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M a t e m á t i c 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 Te m a 0 7 Té c n i c a s d e D e r i v a ç ã o T e m a 0 7 T é c n i c a s d e D e r i v a ç ã o 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 é c n i c a s d e D e r i v a ç ã o . C a d e r n o d e A t i v i d a d e s . A n h a n 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 � S R U W X J X H V D � R X � T X D O T X H U � R X W U R � L G L R P D � P á g . 1 6 P á g . 1 6 P á g . 1 7 P á g . 1 6 P á g . 1 2P á g . 1 1 A C O M P A 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 3 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 : � A i n c l i n a ç ã o d a r e t a t a n g e n t e a p a r t i r d o c á l c u l o d a d e r i v a d a . � A n o t a ç ã o u t i l i z a d a p a r a o c á l c u l o d a d e r i v a d a . � R e g r a s p a r a d e r i v a ç ã o d e f u n ç õ e s . � O c á l c u l o d a s d e r i v a d a s a p a r t i r d e f ó r m u l a s p r e e s t a b e l e c i d a s . � $ � H T X D o m R � G D � U H W D � W D Q J H Q W H � D � X P D � F X U Y D � H P � X P � S R Q W R � H V S H F L ¿ F D G R � 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 � � 4 X D O � D � U H O D o m R � H Q W U H � D � G H U L Y D G D � 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 � D � X P D � F X U Y D � H P � X P � S R Q W R � H V S H F L ¿ F D G R " � & R P R � G H W H U P L Q D U � D � G H U L Y D G D � G H � X P D � I X Q o m R � S R O L Q R P L D O " � � 4 X D O � D � Q R W D o m R � X W L O L ] D G D � S D U D � U H S U H V H Q W D U � R � F i O F X O R � G D � G H U L Y D G D " � & R P R � G H W H U P L Q D U � D � H T X D o m R � G D � U H W D � W D Q J H Q W H � D � X P D � F X U Y D � H P � X P � S R Q W R � H V S H F L ¿ F D G R " C o n t e ú d o C O N V I T E À L E I T U R A 4 T é c n i c a s d e D e r i v a ç ã o I n t r o d u ç ã o 1 R � W H P D � D Q W H U L R U � � Y R F r � H V W X G R X � R � V L J Q L ¿ F D G R � H � R � F i O F X O R � G D � W D [ D � G H � Y D U L D o m R � i n s t a n t â n e a a p a r t i r d e u m p r o c e s s o U H S H W L W L Y R � H � L Q W X L W L Y R � � 1 H V W H � W H P D � � R � Y D O R U � G 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 � � M i � G H ¿ Q L G R � F R P R � G H U L Y D G D � � V H U i � G H W H U P L Q D G R � d e f o r m a d i r e t a p o r m e i o d e f ó r m u l a s . O u s o d e s s a s f ó r m u l a s p e r m i t e o c á l c u l o i m e d i a t o e p r e c i s o d a s d e r i v a d a s . D e r i v a d a e I n c l i n a ç ã o d a R e t a T a n g e n t e & R P R � M i � H V W X G D G R � � R V � S R Q W R V � X W L O L ] D G R V � S D U D � F D O F X O D U � D � W D [ D � G H � Y D U L D o m R � P p G L D � G H V F U H Y H P � X P D � U H W D � s e c a n t e , e m T X H � D � L Q F O L Q D o m R � G H V V D � U H W D � p � R � S U y S U L 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 � � 2 � J U i ¿ F R � G D � ) L J X U D � � � � � P R V W U D � D � U H W D � V H F D Q W H � S D V V D Q G R � S H O R V � S R Q W R V � � [ � , y � � � H � � [ 2 , y 2 ) . O b s e r v e q u e e s s a r e t a p o s s u i i n c l i n a ç ã o m � � T X H � p � L J X D O � j � W D [ D � G H � Y D U L D o m R � P p G L D � F i g u r a 7 . 1 R e t a s e c a n t e . T é c n i c a s d e D e r i v a ç ã o P O R D E N T R O D O T E M A 5 1 R � W H P D � D Q W H U L R U � � Y R F r � H V W X G R X � 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 � X P � S R Q W R � [ � � � ( V V D � W D [ D � G H � Y D U L D o m R � Q m R � W H P � U H O D o m R � F R P � X P D � U H W D � V H F D Q W H � � S R L V � V H � W U D W D � D S H Q D V � G H � X P � S R Q W R � H V S H F t ¿ F R � � [ � , y � ) , e n ã o d e d o i s p o n t o s , c o m o a c o n t e c e n a r e t a V H F D Q W H � � 3 R U � R X W U R � O D G R � � 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 � W H P � U H O D o m R � F R P � D � L Q F O L Q D o m R � G H � X P D � U H W D � t a n g e n t e � � R X � V H M D � � H V W D � I R U Q H F H � R � F R H ¿ F L H Q W H � D Q J X O D U � m t � G H � X P D � U H W D � W D Q J H Q W H � j � I X Q o m R � I � [ � � Q R � S R Q W R � [ � � � 9 H M D � D � ) L J X U D � � � � � F i g u r a 7 . 2 R e t a t a n g e n t e . 3 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 � I R U Q H F 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 � 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 � I R U Q H F H � D � i n c l i n a ç ã o d a r e t a t a n g e n t e . E x e m p l o 7 . 1 : � $ � 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 D � I X Q o m R � S U R G X o m R � 3 � [ � � Q R � S R Q W R � [ � � � � � K R U D V � p � G H � � � � U H D L V � K R U D � � 4 X D O � 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 � D � H V V D � I X Q o m R � 3 � [ � � Q R � S R Q W R � [ � � � � " R e s o l u ç ã o : & R P R � H [ S O L F D G R � � D � L Q F O L Q D o m R � P t � G D � U H W D � W D Q J H Q W H � H P � X P � S R Q W R � H V S H F L ¿ F D G R � � F R P R � R � S R Q W R � [ � � � � � � p � D � S U y S U L 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 V V D � I X Q o m R � Q H V V H � S R Q W R � � R X � V H M D � � m t � � � � � P O R D E N T R O D O T E M A 6 C á l c u l o d a D e r i v a d a d e u m a F u n ç ã o 1 R � W H P D � D Q W H U L R U � � G H ¿ Q L X � V H � 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 � G H � X P D � I X Q o m R � I � [ � � � H P � X P � S R Q W R � [ � � � p � F K D P D G D � G H � G H U L Y D G D � G H � I � [ � � H P � [ � � � 3 D U D � V L P S O L ¿ F D U � H V W D � Q R W D o m R � � D � G H U L Y D G D � G H � I � [ � � H P � [ � s e r á e s c r i t a c o m o f ’ ( x ) � � R X � V H M D � � F R O R F R X � V H � R � D S y V W U R I R � Q D � O H W U D � T X H � U H S U H V H Q W D � D � I X Q o m R � � 2 X W U D � Q R W D o m R � S D U D � R � F i O F X O R � G D � G H U L Y D G D � P X L W R � X W L O L ] D G D � p � ( V V D � Q R W D o m R � p � P X L W R � V X J H V W L Y D � S D U D � D � L G H L D � G H � W D [ D � G H � Y D U L D o m R � � O H P E U D Q G R � � S R U � H [ H P S O R � � D � W D [ D � G H � Y D U L D o m R � P p G L D � x y ' ' . � $ W p � D J R U D � � Y R F r � H V W X G R X � T X H � � S D U D � G H W H U P L Q D U � R � Y D O R U � Q X P p U L F R � G D � G H U L Y D G D � G H � X P D � I X Q o m R � H P � X P � S R Q W R � H V W D E H O H F L G R � � X W L O L ] D � V H � X P � S U R F H V V R � U H S H W L W L Y R � H � D S U R [ L P D G R � T X H � H Q Y R O Y H � R � F i O F X O R � G H � Y i U L D V � W D [ D V � G H � Y D U L D o m R � P p G L D � � ( Q W U H W D Q W R � � V H U i � D S U H V H Q W D G R � X P � S U R F H V V R � H [ D W R � T X H � X W L O L ] D � I y U P X O D V � S D U D � G H W H U P L Q D U � D � G H U L Y D G D � G H � X P D � I XQ o m R � � $ Q W H V � G H � Y H U L ¿ F D U � H V V D V � I y U P X O D V � � F R Q V L G H U H � R � S U R E O H P D � H P � 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 � [ � � � � K R U D V � S D U D � D � I X Q o m R � S U R G X o m R � 3 � [ � � � [ 2 , d a d a e m t o n e l a d a s . A r e s p o s t a d e s s e p r o b l e m a f o i o b t i d a , n o t e m a a n t e r i o r , p o r m e i o d e u m a e s t i m a t i v a � D S y V � R � F i O F X O R � U H S H W L W L Y R � G H � Y i U L D V � W D [ D V � G H � Y D U L D o m R � P p G L D � � 2 � Y D O R U � H Q F R Q W U D G R � � M i � X W L O L ] D Q G R � D � n o t a ç ã o d e d e r i v a d a , f o i : I � µ � � � � � � � � W R Q H O D G D V � K R U D � ( Q W U H W D Q W R � � H V V H � U H V X O W D G R � S R G H � V H U � R E W L G R � G L U H W D P H Q W H � S H O R � F i O F X O R � G R � O L P L W H � 2 � F i O F X O R � G D � G H U L Y D G 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 � � S R U � P H L R � G R � O L P L W H � F K D P D � V H � F i O F X O R � S H O D � G H ¿ Q L o m R � � 2 � F i O F X O R � G D � G H U L Y D G D � S H O D � G H ¿ Q L o m R � Q m R � V H U i � G H V H Q Y R O Y L G R � D T X L � � P D V � S R G H � V H U � H Q F R Q W U D G R � H P � 7 D Q � � � � � � � � � 2 � T X H � V H U i � G H V H Q Y R O Y L G D � Q H V W H � W H P D � p � D � I y U P X O D � U H V X O W D Q W H � G H V V H � F i O F X O R � P O R D E N T R O D O T E M A 7 $ V V L P � � T X D Q G R � D � I X Q o m R � p � 3 � [ � � � [ 2 � � 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 X � D � G H U L Y D G D � � S R V V X L � D � I y U P X O D � P ‘ ( x ) = 2 x . P o r t a n t o , S D U D � 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 � H P � [ � � � G H U L Y D G D � H P � [ � � � � E D V W D � F D O F X O D U � 3 � µ � � � � � 3 � µ � � � � � � � � � � � � � � � W R Q H O D G D V � K R U D 2 E V H U Y H � T X H � R � X V R � G D � I y U P X O D � V L P S O L ¿ F R X � R � S U R F H V V R � G R � F i O F X O R � G D � G H U L Y D G D � H P � X P � S R Q W R � � Q m R � V H Q G R � P D L V � Q H F H V V i U L R � R � F i O F X O R � U H S H W L W L Y R � G H � Y i U L D V � W D [ D V � G H � Y D U L D o m R � P p G L D � � ( Q W U H W D Q W R � � H V V D � I y U P X O D � Q m R � p � ~ Q L F D � � S R L V � D � F D G D � Q R Y D � I X Q o m R � f o r n e c i d a o c á l c u l o d a d e r i v a d a g e r a u m a n o v a f ó r m u l a . $ � V H J X L U � � V m R � D S U H V H Q W D G D V � D O J X P D V � U H J U D V � T X H � S H U P L W H P � D � R E W H Q o m R � G H V V D V � I y U P X O D V � T X D Q G R � D � G H U L Y D G D � p � G H � I X Q o } H V � p o l i n o m i a i s . � R e g r a d a P o t ê n c i a : � S D U D � T X D O T X H U � Q ~ P H U R � U H D O � n � � V H � I � [ � � � [ n , e n t ã o : 2 � T X H � V L J Q L ¿ F D � T X H � � S D U D � F D O F X O D U � D � G H U L Y D G D � G D � I X Q o m R � I � [ � � � [ n � � V X E W U D L � V H � � � G R � H [ S R H Q W H � H � P X O W L S O L F D � V H � R � U H V X O W D G R � S H O R � H [ S R H Q W H � R U L J L Q D O � � E x e m p l o 7 . 2 : � & D O F X O H � D � G H U L Y D G D � G H � I � [ � � � [ 5 � � ( P � V H J X L G D � � G 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 D � I X Q o m R � I � [ � � H P � [ � � � � R e s o l u ç ã o : � R e g r a d a M u l t i p l i c a ç ã o p o r u m a C o n s t a n t e : � V H � F � p � X P D � F R Q V W D Q W H � H � I � [ � � p � X P D � I X Q o m R � G H U L Y i Y H O � � F � � � I � [ � � W D P E p P � p � X P D � I X Q o m R � G H U L Y i Y H O � H P O R D E N T R O D O T E M A 8 E x e m p l o 7 . 3 : � & D O F X O H � D � G H U L Y D G D � G H � I � [ � � � � � [ � � � ( P � V H J X L G D � � G 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 D � I X Q o m R � I � [ � � H P � [ � � � � R e s o l u ç ã o : ĺ f ‘ ( x ) = 3 0 x 2 . 3 D U D � [ � � � I � µ � � � � � � � ĺ � I � µ � � � � � � � ĺ � I � µ � � � � 4 8 0 . � R e g r a d a S o m a o u D i f e r e n ç a : � V H � I � [ � � H � J � [ � � V m R � G X D V � I X Q o } H V � G H U L Y i Y H L V � � D � V R P D � R X � D � G L I H U H Q o D � � I � [ � � r � J � [ � � � W D P E p P � p � X P D � I X Q o m R � G H U L Y i Y H O � H , V V R � V L J Q L ¿ F D � T X H � D � G H U L Y D G D � G H � X P D � V R P D � � R X � G L I H U H Q o D � � p � D � V R P D � � R X � G L I H U H Q o D � � G D V � G H U L Y D G D V � G D V � S D U F H O D V � E x e m p l o 7 . 4 : & D O F X O H � D � G H U L Y D G D � G H � I � [ � � � � [ � � � � � [ � � � � � [ 2 � � ( P � V H J X L G D � � G 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 D � I X Q o m R � I � [ � � H P � [ � � � � R e s o l u ç ã o : O b s e r v a ç ã o : � 4 X D Q G R � D � I X Q o m R � p � F R Q V W D Q W H � � I � [ � � � F � � D � G H U L Y D G D � G D � I X Q o m R � p � L J X D O � D � ] H U R � � S R L V � Q m R � K i � W D [ D � G H � Y D U L D o m R � H P � X P D � I X Q o m R � F R Q V W D Q W H � � M i � T X H � R � Y D O R U � G D � I X Q o m R � Q X Q F D � F U H V F H � H � Q X Q F D � G H F U H V F H � P O R D E N T R O D O T E M A 9 E x e m p l o 7 . 5 : � & D O F X O H � D � G H U L Y D G D � G H � I � [ � � � [ � � � � � � ( P � V H J X L G D � � G 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 D � I X Q o m R � I � [ � � H P � [ � � � � 3 D U D � [ � � � I � µ � � � � � � � � � � ĺ � f ‘ ( 8 ) = 1 6 . E x e m p l o 7 . 6 : � 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 � [ � G H � W U L J R � p � G D G R � S R U � & � [ � � � � [ 2 � � � � � � � � H P � T X H � & � p � G D G R � H P � U H D L V � � 5 � � � H � [ � p � G D G R � H P � W R Q H O D G D V � � W R Q � � � D � � ' 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 � T X D Q G R � [ � � W R Q H O D G D V � b ) Q u a l a i n c l i n a ç ã o m � G D � U H W D � W D Q J H Q W H � j � F X U Y D � & � [ � � T X D Q G R � [ � " F � � $ � I X Q o m R � I � [ � � T X H � G H V F U H Y H � D � U H W D � W D Q J H Q W H � j � F X U Y D � U H S U H V H Q W D G D � S R U � & � [ � � � � [ 2 � � � � � � � Q R � S R Q W R � [ � � R e s o l u ç ã o : D � � 3 D U D � H Q F R Q W U 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 � V H � F D O F X O D U � D � G H U L Y D G D � G H � & � [ � � Q R � S R Q W R � [ � � � 2 E V H U Y H � T X H � � Q H V V H � H [ H P S O R � � D � I X Q o m R � G H S H Q G H � G D � Y D U L i Y H O � [ � � S R U W D Q W R � � D � G H U L Y D G D � G H Y H � V H U � I H L W D � F R P � U H O D o m R � D � H V V D � Y D U L i Y H O � � $ V V L P � � D � G H U L Y D G D � G H � & � [ � � 3 D U D � [ � � � & � µ � � � � � � � � � � ĺ � & � µ � � � � � � � � � � ĺ � C ‘ ( 7 ) = 4 2 r e a i s p o r t o n e l a d a . $ � 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 � � T X D Q G R � [ � � � � W R Q H O D G D V � � p � G H � 5 � � � � � � � � S R U � W R Q H O D G D � E � � $ � L Q F O L Q D o m R � P � G D � U H W D � W D Q J H Q W H � Q R � S R Q W R � [ � � p � D � S U y S U L 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 V V D � I X Q o m R � Q H V W H � S R Q W R � � R X � V H M D � � m = 4 2 . P O R D E N T R O D O T E M A 1 0 F � � & R P R � H V W X G D G R � Q R � 7 H P D � � � � D � I X Q o m R � T X H � U H S U H V H Q W D � X P D � U H W D � p � G D G D� S R U � I � [ � � � m [ � � � b . O v a l o r m , c o n f o r m e o i t e m D Q W H U L R U � � p � L J X D O � D � � � � � $ � I X Q o m R � I � [ � � S D U F L D O P H Q W H � S U R Q W D � p � I � [ � � � 4 2 [ � � � b P a r a d e s c o b r i r o v a l o r d e b , b a s t a o b s e r v a r q u e , q u a n d o d u a s f u n ç õ e s s ã o t a n g e n t e s e n t r e s i , e s t a s p o s s u e m u m p o n t o H P � F R P X P � � 2 E V H U Y H � D � ) L J X U D � � � � � F i g u r a 7 . 3 P o n t o e m c o m u m . 2 E V H U Y H � S H O D � ) L J X U D � � � � � T X H � I � [ � � � � & � [ � ) = y � � � R X � V H M D � � S R V V X H P � R � P H V P R � Y D O R U � H P � [ � . ' H V W D � I R U P D � � F R P R � D � U H W D � W D Q J H Q F L D � D � F X U Y D � Q R � S R Q W R � [ � � � H Q X Q F L D G R � � � H Q W m R � � Q H V W H � S R Q W R � � D V � G X D V � I X Q o } H V � � U H W D � H � F X U Y D � � W r P � R � P H V P R � Y D O R U � � R X � V H M D � � I � � � & � � � � � $ � S D U W L U � G H V V D � L Q I R U P D o m R � � p � S R V V t Y H O � G H W H U P L Q D U � b : I � � � & � � � � � � � � � � b � � � � � � � � 2 � � � � � � � � � � b � � � � � � � b � � � � � � � � � b � � � � $ V V L P � � D � I X Q o m R � T X H � G H V F U H Y H � D � U H W D � W D Q J H Q W H � j � F X U Y D � & � [ � � � � [ 2 � � � � � � � Q R � S R Q W R � [ � � p � f ( x ) = 4 2 x + 3 5 3 . P O R D E N T R O D O T E M A 1 1 I M E - U S P � A c e s s e o s i t e � , 0 ( � 8 6 3 � � & R Q W p P � F R Q F H L W R V � V R E U H � G H U L Y D G D V � D O p P � G H � P X L W R V � H [ H P S O R V � V R E U H � R � t e m a . H á l i n k s q u e d i r e c i o n a m p a r a o u t r o s t e m a s r e l a c i o n a d o s a d e r i v a d a s . D i s p o n í v e l e m : � K W W S � � � H F D O F X O R � L I � X V S � E U � ! � � $ F H V V R � H P � � � � P D L � � � � � � � M a t e m a t i q u ê s � A c e s s e o s i t e � 0 D W H P D W L T X r V � � & R Q W p P � P X L W D V � L Q I R U P D o } H V � V R E U H � R � F i O F X O R � G D � G H U L Y D G D � � $ O p P � G L V V R � � F R Q W p P � P X L W D V � O L V W D V � G H � H [ H U F t F L R V � S D U D � H V W X G R � D i s p o n í v e l e m : � K W W S � � � Z Z Z � P D W H P D W L T X H V � F R P � E U � F R Q W H X G R � S K S " L G � � � ! � � $ F H V V R � H P � � � � P D L � � � � � � � 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 � 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 . N o c a m p o p a r a p e s q u i s a d i g i t e 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 o s c o n c e i t o s s o b r e d e r i v a d a s . D i s p o n í v e l e m : � 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 H P � � � � P D L � � � � � � � D e r i v a d a | T é c n i c a s d e d e r i v a ç ã o 1 | M a t e m á t i c a � A c e s s e o v í d e o : D e r i v a d a | T é c n i c a s d e d e r i v a ç ã o 1 | M a t e m á t i c a � � ( V W H � Y t G H R � F R Q W p P � H [ S O L F D o m R � G D V � W p F Q L F D V � G H � G H U L Y D o m R � � 2 � Y t G H R � X W L O L ] D � X P � I R U P D W R � E H P � G L G i W L F R � D i s p o n í v e l e m : � K W W S � � � Z Z Z � \ R X W X E H � F R P � Z D W F K " Y � + 8 2 � � Y 4 , T , ! � � $ F H V V R � H P � � � � P D L � � � � � � � C á l c u l o 1 / a u l a 1 4 - A D e r i v a d a c o m o u m a F u n ç ã o - p a r t e 1 � A c e s s e o v í d e o : C á l c u l o 1 / a u l a 1 4 - A D e r i v a d a c o m o u m a F u n ç ã o - p a r t e 1 � � ( V W H � Y t G H R � P R V W U D � X P D � W H O H D X O D � P L Q L V W U D G D � Q D � 8 1 , & $ 0 3 � V R E U H � L Q W U R G X o m R � D R � F R Q F H L W R � G H � G H U L Y D G D � D i s p o n í v e l e m : � K W W S � � � Z Z Z � \ R X W X E H � F R P � Z D W F K " Y + . � ' ' [ � * � = 8 ! � � $ F H V V R � H P � � � � P D L � � � � � � � I M E - U S P A C O M P A N H E N A W E B 1 2 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 1 D � / H L W X U D � 2 E U L J D W y U L D � � G H ¿ Q L X � V H � T X H � D � G H U L Y D G D � G H � X P D � F R Q V W D Q W H � p � L J X D O � D � ] H U R � � 3 D U D � F R P S U R Y D U � H V W H � I D W R � � F R Q V W U X D � R � J U i ¿ F R � G D � I X Q o m R � I � [ � � � � � � X P D � I X Q o m R � F R Q V W D Q W H � � ( P � V H J X L G D � � Y H U L ¿ T X H � D � L Q F O L Q D o m R � G D � U H W D � F R P � U H O D o m R � D R � H L [ R � K R U L ] R Q W D O � H � F R Q F O X D � V R E U H � o r e s u l t a d o d e s s a d e r i v a d a . 2 X W U D � F R Q F O X V m R � L Q W H U H V V D Q W H � p � V R E U H � D � I X Q o m R � D ¿ P � � H V W X G D G D � Q R � 7 H P D � � � � $ � I X Q o m R � D ¿ P � W H P � D � I R U P D � I � [ � � m [ � b , e m q u e m U H S U H V H Q W D � R � F R H ¿ F L H Q W H � D Q J X O D U � G D � U H W D � � P � z � � � � H � b � U H S U H V H Q W D � R � F R H ¿ F L H Q W H � O L Q H D U � G D � U H W D � � ' H W H U P L Q H � D � G H U L Y D G D � G D � I X Q o m R � I � [ � m [ � b � F R P � U H O D o m R � j � Y D U L i Y H O � [ � � 2 � T X H � V H � S R G H � G L ] H U � V R E U H � D � G H U L Y D G D � G H V V D � I X Q o m R " Q u e s t ã o 2 $ � 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 � X P D � I X Q o m R � S U R G X o m R � 3 � [ � � Q R � L Q V W D Q W H � W U r V � K R U D V � p � � � � U H D L V � K R U D � � 4 X D O � D � L Q F O L Q D o m R � m d a r e t a W D Q J H Q W H � D � H V V D � I X Q o m R � 3 � [ � � Q R � S R Q W R � [ � � " a ) � P � � � � b ) m = 5 . c ) � P � � � � � d ) � P � � � � � e ) m = 2 5 . 1 3 A G O R A É A S U A V E Z Q u e s t ã o 3 $ � G H U L Y D G D � G D � I X Q o m R � I � [ � � � � � [ � � � [ 2 � � � [ � � � � p � � a ) � I � µ � � [ � � � [ 2 � � [ � � � � b ) � I � µ � � [ � � � [ � � � [ 2 � � � [ � c ) � I � µ � � [ � � � � [ 2 � � � [ � � � � d ) � I � µ � � [ � � � � [ � � � � [ 2 � � � [ � e ) � I � µ � � [ � � � � [ � � � � [ 2 � � � [ � � � � Q u e s t ã o 4 $ � 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 D � I X Q o m R � I � [ � � � � [ � � � [ 2 � � � [ � � � � � H P � [ � � � � � p � a ) � � � � � b ) � � � � � c ) � � � � � d ) � � � � � e ) � � � � � 1 4 A G O R A É A S U A V E Z Q u e s t ã o 5 $ � G H U L Y D G D � G D � I X Q o m R � I � [ � � � � [ � � � � � [ � � � � [ 2 5 � [ � p � a ) � I � µ � � [ � � [ 9 9 � [ � � � [ � � � � � b ) � I � µ � � [ � � � [ � � � � [ � � � [ 2 5 � [ � c ) � I � µ � � [ � � � [ 9 9 � � [ � � � � [ � � � � � d ) � I � µ � � [ � � � � � [ 9 9 � � � � [ � � � � � � [ � � � [ � e ) � I � µ � � [ � � � � � [ 9 9 � � � � [ � � � � � � [ � � � � � A t e n ç ã o : A s Q u e s t õ e s 6 e 7 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 en 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 � [ � � � � [ 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 � [ � 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 8 W L O L ] H � D � G H U L Y D G D � S 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 D � S U R G X o m R � T X D Q G R � R � F D S L W D O � L Q Y H V W L G R � p � [ � � � � Q u e s t ã o 7 D e t e r m i n e a i n c l i n a ç ã o m � G D � U H W D � W D Q J H Q W H � D � H V V D � I X Q o m R � 3 � [ � � Q R � S R Q W R � [ � � � � " 1 5 A t e n ç ã o : A s Q u e s t õ e s d e 8 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 : $ � S U R G X W L Y L G D G H � G H � X P � I X Q F L R Q i U L R � � T X D Q G R � U H O D F L R Q D G D � D R � Q ~ P H U R � [ � G H � K R U D V � W U D E D O K D G D V � � p � G H V F U L W D � S H O D � V H J X L Q W H � I X Q o m R � 3 � [ � � � � � � � [ 2 � � � [ � � � � ( V V D � I X Q o m R � S U R G X W L Y L G D G H � F R Q V L G H U D � D � V L W X D o m R � W t S L F D � G H � X P � G L D � G H � W U D E D O K R � � R X � V H M D � � H P � X P � W X U Q R � G H � D W p � G H ] � K R U D V � G H � W U D E D O K R � � � � d � [ � d � � � � Q u e s t ã o 8 8 W L O L ] H � D � G H U L Y D G D � S 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 D � S U R G X W L Y L G D G H � T X D Q G R � R � L Q V W D Q W H � G H � W U D E D O K R � p � [ � � � � K R U D V � � I n t e r p r e t e o r e s u l t a d o . Q u e s t ã o 9 8 W L O L ] D Q G R � D L Q G D � D � G H U L Y D G D � � G 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 D � S U R G X W L Y L G D G H � T X D Q G R � R � L Q V W D Q W H � G H � W U D E D O K R � p � [ � � � � h o r a s . I n t e r p r e t e o r e s u l t a d o . Q u e s t ã o 1 0 ( Q F R Q W U H � D � I X Q o m R � I � [ � � T X H � G H V F U H Y H � D � U H W D � W D Q J H Q W H � j � F X U Y D � U H S U H V H Q W D G D � S R U � 3 � [ � � � � � � � [ 2 � � � [ � � � � Q R � S R Q W R � [ � � � � K R U D V � A G O R A É A S U A V E Z 1 6 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 � � � � � � � 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 � � � � � H G � � 6 m R � 3 D X O R � � 3 L R Q H L U D � � � � � � � R E F E R Ê N C I A S 1 H V W H � W H P D � � Y R F r � D S U H Q G H X � D � G H W H U P L Q D U � 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 � D � X P D � F X U Y D � H P � X P � S R Q W R � H V S H F L ¿ F D G R � D � p a r t i r d o c á l c u l o d a d e r i v a d a . C o m i s s o , f o i p o s s í v e l e n c o n t r a r a e q u a ç ã o d a r e t a t a n g e n t e a e s s a c u r v a . $ L Q G D � Q H V W H � W H P D � � Y R F r � W D P E p P � H V W X G R X � D � Q R W D o m R � X W L O L ] D G D � Q R � F i O F X O R � G D � G H U L Y D G D � � $ S U H Q G H X � D � X W L O L ] D U � D V � U H J U D V � G H � G H U L Y D o m R � H � D V � I y U P X O D V � S D U D � G H U L Y D o m R � G H � I X Q o } H V � S R O L Q R P L D L V � � 3 R U � ¿ P � � Y R F r � H V W X G R X � D O J X Q V � S U R E O H P D V � T X H � H Q Y R O Y H P � o c á l c u l o d a d e r i v a d a . 1 H V W H Y R F r D S U H Q G H X G H W H U P L Q D U L Q F O L Q D o m R G D H V S H F L ¿ F D G R F I N A L I Z A N D O E s t i m a t i v a : � F i O F X O R � D S U R [ L P D G R � T X H � V H � I D ] � G H � D O J 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 � 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 . 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 . 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 � � H G � 6 m R 3 D X O R � 3 L R Q H L U D � � � � � � E s t i m a t i v a : F i O F X O R D S U R [ L P D G R I D ] G H D O J R � G L O S S Á R I O 1 7 G A B A R I T O Q u e s t ã o 1 R e s p o s t a : 2 � J U i ¿ F R � G D � I X Q o m R � I � [ � � � � � 2 E V H U Y H � T X H � D � U H W D � U H S U H V H Q W D G D � S H O R � J U i ¿ F R � G D � I X Q o m R � Q m R � W H P � L Q F O L Q D o m R � � H V W i � Q D � K R U L ] R Q W D O � � � O R J R � � D � G H U L Y D G D � T X H � U H S U H V H Q W D � 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 � D � H V V D � F X U Y D � W D P E p P � Q m R � W H P � L Q F O L Q D o m R � � S R U W D Q W R � � D � G H U L Y D G D � G H � X P D � I X Q o m R � F R Q V W D Q W H � p � V H P S U H � L J X D O � D � ] H U R � & R P � U H O D o m R � j � I X Q o m R � D ¿ P � I � [ � � m [ � b : 3 R U W D Q W R � � D � G H U L Y D G D � G H � X P D � I X Q o m R � D ¿ P � p � V H P S U H � F R Q V W D Q W H � H � F R U U H V S R Q G H � D R � F R H ¿ F L H Q W H � D Q J X O D U � G D � U H W D � G H V F U L W D � S H O D � I X Q o m R � I � [ � � � Q H V W H � F D V R � � m . 1 8 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 C . & R P R � H [ S O L F D G R � Q D � W H R U L 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 � H P � X P � S R Q W R � � S R U � H [ H P S O R � � Q R � S R Q W R � [ � � � � p � D � S U y S U L 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 V V D � I X Q o m R � Q H V V H � S R Q W R � � R X � V H M D � � m = 1 5 . Q u e s t ã o 3 R e s p o s t a : A l t e r n a t i v a C . Q u e s t ã o 4 R e s p o s t a : A l t e r n a t i v a B . $ � G H U L Y D G D � I � [ � � M i � I R L � G H W H U P L Q D G D � Q D � 4 X H V W m R � � � � I � µ � � [ � � � [ 2 � � � [ � � � 3 D U D � [ � � � I � µ � � � � � � � � � � � 2 � � � � � � � � � � � ĺ � I � ¶ � � � � � � � � � � � � � � � � � ĺ � f ‘ ( 2 ) = 1 7 4 . Q u e s t ã o 5 R e s p o s t a : � $ O W H U Q D W L Y D � ( � 1 9 Q u e s t ã o 6 R e s p o s t a : � , Q L F L D O P H Q W H � � F D O F X O D � V H � D � G H U L Y D G D � G D � I X Q o m R � 3 � [ � � � � [ 2 � � � [ � � � � 3 D U D � [ � � � 3 � µ � � � � � � � � � � � � � � ĺ � 3 � µ � � � � � � � � � � ĺ � P ‘ ( 8 ) = 5 8 . 6 L J Q L ¿ F D � T X H � � Q R � L Q V W D Q W H � H P � T X H � R � F D S L W D O � L Q Y H V W L G R � p � G H � 5 � � � � � � � � � � � � [ � � � � D � S U R G X o m R � G H � F R V P p W L F R V � H V W i � D X P H Q W D Q G R � D � X P D � W D [ D � G H � � � � O L W U R V � S R U � P L O � U H D L V � G H � X Q L G D G H V � L Q Y H V W L G D V � Q u e s t ã o 7 R e s p o s t a : � $ � G H U L Y D G D � 3 � [ � � M i � I R L � G H W H U P L Q D G D � Q D � 4 X H V W m R � � � � P ‘ ( x ) = 6 x + 1 0 . 3 D U D � [ � � � � 3 � µ � � � � � � � � � � � � � � � � ĺ � 3 � µ � � � � � � � � � � � ĺ � P ‘ ( 1 0 ) = 7 0 . A i n c l i n a ç ã o m � G D � U H W D � W D QJ H Q W H � Q R � S R Q W R � [ � � � p � D � S U y S U L 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 V V D � I X Q o m R � Q H V V H � S R Q W R � � R X � V H M D � � m = 7 0 . 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 � � F D O F X O D � V H � D � G H U L Y D G D � G D � I X Q o m R � 3 � [ � � 3 D U D � [ � � � 3 � µ � � � � � � � � � � � � � � � ĺ � 3 � µ � � � � � � � � � � ĺ � P ‘ ( 3 ) = 6 . 6 L J Q L ¿ F D � T X H � R � I X Q F L R Q i U L R � H V W i � D X P H Q W D Q G R � V X D � S U R G X W L Y L G D G H � H P � � � X Q L G D G H V � S R U � K R U D � Q R � L Q V W D Q W H � [ � � � � K R U D V � � , V W R � � Q D � S U i W L F D � � H V W i � U H O D F L R Q D G R � D R � L Q t F L R � G D � M R U Q D G D � G H � W U D E D O K R � � V L W X D o m R � H P � T X H � R � I X Q F L R Q i U L R � H V W i � P H Q R V � F D Q V D G R � 2 0 Q u e s t ã o 9 R e s p o s t a : � , Q L F L D O P H Q W H � � F D O F X O D � V H � D � G H U L Y D G D � G D � I X Q o m R � 3 � [ � � 6 L J Q L ¿ F D � T X H � R � I X Q F L R Q i U L R � H V W i � G L P L Q X L Q G R � V X D � S U R G X W L Y L G D G H � H P � � � X Q L G D G H V � S R U � K R U D � Q R � L Q V W D Q W H � [ � � � � K R U D V � � , V W R � � Q D � S U i W L F D � � H V W i � U H O D F L R Q D G R � j � S U R [ L P L G D G H � G R � W p U P L Q R � G D � M R U Q D G D � G H � W U D E D O K R � � V L W X D o m R � H P � T X H � R � I X Q F L R Q i U L R � H V W i � P D L V � c a n s a d o . Q u e s t ã o 1 0 R e s p o s t a : � $ � I X Q o m R � T X H � U H S U H V H Q W D � X P D � U H W D � p � G D G D � S R U � I � [ � � � m [ � � � b . O v a l o r m � � F R Q I R U P H � 4 X H V W m R � � � � p � - 6 � � S R L V � 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 � F R U U H V S R Q G H � j � L Q F O L Q D o m R � G D � U H W D � W D Q J H Q W H � Q R � P H V P R � S R Q W R � � � $ � I X Q o m R � I � [ � � S D U F L D O P H Q W H � S U R Q W D � p � I � [ � � � - 6 [ � � � b P a r a d e s c o b r i r o v a l o r d e b b a s t a o b s e r v a r q u e , q u a n d o d u a s f u n ç õ e s s ã o t a n g e n t e s e n t r e s i , e l a s p o s s u e m u m p o n t o H P � F R P X P � � R X � V H M D � � I � � � � � 3 � � � � � ' H V W D � I R U P D � $ V V L P � � D � I X Q o m R � T X H � G H V F U H Y H � D � U H W D � W D Q J H Q W H � j � F X U Y D � 3 � [ � � � � � [ 2 � � � [ � � � � Q R � S R Q W R � [ � � p � f ( x ) = - 6 x + 9 3 , 5 .
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