CADERNO DE QUESTOES E RESPOSTAS MATEMATICA AULA 6

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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 6
Ta x a d e V a r i a ç ã o M é d i a e I n s t a n t â n e a . O C o n c e i t o d e D e r i v a d a
7 H P D � � �
Ta x a d e V a r i a ç ã o M é d i a e I n s t a n t â n e a . O C o n c e i t o d e D e r i v a d a
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 : T a x a d e V a r i a ç ã o M é d i a e I n s t a n t â n e a . O C o n c e i t o d e D e r i v a d a . 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 �
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P á g . 1 8 P á g . 1 8
P á g . 1 7
P á g . 1 3P á g . 1 2
A C O M PA N H E N A W E B
P á g . 3
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P á g . 4
P O R D E N T R O D O T E M A
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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 t a x a d e v a r i a ç ã o m é d i a .
• A p l i c a ç ã o d a t a x a d e v a r i a ç ã o m é d i a .
• O c á l c u l o d a t a x a d e v a r i a ç ã o i n s t a n t â n e a a p a r t i r d o c á l c u l o r e p e t i d o d e t a x a d e v a r i a ç ã o m é d i a .
• $ � G H ¿ Q L o m R � G D � W D [ D � G H � Y D U L D o m R � S R U � P H L R � G R � X V R � G H � O L P L W H V � � D � G H U L Y D G D �
• P r o b l e m a s q u e e n v o l v a m t a x a s d e v a r i a ç ã o m é d i a e i n s t a n t â n e a .
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 L J Q L ¿ F D � W D [ D � G H � Y D U L D o m R � P p G L D " � 2 � T X H � p � S U H F L V R � S D U D � F D O F X O i � O D "
• 2 � T X H � V L J Q L ¿ F D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D "
• & R P R � F D O F X O D U � D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � D � S D U W L U � G D � W D [ D � G H � Y D U L D o m R � P p G L D "
• 4 X D O � D � U H O D o m R � H Q W U H � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � H � 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
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T a x a d e V a r i a ç ã o M é d i a e I n s t a n t â n e a . O C o n c e i t o d e D e r i v a d a
I n t r o d u ç ã o
3 R G H � V H � U H S U H V H Q W D U � D � Y D U L D o m R � G H � X P D � T X D Q W L G D G H � H P � U H O D o m R � D � R X W U D � S R U � P H L R � G H � X P D � U D ] m R � G H Q R P L Q D G D � t a x a 
d e v a r i a ç ã o . A t a x a d e v a r i a ç ã o p o d e s e r m é d i a o u i n s t a n t â n e a � � $ V � W D [ D V � G H � Y D U L D o m R � R F R U U H P � H P � P X L W D V � V L W X D o } H V �
S U i W L F D V � H P � 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 � � 3 R U � H [ H P S O R � � D � Y H O R F L G D G H � F R P � T X H � X P D � H P S U H V D � S U R G X ] � X P �
S U R G X W R � R X � D � U D ] m R � H Q W U H � D � T X D Q W L G D G H � S U R G X ] L G D � H � R � F D S L W D O � L Q Y H V W L G R � F R U U H V S R Q G H P � D � W D [ D V � G H � Y D U L D o m R �
T a x a d e V a r i a ç ã o M é d i a
$ � W D [ D � G H � Y D U L D o m R � P p G L D � p � R E W L G D � S H O D � G L Y L V m R � G H � G X D V � J U D Q G H ] D V � T X H � � H P � V L W X D o } H V � S U i W L F D V � � W r P � X Q L G D G H V � G H �
m e d i d a s . A t a x a d e v a r i a ç ã o p o d e s e r c a l c u l a d a p a r a q u a l q u e r f u n ç ã o . A s s i m , p o r e x e m p l o , a f u n ç ã o P ( x ) , e m q u e P é a 
T X D Q W L G D G H � S U R G X ] L G D � H � [ � p � R � W H P S R � � W H U i �
T a x a d e v a r i a ç ã o m é d i a = 
S e a p r o d u ç ã o d e u m a e m p r e s a e m t o n e l a d a s é d a d a p o r P ( x ) = x
2
, e m q u e x é o t e m p o e m h o r a s , a p ó s 2 h o r a s a 
H P S U H V D � S U R G X ] L U i � 3 � � � � � �
2
 � � W R Q H O D G D V � � $ S y V � � � K R U D V � � D � H P S U H V D � S U R G X ] L U i � 3 � � � � � �
2
 = 1 6 t o n e l a d a s . D e s t a f o r m a , a 
t a x a d e v a r i a ç ã o m é d i a d a p r o d u ç ã o d e s s a e m p r e s a s e r á :
T a x a d e v a r i a ç ã o m é d i a = 
r a st o n e d a s / h o6
2
2 1
2
46 1
24
P ( 2 )P ( 4 )
 
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( V W H � U H V X O W D G R � P R V W U D � T X H � � Q R � L Q W H U Y D O R � G H � � � D W p � � � K R U D V � � D � H P S U H V D � S U R G X ] L X � H P � P p G L D � � � W R Q H O D G D V � S R U � K R U D �
2 � J U i ¿ F R � G D � ) L J X U D � � � � � P R V W U D � D � I X Q o m R � S U R G X o m R � H � D � U H W D � T X H � U H S U H V H Q W D � D � W D [ D � G H � Y D U L D o m R � P p G L D � � 2 E V H U Y H � T X H �
a t a x a d e v a r i a ç ã o m é d i a , n o i n t e r v a l o d e 2 a 4 h o r a s , a p r o x i m o u o c o m p o r t a m e n t o d a c u r v a p r o d u ç ã o d e f o r m a 
l i n e a r , o u s e j a , p o r u m a r e t a . P o r t a n t o , n e s t e c a s o , a t a x a d e v a r i a ç ã o m é d i a r e p r e s e n t a a p e n a s u m c o m p o r t a m e n t o 
D S U R [ L P D G R � S D U D � D � Y D U L D o m R � G D � S U R G X o m R � S R U � K R U D � � 3 R U � X P D � V L P S O H V � D Q i O L V H � G R � J U i ¿ F R � S H U F H E H � V H � T X H � � H P � [ � � � D �
r e t a e s t á d i s t a n t e d a c u r v a p r o d u ç ã o , m o s t r a n d o q u e a t a x a d e v a r i a ç ã o m é d i a é a p e n a s u m a a p r o x i m a ç ã o p a r a a 
v a r i a ç ã o d a p r o d u ç ã o p o r h o r a .
P O R D E N T R O D O T E M A
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F i g u r a 6 . 1 T a x a d e V a r i a ç ã o M é d i a .
� 2 X W U R � H [ H P S O R � L P S R U W D Q W H � p � P R V W U D G R � Q D � ) L J X U D � � � � � � H P � T X H � D � F X U Y D � U H S U H V H Q W D � D � T X D Q W L G D G H � 4 � H P � W R Q H O D G D V � G H �
F H U H D L V � D U P D ] H Q D G R V � H P � X P � s i l o � H � [ � U H S U H V H Q W D � R � Q ~ P H U R � G H � G L D V � W U D Q V F R U U L G R V � G R � D U P D ] H Q D P H Q W R � � 1 H V W H � H [ H P S O R � �
p a r a c a l c u l a r a t a x a d e v a r i a ç ã o m é d i a , c o n s i d e r e a q u a n t i d a d e e m x = 2 d i a s d e Q ( 2 ) = 8 , 2 t o n e l a d a s e e m x = 7 d i a s 
d e Q ( 7 ) = 2 7 t o n e l a d a s . D e s t a f o r m a :
T a x a d e v a r i a ç ã o m é d i a = 
t o n . / d i a s� � � �
5
1 8 , 8
5
8 , 27 2
27
Q ( 2 )Q ( 7 )
 
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( V W H � U H V X O W D G R � P R V W U D � T X H � � Q R � L Q W H U Y D O R � G H � � � D W p � � � G L D V � � H V V H � V L O R � W H Y H � X P � D X P H Q W R � P p G L R � Q R � D U P D ] H Q D P H Q W R � G H � F H U H D L V �
G H � � � � � � W R Q H O D G D V � S R U � G L D � � 1 R � J U i ¿ F R � G D � ) L J X U D � � � � � � p � S R V V t Y H O � S H U F H E H U � T X H � D � U H W D � T X H � U H S U H V H Q W D � D � W D [ D � G H � Y D U L D o m R �
P p G L D � D S U R [ L P R X � R � F R P S R U W D P H Q W R � G D � F X U Y D � 4 � [ � � � 3 R G H � V H � W H U � D � I D O V D � L P S U H V V m R� � D � S D U W L U � G D � W D [ D � G H � Y D U L D o m R � P p G L D � �
T X H � D � T X D Q W L G D G H � G H � F H U H D L V � Q R � V L O R � V H P S U H � D X P H Q W R X � D R � S R Q W R � T X H � � S H O R � J U i ¿ F R � G D � ) L J X U D � � � � � � p � S R V V t Y H O � S H U F H E H U �
T X H � K R X Y H � G H F U p V F L P R � Q D � T X D Q W L G D G H � G H � F H U H D L V � � S R U � H [ H P S O R � � H Q W U H � R � T X D U W R � H � R � T X L Q W R � G L D � � ' D t � D � L P S R U W k Q F L D � G H �
c o m p r e e n d e r q u e a r e t a q u e r e p r e s e n t a a t a x a d e v a r i a ç ã o m é d i a d á a p e n a s u m a n o ç ã o a p r o x i m a d a d o c o m p o r t a m e n t o 
I X Q o m R � � j V � Y H ] H V � F R P � X P D � D S U R [ L P D o m R � Q m R � P X L W R � E R D � � F R P R � Q H V W H � H [ H P S O R �
P O R D E N T R O D O T E M A
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F i g u r a 6 . 2 T a x a d e V a r i a ç ã o M é d i a e o c o m p o r t a m e n t o d a c u r v a Q ( x ) .
$ � U H W D � T X H � U H S U H V H Q W D � D � W D [ D � G H � Y D U L D o m R � P p G L D � F K D P D � V H � U H W D � s e c a n t e � � $ � L Q F O L Q D o m R � G D � U H W D � V H F D Q W H � � R X � R � F R H ¿ F L H Q W H �
D Q J X O D U � G D � U H W D � V H F D Q W H � � p � D � S U y S U L D � W D [ D � G H � Y D U L D o m R � P p G L D � � 3 D U D � F R P S U H H Q G H U � H V W H � U H V X O W D G R � � O H P E U H � V H � G D � I y U P X O D � G R �
F R H ¿ F L H Q W H � D Q J X O D U � P � G D � U H W D � H V W X G D G D � Q R � 7 H P D � � �
E s s a f ó r m u l a f o r n e c e a i n c l i n a ç ã o d a r e t a d a d o s d o i s p o n t o s ( x
1
, y
1
) e ( x
2
, y
2
) . M a s a t a x a d e v a r i a ç ã o m é d i a t a m b é m é 
d a d a p o r d o i s p o n t o s :
T a x a d e v a r i a ç ã o m é d i a = 
P O R D E N T R O D O T E M A
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2 E V H U Y H � � H Q W m R � � T X H � D � L Q F O L Q D o m R � P � G D � U H W D � V H F D Q W H � W H P � R � P H V P R � F i O F X O R � G D � W D [ D � G H � Y D U L D o m R � P p G L D � � / R J R � � P � � W D [ D � G H �
Y D U L D o m R � P p G L D � � 2 E V H U Y H � R � J U i ¿ F R � G D � ) L J X U D � � � � �
F i g u r a 6 . 3 R e t a s e c a n t e .
T a x a d e V a r i a ç ã o I n s t a n t â n e a
A i n d a c o m r e l a ç ã o a o e x e m p l o d a e m p r e s a e m q u e a p r o d u ç ã o é d a d a p o r P ( x ) = x
2
� � S R G H � V H � F D O F X O D U � D � W D [ D � G H �
Y D U L D o m R � G D � S U R G X o m R � H P � X P � L Q V W D Q W H � H V S H F t ¿ F R � � 3 D U D � L V V R � � F R Q V L G H U H � D � P H V P D � I X Q o m R � S U R G X o m R � 3 � [ � � � [
2
 e o i n s t a n t e 
[ � � � � K R U D V � �
, Q L F L D O P H Q W H � � S D U D � H V V H � F i O F X O R � � F R Q V L G H U H � D � W D [ D � G H � Y D U L D o m R � P p G L D � Q R � L Q W H U Y D O R � � � D W p � � � � � K � � H P � T X H � K � U H S U H V H Q W D � R �
W D P D Q K R � G H V V H � L Q W H U Y D O R � � ¬ � P H G L G D � T X H � R � Y D O R U � K � G L P L Q X L � � D � G L V W k Q F L D � G H � � � D W p � � � K � W D P E p P � G L P L Q X L � H � D � W D [ D � G H � Y D U L D o m R �
P p G L D � V H � D S U R [ L P D � G D � W D [ D � G H � Y D U L D o m R � Q R � L Q V W D Q W H � [ � � � � � 3 D U D � H V W X G D U � H V W H � F R P S R U W D P H Q W R � � F R Q V L G H U H � D V � V H J X L Q W H V �
U H G X o } H V � S D U D � R � Y D O R U � G H � K �
• 3 D U D � K � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � D W p � � � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
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0 , 1
0 , 6 1
0 , 1
33 , 1
33 , 1
P ( 3 )P ( 3 , 1 )
22
 
= 6 , 1 t o n e l a d a s p o r h o r a .
P O R D E N T R O D O T E M A
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• 3 D U D � K � � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � D W p � � � � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
�
�
0 , 0 1
0 , 0 6 0 1
0 , 0 1
33 , 0 1
33 , 0 1
P ( 3 )P ( 3 , 0 1 )
22
 
=
 6 , 0 1 t o n e l a d a s p o r h o r a .
• 3 D U D � K � � � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � D W p � � � � � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
�
�
0 , 0 0 1
0 , 0 0 6 0 0 1
0 , 0 0 1
33 , 0 0 1
33 , 0 0 1
P ( 3 )P ( 3 , 0 0 1 )
22
 
= 6 , 0 0 1 t o n e l a d a s p o r h o r a .
3 D U D � R V � W U r V � Y D O R U H V � G H � K � F R Q V L G H U D G R V � � � � � � � � � � � � H � � � � � � � � S H U F H E H � V H � T X H � � j � P H G L G D � T X H � K � G L P L Q X L � � R X � V H M D � � j � P H G L G D �
T X H � D � G L V W k Q F L D � H Q W U H � � � H � � � � � K � G L P L Q X L � � � D � W D [ D � G H � Y D U L D o m R � P p G L D � D S U R [ L P D � V H � G H � 6 t o n e l a d a s p o r h o r a .
3 D U D � J D U D Q W L U � T X H � D � W D [ D � G H � Y D U L D o m R � Q R � L Q V W D Q W H � [ � � � � V H M D � G H � � � W R Q H O D G D V � S R U � K R U D � � G H Y H � V H � F R Q V L G H U D U � D � R E V H U Y D o m R � G D �
Y D U L D o m R � G H � K � F R P � Y D O R U H V � Q H J D W L Y R V � � R X � V H M D � � D S U R [ L P D U � G H � � � S H O R � L Q W H U Y D O R � � � � � K � D W p � � � � H Q W m R �
• 3 D U D � K � � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � � � D W p � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
�
�
0 , 1
0 , 5 9
0 , 1
2 , 93
2 , 93
P ( 2 , 9 )P ( 3 )
22
 
= 5 , 9 t o n e l a d a s p o r h o r a .
• 3 D U D � K � � � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � � � � D W p � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
�
�
0 , 0 1
0 , 0 5 9 9
0 , 0 1
2 , 9 93
2 , 9 93
P ( 2 , 9 9 )P ( 3 )
22
 
= 5 , 9 9 t o n e l a d a s p o r h o r a .
• 3 D U D � K � � � � � � � � � � R � L Q W H U Y D O R � F R Q V L G H U D G R � V H U i � � � � � � � D W p � � � � S R U W D Q W R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � V H U i �
T a x a d e v a r i a ç ã o m é d i a = 
 
�
 
�
�
0 , 0 0 1
0 , 0 0 5 9 9 9
0 , 0 0 1
2 , 9 9 93
2 , 9 9 93
P ( 2 , 9 9 9 )P ( 3 )
22
 
= 5 , 9 9 9 t o n e l a d a s p o r h o r a .
1 R Y D P H Q W H � � j � P H G L G D � T X H � R V � Y D O R U H V � G H � K � D S U R [ L P D P � V H � G H � ] H U R � � R � L Q W H U Y D O R � � � � � K � D W p � � � G L P L Q X L � H � D � W D [ D � G H � Y D U L D o m R �
P p G L D � D S U R [ L P D � V H � G H � 6 t o n e l a d a s p o r h o r a .
P O R D E N T R O D O T E M A
�
3 R U W D Q W R � � S R G H � V H � G L ] H U � T X H � Q R � L Q V W D Q W H � [ � � � � D � W D [ D � G H � Y D U L D o m R � p � G H � � � W R Q H O D G D V � S R U � K R U D � � 2 � S U R F H G L P H Q W R � G H � W R U Q D U � K �
� R � W D P D Q K R � G R � L Q W H U Y D O R � � S U y [ L P R � G H � ] H U R � F R U U H V S R Q G H � D R � F i O F X O R � G H � X P � O L P L W H �
2 � F i O F X O R � S U H F L V R � G R � Y D O R U � G H V V H � O L P L W H � Q m R � V H U i � G L V F X W L G R � Q H V W H � W H P D � � P D V � H O H � U H V X P H � W R G R V � R V � F i OF X O R V � U H D O L ] D G R V �
D Q W H U L R U P H Q W H � S D U D � R V � Y i U L R V � Y D O R U H V � G H � K � � 2 � F i O F X O R � S U H F L V R � G H V V H � O L P L W H � F K D P D � V H � d e r i v a d a , e s e u c á l c u l o s e r á e s t u d a d o 
n o p r ó x i m o t e m a .
3 D U D � G H W H U P L Q D U � D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � � G H Y H P � V H � J H U D U � Y i U L R V � Y D O R U H V � S D U D � K � � F D G D � Y H ] � P H Q R U H V � � H � G H W H U P L Q D U �
o v a l o r n o q u a l a t a x a d e v a r i a ç ã o m é d i a s e a p r o x i m a .
E x e m p l o 6 . 1 : 2 � F X V W R � & � � S D U D � V H � E H Q H ¿ F L D U � X P D � T X D Q W L G D G H � T � G H � W U L J R � � p � G D G R � S R U � & � T � � � � T
2
� � � � � � � � H P � T X H � & � p � G D G R �
e m r e a i s ( R $ ) e q é d a d o e m t o n e l a d a s ( t o n . ) .
a ) D e t e r m i n e a t a x a d e v a r i a ç ã o m é d i a d o c u s t o p a r a o i n t e r v a l o d e 1 a t é 6 t o n e l a d a s . 
b ) Q u a l a i n c l i n a ç ã o d a r e t a s e c a n t e a s s o c i a d a à t a x a d e v a r i a ç ã o m é d i a o b t i d a n o i t e m a " � �
F � � ' H W H U P L Q H � D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � G R � F X V W R � S D U D � T � � � � � � 8 W L O L ] H � � S D U D � D V � H V W L P D W L Y D V � � K � � “ � � � � � � K � � “ � � � � � � � K � �
“ � � � � � � � �
R e s o l u ç ã o :
a ) T a x a d e v a r i a ç ã o m é d i a = 
�˜�˜
 
�
�
5
5 0 0 )1� (�5 0 06�
16
C ( 1 )C ( 6 )
22
5
1 0 5
5
�1 0 8
5
5 0 0�1��5 0 06 ��
 
�
 
˜�˜
 2 1 R $ / t o n .
2 X � V H M D � � D � W D [ D � G H � Y D U L D o m R � P p G L D � G R � F X V W R � S D U D � E H Q H ¿ F L D U � G H � � � D W p � � � W R Q H O D G D V � G H � W U L J R � p � G H � 5 � � � � � � � � S R U � W R Q H O D G D � � 2 �
T X H � V L J Q L ¿ F D � T X H � � H P � P p G L D � � 5 � � � � � � � � V m R � J D V W R V � S R U � W R Q H O D G D � G H � W U L J R � S D U D � I D ] H U � R � E H Q H ¿ F L D P H Q W R � �
b ) C o m o o b s e r v a d o n a t e o r i a , a i n c l i n a ç ã o d a r e t a s e c a n t e n o i n t e r v a l o d e 1 a t é 6 t o n e l a d a s é o p r ó p r i o v a l o r d a t a x a d e 
v a r i a ç ã o i n s t a n t â n e a , o u s e j a , m
s e c a n t e
= 2 1 .
P O R D E N T R O D O T E M A
� �
c ) 3 D U D � G H W H U P L Q D U � D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � H P � [ � � � � � D Q D O L V D � V H � R � F R P S R U W D P H Q W R � S D U D � Y D O R U H V � G H � K � S R V L W L Y R V � H �
Y D O R U H V � Q H J D W L Y R V �
3 D U D � Y D O R U H V � S R V L W L Y R V � G H � K � � K � � � � � � � K � � � � � � � � K � � � � � � � � � � R V � L Q W H U Y D O R V � � � � D W p � � � � K � � V H U m R � � U H V S H F W L Y D P H Q W H � � 4 a t é 4 , 1 � � 4 
a t é 4 , 0 1 � � H � 4 a t é 4 , 0 0 1 .
• h = 0 , 1 , o u s e j a , i n t e r v a l o d e 4 a t é 4 , 1 .
• h = 0 , 0 1 , o u s e j a , i n t e r v a l o d e 4 a t é 4 , 0 1 .
• h = 0 , 0 0 1 , o u s e j a , i n t e r v a l o d e 4 a t é 4 , 0 0 1 .
2 E V H U Y H � T X H � � F R P � D � U H G X o m R � G R � Y D O R U � K � � D � W D [ D � G H � Y D U L D o m R � P p G L D � Q R � L Q W H U Y D O R � G H � � � D W p � � � � � K � D S U R [ L P D � V H � F D G D � Y H ] � P D L V �
G R � Y D O R U � G H � � � � 5 � � W R Q � � $ J R U D � � G H Y H � V H � D Q D O L V D U � R � F R P S R U W D P H Q W R � G D � W D [ D � G H � Y D U L D o m R � S D U D � Y D O R U H V � Q H J D W L Y R V � G H � K � � K � �
� � � � � � K � � � � � � � � � K � � � � � � � � � � � ( Q W m R � � R V � L Q W H U Y D O R V � � � � K � D W p � � � � V H U m R � � U H V S H F W L Y D P H Q W H � � 3 , 9 a t é 4 � � 3 , 9 9 a t é 4 � � H � 3 , 9 9 9 a t é 4 .
P O R D E N T R O D O T E M A
� �
• h = - 0 , 1 , o u s e j a , i n t e r v a l o d e 3 , 9 a t é 4 .
• h = - 0 , 0 1 , o u s e j a , i n t e r v a l o d e 3 , 9 9 a t é 4 .
• h = - 0 , 0 0 1 , o u s e j a , i n t e r v a l o d e 3 , 9 9 9 a t é 4 .
2 � Y D O R U � G D � W D [ D � G H � Y D U L D o m R � P p G L D � Q R � L Q W H U Y D O R � � � K � D W p � � � W D P E p P � V H � D S U R [ L P D � G H � � � � 5 � � W R Q � � T X D Q G R � Y D U L D � R � Y D O R U � G H � K � �
$ V V L P � � j � P H G L G D � T X H � R V � G R L V � L Q W H U Y D O R V � � � � K � D W p � � � H � � � D W p � � � K � � W R U Q D P � V H � S H T X H Q R V � � R � Y D O R U � G D � W D [ D � G H � Y D U L D o m R � P p G L D �
t e n d e a R $ 2 4 , 0 0 p o r t o n e l a d a . 
$ � S D U W L U � G R V � F i O F X O R V � D Q W H U L R U H V � � S R G H � V H � D ¿ U P D U � T X H � D � W D [ D � G H � Y D U L D o m R � L Q V W D Q W k Q H D � T X D Q G R � T � � � � p � G H � R $ 2 4 , 0 0 p o r 
t o n e l a d a .
P O R D E N T R O D O T E M A
� �
A c e s s e o s i t e I M E - U S P .
• C o n t é m e x e m p l o s s o b r e t a x a d e v a r i a ç ã o m é d i a , a l é m d e l i n k s p a r a o u t r o s a s s u n t o s c o m o a 
t a x a d e v a r i a ç ã o i n s t a n t â n e a .
L i n k : < h t t p : / / e c a l c u l o . i f . u s p . b r / d e r i v a d a s / p o p u p s / t x _ v a r _ m e d i a . h t m > . A c e s s o e m : 5 m a i . 2 0 1 4 .
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 . A p a r e c e r ã o v á r i o s e - b o o k s q u e c o n t ê m c o n c e i t o s 
s o b r e d e r i v a d a s ( t a x a d e v a r i a ç ã o i n s t a n t â n e a ) .
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 ! . A c e s s o e m : 5 m a i . 2 0 1 4 .
C á l c u l o I – T a x a d e V a r i a ç ã o .
• ( V W H � Y t G H R � P R V W U D � D � Q R o m R � L Q W X L W L Y D � V R E U H � D � W D [ D � Y D U L D o m R � X W L O L ] D Q G R � H [ H P S O R V � V L P S O L ¿ F D G R V �
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 � � 3 9 J � P B � O V ! . A c e s s o e m : 5 m a i . 2 0 1 4 .
T e m p o : 1 2 : 2 6
S
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
$ � W D [ D � G H � Y D U L D o m R � P p G L D � H � L Q V W D Q W k Q H D � D S D U H F H � H P � P X L W D V � V L W X D o } H V � G R � F R W L G L D Q R � G D V � S H V V R D V � � 2 � G H W D O K H � p � T X H � H V V D V � V L W X D o } H V �
Q m R � D S D U H F H P � H [ S O L F L W D P H Q W H � F R P � H V V H V � Q R P H V � � ) R U Q H o D � H [ H P S O R V � S U i W L F R V � S D U D � F D G D � X P D � G H V V D V � G X D V � W D [ D V � G H � Y D U L D o m R �
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 :
A p r o d u ç ã o s e m a n a l d e u m a e m p r e s a e m t o n e l a d a s é d a d a p o r P ( x ) = 4 x
2
, e m q u ex r e p r e s e n t a o t e m p o e m h o r a s e m q u e l i n h a 
G H � S U R G X o m R � I X Q F L R Q D � G H � I R U P D � L Q L Q W H U U X S W D � � 2 � J U i ¿ F R � � D � V H J X L U � � P R V W U D � D � H Y R O X o m R � G D � S U R G X o m R �
� �
A G O R A É A S U A V E Z
Q u e s t ã o 2
& R P � E D V H � Q R � J U i ¿ F R � � S R G H � V H � G L ] H U � T X H � D � W D [ D � G H � Y D U L D o m R � P p G L D � G D � S U R G X o m R � � F R Q V L G H U D Q G R � R � L Q W H U Y D O R � G H � � � � D � � � � K R U D V � � p � G H �
a ) 1 0 0 t o n e l a d a s / h o r a s .
b ) 2 0 0 t o n e l a d a s / h o r a s .
c ) � � � � � W R Q H O D G D V � K R U D V �
d ) 4 0 0 t o n e l a d a s / h o r a s .
e ) 5 0 0 t o n e l a d a s / h o r a s .
Q u e s t ã o 3
A t a x a d e v a r i a ç ã o m é d i a e m t o n e l a d a s p o r h o r a � G D � S U R G X o m R � � F R Q V L G H U D Q G R � R � L Q W H U Y D O R � G H � � � � D � � � � K R U D V � � p � G H �
a ) 8 0 t o n e l a d a s / h o r a s .
b ) 1 8 0 t o n e l a d a s / h o r a s .
c ) 2 8 0 t o n e l a d a s / h o r a s .
d ) � � � � � W R Q H O D G D V � K R U D V �
e ) 4 8 0 t o n e l a d a s / h o r a s .
� �
A G O R A É A S U A V E Z
Q u e s t ã o 4
1 D � W H R U L D � � Y H U L ¿ F R X � V H � T X H � D � L Q F O L Q D o m R � G D � U H W D � V H F D Q W H � H V W i � U H O D F L R Q D G D � j � W D [ D � G H � Y D U L D o m R � P p G L D � � $ � S D U W L U � G L V V R � � D � L Q F O L Q D o m R � G D �
r e t a s e c a n t e q u e p a s s a p e l o s p o n t o s ( 2 0 , 1 6 0 0 ) e ( 4 0 , 6 4 0 0 ) é :
a ) 8 0 .
b ) 1 0 0 .
c ) 2 0 0 .
d ) 2 4 0 .
e ) 2 8 0 .
Q u e s t ã o 5
A t a x a d e v a r i a ç ã o m é d i a e m t o n e l a d a s p o r h o r a � G D � S U R G X o m R � � F R Q V L G H U D Q G R � R � L Q W H U Y D O R � G H � � � � D � � � � K R U D V � � p � G H �
a ) 1 8 8 t o n e l a d a s / h o r a s .
b ) 2 8 8 t o n e l a d a s / h o r a s .
c ) � � � � � W R Q H O D G D V � K R U D V �
d ) 4 8 8 t o n e l a d a s / h o r a s .
e ) 5 8 8 t o n e l a d a s / h o r a s .
' L F D � � 2 V � Y D O R U H V � [ � � � � � H � [ � � � � � Q m R � H V W m R � H [ S O t F L W R V � Q R � J U i ¿ F R � � ' H V W D � I R U P D � � G H Y H � V H � X W L O L ] D U � D � I X Q o m R � S U R G X o m R � 3 � [ � � � � [
2
 p a r a 
G H W H U P L Q i � O R V �
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 � G H � F R V P p W L F R V � � F R Q V L G H U R X � V H � D � S U R G X o m R � F R P R � I X Q o m R � G R � F D S L W D O � L Q Y H V W L G R � H P � H T X L S D P H Q W R V � H � H V W D E H O H F H X � V H �
3 � T � � � � T
2
� � � � � H P � T X H � D � S U R G X o m R � 3 � p � G D G D � H P � O L W U R V � H � R � F D S L W D O � L Q Y H V W L G R � T � p � G D G R � H P � P L O K D U H V � G H � U H D L V �
� �
Q u e s t ã o 6
D � � 4 X D O � D � W D [ D � G H � Y D U L D o m R � P p G L D � G D � S U R G X o m R � Q R � L Q W H U Y D O R � G H � F D S L W D O � L Q Y H V W L G R � G H � � � D � � � P L O � U H D L V "
E � � 4 X D O � R � V L J Q L ¿ F D G R � S U i W L F R � S D U D � D � H P S U H V D � G H V V D � W D [ D � G H � Y D U L D o m R "
Q u e s t ã o 7
D e t e r m i n e a t a x a d e v a r i a ç ã o m é d i a d a p r o d u ç ã o p a r a o s i n t e r v a l o s :
I . 8 a 8 , 1 m i l r e a i s d e c a p i t a l i n v e s t i d o .
I I . 8 a 8 , 0 1 m i l r e a i s d e c a p i t a l i n v e s t i d o .
I I I . 8 a 8 , 0 0 1 m i l r e a i s d e c a p i t a l i n v e s t i d o .
Q u e s t ã o 8
D e t e r m i n e a t a x a d e v a r i a ç ã o m é d i a d a p r o d u ç ã o p a r a o s i n t e r v a l o s :
, � � � � � � D � � � P L O � U H D L V � G H � F D S L W D O � L Q Y H V W L G R �
, , � � � � � � � D � � � P L O � U H D L V � G H � F D S L W D O � L Q Y H V W L G R �
, , , � � � � � � � � D � � � P L O � U H D L V � G H � F D S L W D O � L Q Y H V W L G R �
Q u e s t ã o 9
A p a r t i r d a s r e s p o s t a s o b t i d a s n a s Q u e s t õ e s 7 e 8 , d e t e r m i n e a t a x a v a r i a ç ã o i n s t a n t â n e a d a p r o d u ç ã o q u a n d o o c a p i t a l i n v e s t i d o 
é q = 8 . E x p l i q u e s u a r e s p o s t a .
A G O R A É A S U A V E Z
� �
Q u e s t ã o 1 0
4 X D O � R � V L J Q L ¿ F D G R � G R � O L P L W H � D � V H J X L U �
8h8
P ( 8 )) hP ( 8
L i m
0h
��
��
o
4 X D O � R � U H V X O W D G R � T X H � H V V H � O L P L W H � I R U Q H F H "
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 � V R E U H � R � F R Q F H L W R � G D � W D [ D � G H � Y D U L D o m R � P p G L D � H � F R P R � F D O F X O i � O D � � $ O p P � L V V R � � H V W X G R X �
p r o b l e m a s a p l i c a d o s q u e e n v o l v e m t a x a d e v a r i a ç ã o m é d i a . V o c ê t a m b é m a p r e n d e u o q u e é t a x a d e v a r i a ç ã o i n s t a n t â n e a 
H � F R P R � F D O F X O i � O D � D � S D U W L U � G D V � W D [ D V � G H � Y D U L D o m R � P p G L D � � 5 H V R O Y H X � � W D P E p P � � S U R E O H P D V � T X H � H Q Y R O Y H P � W D [ D V � G H � Y D U L D o m R �
L Q V W D Q W k Q H D � � 3 R U � ¿ P � � Y R F r � L Q L F L R X � R � H V W X G R � G D � G H U L Y D G D �
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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 . 2 . e d . S ã o P a u l o : 
& H Q J D J H � / H D U Q L Q J � � � � � � �
7 $ 1 � � 6 R R � 7 D Q J � � 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 . 5 . e d . S ã o P a u l o : P i o n e i r a , 2 0 0 1 .
L Q V W D Q W k Q H D � � 3 R U � ¿ P � � Y R F r � L Q L F L R X � R � H V W X G R � G D � G H U L Y D G D �
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S e c a n t e : U m a r e t a s e c a n t e i n t e r c e p t a u m a c u r v a e m d o i s p o n t o s o u m a i s .
S i l o : � & R Q V W U X o m R � W L S L F D P H Q W H � F L O t Q G U L F D � X W L O L ] D G D � S D U D � S U H S D U D o m R � � F R Q V H U Y D o m R � H � D U P D ] H Q D J H P � G H � F H U H D L V � � V X E V W k Q F L D V �
P L Q H U D L V � R X � J U m R V �
T o n e l a d a s : � 8 Q L G D G H � G H � P H G L G D � G H � P D V V D � T X H � U H S U H V H Q W D � � � � � � � T X L O R J U D P D V �
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T o n e l a d a s : � 8 Q L G D G H � G H � P H G L G D � G H � P D V V D � T X H � U H S U H V H Q W D � � � � � � � T X L O R J U D P D V �
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R e s p o s t a : � 5 H V S R V W D � S H V V R D O � � 6 X J H V W } H V �
T a x a d e V a r i a ç ã o M é d i a
O s i t e � , & $ 5 5 2 6 � S X E O L F R X � � H P � � � � � � � � � � � � � T X H � Q R � % U D V L O � V m R � U R X E D G R V � � � � � � � F D U U R V � S R U � G L D � � ( V V H � G D G R � p � X P D � W D [ D � G H �
v a r i a ç ã o , p o i s r e p r e s e n t a a r e l a ç ã o e n t r e o n ú m e r o d e r o u b o s d e c a r r o s e o n ú m e r o d e d i a s . A t a x a d e v a r i a ç ã o é m é d i a , 
M i � T X H � � F R P R � D � S U y S UL D � P D W p U L D � L Q I R U P D � � G H � M D Q H L U R � D � M X Q K R � G H � � � � � � I R U D P � U R X E D G R V � � � � � � � � � Y H t F X O R V � � H � R � L Q W H U Y D O R � G H �
t e m p o n e s s e c a s s o é d e 1 8 1 d i a s . A s s i m , 
1 2 5 6
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 c a r r o s p o r d i a .
S i t e I C A R R O S : � K W W S � � � Z Z Z � L F D U U R V � F R P � E U � Q R W L F L D V � J H U D O � E U D V L O � W H P � P D L V � G H � � � � � � Y H L F X O R V � U R X E D G R V � S R U � G L D � � � � � � � �
h t m l > . A c e s s o e m : 0 5 / 0 5 / 2 0 1 4 .
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T a x a d e V a r i a ç ã o I n s t a n t â n e a
• $ � Y H O R F L G D G H � I R U Q H F L G D � S H O R � Y H O R F t P H W U R � G H � X P � F D U U R � p � X P D � W D [ D � G H � Y D U L D o m R � � M i � T X H � U H S U H V H Q W D � D � U H O D o m R � H Q W U H �
H V S D o R � H � W H P S R � � ( V V D � W D [ D � G H � Y D U L D o m R � p � L Q V W D Q W k Q H D � � S R L V � Q m R � F R Q V L G H U D � H V S D o R � L Q L F L D O � H � ¿ Q D O � Q H P � W H P S R � L Q L F L D O � H �
¿ Q D O � � 7 H P � V H � D S H Q D V � D � Y H O R F L G D G H � H P � F D G D � L Q V W D Q W H � F R Q V L G H U D G R �
Q u e s t ã o 2
R e s p o s t a : $ O W H U Q D W L Y D � %
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R e s p o s t a : A l t e r n a t i v a C
T a x a d e v a r i a ç ã o m é d i a = 
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R e s p o s t a : A l t e r n a t i v a D
2 V � S R Q W R V � V m R � � � � � � � � � � � H � � � � � � � � � � � � ( Q W m R � � D � L Q F O L Q D o m R � G D � U H W D � V H F D Q W H � � R X � F R H ¿ F L H Q W H � D Q J X O D U � G D � U H W D � V H F D Q W H � � p �
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 A
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 t o n e l a d a s p o r h o r a .
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T a x a d e v a r i a ç ã o m é d i a = 
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 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d a . 
E � � 1 R � L Q W H U Y D O R � G H � L Q Y H V W L P H Q W R � H Q W U H � � � P L O � D � � � P L O � U H D L V � � D � S U R G X o m R � H P � P p G L D � L U i � D X P H Q W D U � � � � O L W U R V � S D U D � F D G D � P L O �
r e a i s i n v e s t i d o . A s s i m , a e m p r e s a p o d e e s t i m a r q u e , a p a r t i r d o i n v e s t i m e n t o m a i o r q u e 7 m i l r e a i s p a r a c a d a m i l r e a i s 
i n v e s t i d o , a p r o d u ç ã o i r á a u m e n t a r e m 4 8 l i t r o s d e c o s m é t i c o s .
Q u e s t ã o 7
R e s p o s t a :
I . I n t e r v a l o d e 8 a t é 8 , 1 .
 
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
I I . I n t e r v a l o d e 8 a t é 8 , 0 1 .
 
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
G A B A R I T O
 
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I I . I n t e r v a l o d e 8 a t é 8 , 0 0 1 .
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
Q u e s t ã o 8
R e s p o s t a : 
I . I n t e r v a l o d e 7 , 9 a t é 8
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
I I . I n t e r v a l o d e 7 , 9 9 a t é 8 .
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
I I I . I n t e r v a l o d e 7 , 9 9 9 a t é 8 .
 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
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Q u e s t ã o 9
R e s p o s t a : N a Q u e s t ã o 7 , a s r e p o s t a s o b t i d a s f o r a m : 
• , Q W H U Y D O R � G H � � � D W p � � � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � �
• , Q W H U Y D O R � G H � � � D W p � � � � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � � �
• , Q W H U Y D O R � G H � � � D W p � � � � � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � � � �
2 E V H U Y H � T X H � � j � P H G L G D � T X H � R V � L Q W H U Y D O R V � V H � W R U Q D P � P H Q R U H V � � D � W D [ D � G H � Y D U L D o m R � P p G L D � D S U R [ L P D � V H � G H � � � �
N a Q u e s t ã o 8 , a s r e p o s t a s o b t i d a s f o r a m : 
• , Q W H U Y D O R � G H � � � � � D W p � � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � �
• , Q W H U Y D O R � G H � � � � � � D W p � � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � � �
• , Q W H U Y D O R � G H � � � � � � � D W p � � � � 7 D [ D � G H � Y D U L D o m R � P p G L D � � � � � � � �
$ Q D O R J D P H Q W H � � j � P H G L G D � T X H � R V � L Q W H U Y D O R V � W R U Q D P � V H � P H Q R U H V � � D � W D [ D � G H � Y D U L D o m R � P p G L D � W D P E p P � V H � D S U R [ L P D � G H � � � �
3 R U W D Q W R � � U H G X ] L Q G R � V H � R V � L Q W H U Y D O R V � H P � D P E R V � R V � V H Q W L G R V � � S R U � Y D O R U H V � P H Q R U H V � H � P D L R U H V � T X H � � � � � D � W D [ D � G H � Y D U L D o m R �
m é d i a v a i s e a p r o x i m a n d o d e 4 8 . A s s i m , a t a x a d e v a r i a ç ã o i n s t a n t â n e a , q u a n d o q = 8 , é d e 4 8 l i t r o s p o r m i l u n i d a d e s 
d e r e a i s i n v e s t i d o .
Q u e s t ã o 1 0
R e s p o s t a : O 
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r e p r e s e n t a a t a x a d e v a r i a ç ã o i n s t a n t â n e a e m q = 8 . O c á l c u l o p r e c i s o d e s s e l i m i t e r e s u m e 
W R G R V � R V � F i O F X O R V � U H D O L ] D G R V � Q D V � 4 X H V W } H V � � � H � � � � 3 R U W D Q W R � � S H O D � 4 X H V W m R � � � � R � U H V X O W D G R � G H V V H � O L P L W H � p �
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 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
A s s i m , a e m p r e s a s a b e p r e c i s a m e n t e q u e , q u a n d o o c a p i t a l i n v e s t i d o f o r 8 m i l r e a i s , a p r o d u ç ã o d e c o s m é t i c o s e s t á 
a u m e n t o e m 4 8 l i t r o s p o r m i l u n i d a d e s d e r e a i s i n v e s t i d o .
G A B A R I T O