ � Q m R � D S D U H F H � Q D � I X Q o m R � P D U J L Q D O � � , V V R � V L J Q L ¿ F D � T X H � � S D U D � T X D O T X H U �
v a l o r d e x , o c u s t o m a r g i n a l s e m a n t e r á c o n s t a n t e c o m v a l o r i g u a l a m � � $ V V L P � � R � F X V W R � G H � S U R G X o m R � D X P H Q W D � V H P S U H � D �
u m a m e s m a t a x a m � � L Q G H S H Q G H Q W H P H Q W H � G D � T X D Q W L G D G H � [ � S U R G X ] L G D �
Q u e s t ã o 2
R e s p o s t a : A l t e r n a t i v a E .
3 U L P H L U R � � F D O F X O D � V H � R � F X V W R � S D U D � D � S U R G X o m R � G H � � � � S H o D V �
& � � � � � � � ˜ 1 1
3
± � ˜ 1 1
�
� � � ˜ � � � � � Ÿ & � � � � � � � � � � � � � � � � � � � �
� �
G A B A R I T O
Ÿ C ( 1 1 ) = 1 6 2 , 2 r e a i s .
( P � V H J X L G D � � R � F X V W R � S D U D � D � S U R G X o m R � G H � � � � S H o D V � �
& � � � � � � � ˜ � �
3
± � ˜ � �
�
� � � ˜ � � � � � Ÿ & � � � � � � � � � � � � � � � � � � � �
Ÿ C ( 1 2 ) = 1 7 9 , 6 r e a i s
2 � F X V W R � S D U D � D � I D E U L F D o m R � G D � � �
a
� S H o D � p �
& � � � � � ± � & � � � � � � � � � � � � � � � � � � � 1 7 , 4 r e a i s .
P o r t a n t o , o c u s t o p a r a f a b r i c a r a 1 2
a
 p e ç a é R $ 1 7 , 4 0 .
Q u e s t ã o 3
R e s p o s t a : $ O W H U Q D W L Y D � ' �
C
'
� [ � 
� � � � � � � � � �� 5
x d
d
x
x d
d
� 3x
x d
d
�x
x d
d
� � �� 5� � [x �� � � [
x d
d
�3�3
��� ���
Ÿ
 C
'
( x ) = 0 , 6 x
2
- 8 x + 3 0 .
Q u e s t ã o 4
R e s p o s t a : � $ O W H U Q D W L Y D � % �
$ � I X Q o m R � F X V W R � P D U J L Q D O � R E W L G D � Q R � L W H P � D Q W H U L R U � p � &
'
� [ � � � � � [
�
� � [ � � � �
3 D U D � [ � � � � &
'
� � � � � � � � ˜ 1 1
�
� � ˜ � � � � �
Ÿ
 C
'
( 1 1 ) = 1 4 , 6 r e a i s .
Q u e s t ã o 5
R e s p o s t a : A l t e r n a t i v a C .
1 D � 4 X H V W m R � � � � F D O F X O R X � V H � R � F X V W R � S D U D � D � I D E U L F D o m R � G D � � �
a
� S H o D � � V H Q G R � R � U H V X O W D G R � R E W L G R � G H � 5 � � � � � � � � � , V W R � V L J Q L ¿ F D �
T X H � � Q R � Q t Y H O � G H � S U R G X o m R � G H � � � � S H o D V � � R � F X V W R � D G L F L R Q D O � S D U D � D � S U R G X o m R � G H � P D L V � X P D � X Q L G D G H � p � G H � 5 � � � � � � � � � R X � V H M D � �
i s s o r e p r e s e n t a o c u s t o m a r g i n a l .
3 R U W D Q W R � � R � F X V W R � P D U J L Q D O � � T X D Q G R � R � Q t Y H O � G H � S U R G X o m R � p � [ � � � S H o D V � � p � G H � 5 � � � � � � � �
� �
Q u e s t ã o 6
R e s p o s t a : D � $ � I X Q o m R � O X F U R �
/ � 5 � ± � & � Ÿ / � � � � � � � � [
�
� � � [ � ± � � � [ � � � � � � � Ÿ � / � � � � � � [
�
� � � [ ± � [ � � � � � � �
Ÿ L ( x ) = - 0 , 0 0 5 x
2
+ 1 8 x - 5 0 0 0
E � � $ � I X Q o m R � O X F U R � P D U J L Q D O � � 3 D U D � H V V H � F i O F X O R � � X W L O L ] H � D � G H U L Y D G D �
L ' � [ � 
� � � � � � � �� � � �
x d
d
x
x d
d
� 1x
x d
d
� � � � �� � � �� � [� � � � � [�
x d
d
��
��� ��
Ÿ L ' ( x ) = - 0 , 0 1 x + 1 8 .
Q u e s t ã o 7
R e s p o s t a : � � 3 D U D � [ � � � � � / ' � � � � � � � � � � ˜ � � � � � � � Ÿ L ' � � � � � � � � � � Ÿ L ' ( 6 0 0 ) = 1 2 .
, V W R � V L J Q L ¿ F D � T X H � � Q D � Y H Q G D � G D � � � �
a
� W R D O K D � � K D Y H U i � X P � D X P H Q W R � G H � 5 � � � � � � � � Q R � O X F U R � G D � H P S U H V D �
� � 3 D U D � [ � � � � � � / ' � � � � � � � � � � � ˜ � � � � � � � � Ÿ L ' � � � � � � � � � � � � � �
Ÿ L ' ( 1 9 5 0 ) = - 1 , 5 .
, V W R � V L J Q L ¿ F D � T X H � � Q D � Y H Q G D � G D � � � � �
a
� W R D O K D � � K D Y H U i � X P D � U H G X o m R � G H � 5 � � � � � � � Q R � O X F U R � G D � H P S U H V D �
Q u e s t ã o 8
R e s p o s t a : , Q L F L D O P H Q W H � � p � Q H F H V V i U L R � G H W H U P L Q D U � D � I X Q o m R � U H F H L W D � P D U J L Q D O � � ' H U L Y D Q G R � D � I X Q o m R � U H F H L W D � �
5 ' � [ � 
� � � � � �x
x d
d
� �x
x d
d
� � � � �� � [� � � � � [�
x d
d
��
�� �
 Ÿ R ' ( x ) = - 0 , 0 1 x + 2 0 .
� � 3 D U D � [ � � � � � � 5 ' � � � � � � � � � � � ˜ � � � � � � � � Ÿ � 5 ' � � � � � � � � � � � � � � Ÿ 
R ' ( 1 9 5 0 ) = 0 , 5 .
G A B A R I T O
� �
G A B A R I T O
( P � V H J X L G D � � G H W H U P L Q D U � D � I X Q o m R � F X V W R � P D U J L Q D O � � ' H U L Y D Q G R � D � I X Q o m R � F X V W R � �
& � [ � � � � [ � � � � �
C ' � [ � 
� � � � � �� � � �
x d
d
x
x d
d
�� � ��x �
x d
d
� �
 Ÿ C ' ( x ) = 2 . 
$ � I X Q o m R � F X V W R � P D U J L Q D O � p � F R Q V W D Q W H � � L Q G H S H Q G H � G D � T X D Q W L G D G H � [ � � 3 R U W D Q W R � � S D U D � [ � � � � � � � � R � F X V W R � P D U J L Q D O � p � G H � R $ 2 , 0 0 .
$ V V L P � � F R P R � S D U D � [ � � � � � � � D � U H F H L W D � H � R � F X V W R � P D U J L Q D O � V m R � � U H V S H F W L Y D P H Q W H � � 5 � � � � � � � H � 5 � � � � � � � � H Q W m R � R � O X F U R � P D U J L Q D O � p �
L ' � [ � � � � � � � ± � � � � � � � � � � � � � U H D L V
, V W R � V L J Q L ¿ F D � T X H � � Q D � Y H Q G D � G D � � � � �
a
� W R D O K D � � K D Y H U i � X P � D X P H Q W R � G H � 5 � � � � � � � Q D � U H F H L W D � G D � H P S U H V D � H � X P � D X P H Q W R � G R �
F X V W R � G H � 5 � � � � � � � � ( Q W U H W D Q W R � � F R P R � R � D X P H Q W R � G R � J D V W R � p � P D L R U � T X H � R � D X P H Q W R � G D � U H F H L W D � � Q m R � K D Y H U i � D X P H Q W R � Q R � O X F U R � �
H � V L P � G H F U p V F L P R � G H � 5 � � � � � � � � � ( V V H � U H V X O W D G R � F R U U H V S R Q G H � D R � P H V P R � R E W L G R � Q D � 4 X H V W m R � � �
Q u e s t ã o 9
R e s p o s t a : � & R P R � R � F i O F X O R � G D � G H U L Y D G D � G D � I X Q o m R � U H F H L W D � M i � I R L � G H W H U P L Q D G R � Q D � 4 X H V W m R � � � � W H P � V H �
5 ' � [ � � � � � � � � [ � � � �
& R P R � H V W X G D G R � � G H Y H � V H � L J X D O D U � D � I X Q o m R � T X H � I R L � G H U L Y D G D � D � ] H U R � S D U D � R E W H U � D � T X D Q W L G D G H � [ � T X H � I R U Q H F H � D � U H F H L W D � P i [ L P D � �
� � � � � � � � [ � � � �
� � � � [ � � � � �
X = 2 0 0 0 .
3 R U W D Q W R � � D � U H F H L W D � P i [ L P D � R F R U U H � T X D Q G R � [ � � � � � � � X Q L G D G H V � F R P � X P � Y D O R U � F R U U H V S R Q G H Q W H � D � �
5 � � � � � � � � � � � � � � ˜ � � � �
�
� � � ˜ � � � � Ÿ � 5 � � � � � � � � � � � � � � � � � � � � � Ÿ
R ( 2 0 0 0 ) = 2 0 0 0 0 r e a i s . 
3 R U W D Q W R � � S D U D � D � H P S U H V D � P D [ L P L ] D U � D � U H F H L W D � � H O D � G H Y H � Y H Q G H U � � � � � � W R D O K D V � G H � P H V D � � J H U D Q G R � X P � O X F U R � G H � 5 � � � � � � � � � � � �
� �
Q u e s t ã o 1 0
R e s p o s t a : � & R P R � R � F i O F X O R � G D � G H U L Y D G D � G D � I X Q o m R � O X F U R � M i � I R L � G H W H U P L Q D G R � Q D � 4 X H V W m R � � � � W H P � V H �
L ' � [ � � � � � � � � [ � � � �
, J X D O D Q G R � D � I X Q o m R � T X H � I R L � G H U L Y D G D � D � ] H U R � �
� � � � � � � � [ � � � �
� � � � [ � � � � �
X = 1 8 0 0 .
3 R U W D Q W R � � R � O X F U R � P i [ L P R � R F R U U H � T X D Q G R � [ � � � � � � � X Q L G D G H V � F R P � X P � Y D O R U � F R U U H V S R Q G H Q W H � D �
/ � � � � � � � � � � � � � � ˜ � � � �
�
� � � ˜ � � � � � � � � � Ÿ � / � � � � � � � � � � � � � � � � � � � � � � � � � � Ÿ L ( 1 8 0 0 ) = 1 1 2 0 0 r e a i s . 
3 R U W D Q W R � � S D U D � D � H P S U H V D � P D [ L P L ] D U � R � O X F U R � � H O D � G H Y H � S U R G X ] L U � H � Y H Q G H U � � � � � � � W R D O K D V � G H � P H V D � � J H U D Q G R � X P � O X F U R � G H �
5 � � � � � � � � � � � �
& R P S D U D Q G R � F R P � D � 4 X H V W m R � � � � R � Y D O R U � G H � [ � T X H � P D [ L P L ] D � D � U H F H L W D � Q m R � p � R � P H V P R � T X H � P D [ L P L ] D � R � O X F U R � � , V W R � D F R Q W H F H �
S R U T X H � D � I X Q o m R � O X F U R � H V W i � U H O D F L R Q D G D � j � I X Q o m R � F X V W R � � S R U � P H L R � G D � I y U P X O D � / 5 � & �
G A B A R I T O

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