application of derivatives part presentation

8/3/2019 Application of Derivatives Part Presentation

http://slidepdf.com/reader/full/application-of-derivatives-part-presentation 1/67

A p p l i c a t i o n s o f D e r i v a t i v e s

T h e o r y w i t h p r o b l e m s

V i d y a l a n k a r I n s t i t u t e

I I T J E E

L e c t u r e @ V i d y a l a n k a r I n s t i t u t e A p p l i c a t i o n s o f D e r i v a t i v e s

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M o n o t o n o c i t y

C o n c a v i t y & P o i n t o f i n e c t i o n

O u t l i n e


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M o n o t o n o c i t y a t a p o i n t

G r a p h s p r e s e n t i n g M o n o t o n o c i t y a t a p o i n t

M o n o t o n o c i t y i n a n i n t e r v a l

M o n o t o n o c i t y u s i n g d e r i v a t i v e s

O u t l i n e

1






2


C o n c a v i t y

P o i n t o f i n e c t i o n

T y p e s o f i n e c t i o n p o i n t s


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D e n i t i o n

A f u n c t i o n i f i n c r e a s i n g i n i t s e n t i r e d o m a i n O R i f d e c r e a s i n g i n i t s

e n t i r e d o m a i n , t h e n i t s c a l l e d m o n o t o n e f u n c t i o n .

e . g . e

x ,e

−x ,

x +

s i n x a r e m o n o t o n i c f u n c t i o n ( i n c r e a s i n g a n d

d e c r e a s i n g r e s p e c t i v e l y )

F a c t

S o f u n c t i o n s t h a t a r e b o t h i n c r e a s i n g a n d d e c r e a s i n g i n i t s e n t i r e

d o m a i n a r e c a l l e d n o n - m o n o t o n o u s .

e . g . e

x + e

− x

, s i n x a r e n o t m o n o t o n i c o n R


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M o n o t o n i c a t a p o i n t x = a

A f u n c t i o n i s m o n o t o n i c a t a p o i n t t h e n e i t h e r i t s m o n o t o n i c a l l y

i n c r e a s i n g o r m o n o t o n i c a l l y d e c r e a s i n g , S o

D e n i t i o n

f i s m o n o t o n i c a l l y i n c r e a s i n g

a

a t p o i n t x = a i f f o r s m a l l e s t p o s s i b l e

p o s i t i v e h

f ( a −

h ) < f ( a ) < f ( a + h )

A n d f i s m o n o t o n i c a l l y d e c r e a s i n g a t p o i n t x =

a i f f o r s m a l l e s t

p o s s i b l e p o s i t i v e h

f (

a −

h ) >

f (

a ) >

f (

a +

h )

a

T h i s d e n i t i o n h o l d s i r r e s p e c t i v e o f t h e f u n c t i o n i s c o n t i n o u s o r n o t


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O u t l i n e

1






2


C o n c a v i t y




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G r a p h s o f M o n o t o n o c i t y a t a p o i n t

M o n o t o n o c i t y a t a p o i n t x = a c a n b e s e e n d i s t i n c t l y u s i n g t h e

f o l l o w i n g g r a p h s w i t h t h e f o l l o w i n g c a t e g o r i e s

1

G r a p h s - D i e r e n t i a b l e a t p o i n t x = a

2

G r a p h s - N o t d i e r e n t i a b l e a t p o i n t x =

a b u t c o n t i n u o u s

3

G r a p h s - N o t c o n t i n u o u s a t x = a ( R e m o v a b l e d i s c o n t i n u i t y )

4

G r a p h s - N o t c o n t i n u o u s a t x = a ( N o n - R e m o v a b l e )


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G r a p h s - D i e r e n t i a b l e a t x = a

M o n o t o n i c a t a p o i n t

Monotonic Increasing

concave upwards flat concave downwards

concave upwards flat concave downwards

Monotonic Decreasing


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G r a p h s - N o t d i e r e n t i a b l e b u t c o n t i n u o u s a t x = a


Monotonic Increasing

Monotonic Decreasing


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N o t C o n t i n u o u s a t x = a ( N o n - R e m o v a b l e )



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P r o b l e m s

E x a m p l e

C h e c k i f t h e f o l l o w i n g f u n c t i o n s a r e m o n o t o n i c a t x

=a

a a a a


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O u t l i n e

1






2


C o n c a v i t y




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M o n o t o n i c i t y i n a n i n t e r v a l

I n c r e a s i n g / D e c r e a s i n g f u n c t i o n i . e . N o n - d e c r e a s i n g / N o n - i n c r e a s i n g f u n c t i o n

I n c r e a s i n g f u n c t i o n / D e c r e a s i n g f u n c t i o n o n

a n i n t e r v a l [

a ,

b ]

A f u n c t i o n f i s m o n o t o n i c a l l y i n c r e a s i n g

a

i f

x

1

<x

2

=⇒f

(x

1

) ≤f

(x

2

),∀x

1

,x

2

∈ [a ,

b ]

a n d

f i s m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n i f

x

1

<x

2

=⇒f

(x

1

) ≥f

(x

2

),∀x

1

,x

2

∈ [a ,

b ]

a

A f u n c t i o n i n c r e a s i n g i s a l s o m e a n t b y f u n c t i o n

i s n o n - d e c r e a s i n g



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S t r i c t l y i n c r e a s i n g / d e c r e a s i n g f u n c t i o n s

S t r i c t l y I n c r e a s i n g / D e c r e a s i n g f u n c t i o n o n

a n i n t e r v a l [

a ,

b ]

A f u n c t i o n f i s s t r i c t l y m o n o t o n i c a l l y

s t r i c t l y i n c r e a s i n g i f

x

1

< x

2

=⇒ f ( x

1

) < f ( x

2

),∀x

1

, x

2

∈ [a , b ]

a n d

f i s m o n o t o n i c a l l y s t r i c t l y d e c r e a s i n g i f

x

1

<x

2

=⇒f

(x

1

) >f

(x

2

),∀x

1

,x

2

∈ [a ,

b ]



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http://goback/








O u t l i n e

1






2


C o n c a v i t y





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http://goback/








M o n o t o n i c i t y u s i n g D e r i v a t i v e s

D e n i t i o n

A f u n c t i o n i s i n c r e a s i n g i f f

(x

) ≥ 0 i . e . f

(x

) >0 o r f

(x

) =0 o n a n

i n t e r v a l o f t h e D o m a i n

D e n i t i o n

A f u n c t i o n i s d e c r e a s i n g i f f

(x

) ≤ 0 i . e . f

(x

) <0 o r f

(x

) =0 o n a n

i n t e r v a l o f t h e D o m a i n

D e n i t i o n

A f u n c t i o n i s s t r i c t l y i n c r e a s i n g i f f (x

) ≥ 0 i . e . f (x

) >0 o n a n i n t e r v a l

o r f

(x

) =0 a t d i s c r e t e o r c o u n t a b l e n u m b e r o f p o i n t s

D e n i t i o n

A f u n c t i o n i s s t r i c t l y d e c r e a s i n g i f f

(x

)

≤0 i . e . f

(x

) <0 o n a n i n t e r v a l

o r f (x ) = 0 a t a d i s c r e t e o r c o u n t a b l e

a

n u m b e r o f p o i n t s



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P r o b l e m s

E x a m p l e s

1

C h e c k i f t h e f u n c t i o n φ ( x ) = x + s i n x i s m o n o t o n i c o r n o t ?

2

F i n d t h e i n t e r v a l w h e r e f

(x

) =x

2

e

−x

i s m o n o t o n i c ?

3

F i n d t h e i n t e r v a l i n w h i c h f u n c t i o n f ( x ) = x

x

i s m o n o t o n i c

4

D e t e r m i n e t h e i n t e r v a l o f m o n o t o n o c i t y o f f (x ) = 2 x

2 −l n x

5

P r o v e t a n x >

x +

x

3 /3 f o r x

∈ (0 ,π /2

)

6

W h i c h o f e

π a n d

π e

a r e g r e a t e r

7

U s e t h e f u n c t i o n (

s i n x ) s i n x ,

x ∈ (

0 ,π ) t o d e t e r m i n e t h e b i g g e r

o f t h e t w o n u m b e r s (1 /2 ) e

a n d ( 1 /e

2 )


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C o n c a v i t y



O u t l i n e

1






2


C o n c a v i t y




C o n c a v i t y

http://find/

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C o n c a v i t y

D e n i t i o n

A c u r v e i s c o n c a v e u p w a r d s i n (

a ,

b )

i f

(x

) >0 a n d c o n c a v e

d o w n w a r d s i n

(a

,b

)i f ( x

) <0

R a d i u s o f c u r v a t u r e

F o r a c u r v e g i v e n b y y = f ( x )

R =[1 + (d y /d x )

2

]3 /2

|d

2

y /d x

2 |


C o n c a v i t y

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O u t l i n e

1






2


C o n c a v i t y




C o n c a v i t y

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P o i n t o f I n e c t i o n

D e n i t i o n

P o i n t o f i n e c t i o n i s t h e p o i n t w h e r e t h e c o n c a v i t y o f t h e c u r v e

c h a n g e s d i r e c t i o n , f r o m c o n c a v e u p w a r d t o d o w n w a r d o r v i c e v e r s a

E q u i v a l e n t d e n i t i o n s ,

P o i n t o f i n e c t i o n i s t h e p o i n t w h e r e t h e s e c o n d o r d e r d e r i v a t i v e

c h a n g e s s i g n ( i f i t e x i s t s ) O R

I s a p o i n t w h e r e t h e t a n g e n t c r o s s e s t h e c u r v e a t t h a t p o i n t

F o r x ∈ (

a ,

b )

, f (λ

a + (

1 −λ )

b ) > λ

f (

a ) + (

1 −λ )

f (

b ) ⇐=

c o n c a v e

u p w a r d s a n d r e s p e c t i v e l y c o n c a v e d o w n w a r d s

T y p e s o f i n e c t i o n p o i n t

1

S a d d l e p o i n t o r s t a t i o n a r y p o i n t o f i n e c t i o n : i s a i n e c t i o n p o i n t a t

w h i c h f

( x ) = 0

2

N o n - s t a t i o n a r y p o i n t o f i n e c t i o n : i s a p o i n t w h e r e f

( x )= 0


C o n c a v i t y

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O u t l i n e

1






2


C o n c a v i t y





C o n c a v i t y

http://find/

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S t a t i o n a r y P o i n t o f I n e c t i o n

f

( x ) c h a n g e s s i g n a t x = a a n d f

(a ) = 0



C o n c a v i t y

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N o n - S t a t i o n a r y P o i n t o f I n e c t i o n

f

( x ) c h a n g e s s i g n a t x = a b u t f

(a ) = 0


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C o n t i n u o u s f u n c t i o n s E x t r e m a


S o m e c o u n t e r e x a m p l e s s h o w i n g c o n v e r s e m i g h t n o t b e t r u e

f

( c ) i s n o t a g o o d c r i t e r i o n t o d e c i d e p o i n t o f i n e c t i o n t h o u g h

m o s t o f t h e p o i n t s o f i n e c t i o n o c c u r s a t p o i n t w h e r e f

( x ) = 0

f

(c

) =0

=

⇒c i s p o i n t o f i n e c t i o n e . g . f

(x

) =x

4

f ( c ) i s u n d e n e d s t i l l c i s a p o i n t o f i n e c t i o n e . g . f ( x ) = x

1

/3

E x a m p l e s

F i n d t h e p o i n t s o f i n e c t i o n f o r t h e f o l l o w i n g f u n c t i o n s a n d s t a t e i f t h e i t s a

s t a t i o n a r y o r n o n - s t a t i o n a r y i n e c t i o n p o i n t

1

f ( x ) = x

3 −x

2 + x

2

g ( x ) = x

3 −x + 1

3

D o e s f u n c t i o n h h a s a n i n e c t i o n p o i n t ? I s t h e r e a p o i n t w h e r e t h e

s e c o n d d e r i v a t i v e b e c o m e s z e r o , a n d d o e s t h e s e c o n d d e r i v a t i v e c h a n g e s

s i g n a b o u t t h a t p o i n t o f z e r o ? h ( x ) = x

4

−x


L o c a l e x t r e m a - F i r s t d e r i v a t i v e t e s t

S e c o n d D e r i v a t i v e t e s t

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P a r a m e t r i c f o r m

T y p e s i n C o n t i n u o u s b u t n o t d i e r e n t i a b l e

O u t l i n e

3










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L o c a l M a x i m a & L o c a l M i n i m a

C o n t i n u o u s f u n c t i o n s - F i r s t d e r i v a t i v e t e s t ( E x i s t e n c e o f E x t r e m u m )

T h e o r e m

F o r a c o n t i n u o u s f u n c t i o n t h e n e c e s s a r y c o n d i t i o n f o r e x i s t e n c e o f

e x t r e m u m a t x = c i s f

( c ) = 0 o r f

( c ) d o e s n o t e x i s t .

S u c i e n t c o n d i t i o n i s t h a t f (x ) c h a n g e s s i g n a b o u t p o i n t x = c

i . e . f

(x

)i s p o s i t i v e f o r x

<c a n d n e g a t i v e f o r x

>c O R f

(x

)i s

n e g a t i v e f o r x < c a n d p o s i t i v e f o r x > c

W h a t i f f

(x

)d o e s n o t c h a n g e s i g n a b o u t p o i n t x

=c ?

T h e n t h e f u n c t i o n d o e s n o t a t t a i n e x t r e m u m a t x =

c , t h i s p o i n t i s

p o i n t o f i n e c t i o n p r o v i d e d f

(c ) e x i s t s ( i . e . i n p r e s e n t c a s e o n l y

w h e n f

(c

) =0 )





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C o n t i n u o u s f u n c t i o n s - F i r s t d e r i v a t i v e t e s t

D e n i t i o n

C r i t i c a l p o i n t s : P o i n t s w h e r e f

(x ) i s z e r o o r f

( x ) i s u n d e n e d

W o r k i n g R u l e

I n p r o b l e m s o l v i n g , f o r g i v e n f (

x )

1

F i n d c r i t i c a l p o i n t s r s t , n d a l l x = c s u c h t h a t f

( c ) = 0 o r f

( c )d o e s n ' t e x i s t

2

C h e c k t o t h e l e f t a n d r i g h t f o r c h a n g e o f s i g n i n f

(x

)f o r x

<c a n d x

>c

3

I f f ( x

) >0 f o r x

<c a n d f ( x

) <0 f o r x

>c t h e n c i s p o i n t o f l o c a l

m a x i m a ( a n d c o r r e s p o n d i n g l y f o r l o c a l m i n i m a )





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C o n t i n u o u s f u n c t i o n s - F i r s t d e r i v a t i v e t e s t

D e n i t i o n

C r i t i c a l p o i n t s : P o i n t s w h e r e f

(x ) i s z e r o o r f

( x ) i s u n d e n e d

W o r k i n g R u l e

I n p r o b l e m s o l v i n g , f o r g i v e n f (

x )

1

F i n d c r i t i c a l p o i n t s r s t , n d a l l x = c s u c h t h a t f

( c ) = 0 o r f

( c )d o e s n ' t e x i s t

2

C h e c k t o t h e l e f t a n d r i g h t f o r c h a n g e o f s i g n i n f

(x

)f o r x

<c a n d x

>c

3

I f f ( x

) >0 f o r x

<c a n d f ( x

) <0 f o r x

>c t h e n c i s p o i n t o f l o c a l

m a x i m a ( a n d c o r r e s p o n d i n g l y f o r l o c a l m i n i m a )

E x a m p l e

U s i n g t h e r s t d e r i v a t i v e t e s t , n d t h e e x t r e m a o f f ( x ) = 3

3

√x

2

−x

2





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O u t l i n e

3










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http://goback/






C o n t i n u o u s f u n c t i o n s - S e c o n d d e r i v a t i v e t e s t ( E x i s t e n c e o f e x t r e m u m )

I f a f u n c t i o n f (

x )

i s d i e r e n t i a b l e t w i c e t h e n f o r e x t r e m u m t o e x i s t

T h e o r e m

F o r a t w i c e d i e r e n t i a b l e f u n c t i o n , t h e n e c e s s a r y c o n d i t i o n f o r

e x i s t e n c e o f e x t r e m u m a t x =

c i s f

(c

) =0 o r f

(c

)d o e s n o t e x i s t .

S u c i e n t c o n d i t i o n i s t h a t f

( c ) < 0 i m p l i e s c i s p o i n t o f l o c a l

m a x i m a a n d f (c

) >0 i m p l i e s c i s p o i n t o f l o c a l m i n i m a .






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C o n t i n u o u s f u n c t i o n s - S e c o n d d e r i v a t i v e t e s t ( E x i s t e n c e o f e x t r e m u m )

E x t e n s i o n o f t h e t h e o r e m i n c a s e o f f

(c

) =0

W h a t i f f

( c ) = 0 t h e n w e f o l l o w t h e f o l l o w i n g g e n e r a l i z e d p r o c e s s ,

I f f

( c ) = f

( c ) = f

( c ) = · · ·= f

(n − 1 )( c ) = 0 b u t f

(n )(c ) = 0 t h e n

n i s e v e n

a n d f

( n )(c

) <0

=⇒ c i s p o i n t o f m a x i m a

f

(n )(c

) >0

=⇒c i s p o i n t o f m i n i m a

n i s o d d

t h e n t h e r e i s n o e x t r e m u m a t x =

c






http://find/

http://goback/




O u t l i n e

3











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http://goback/




L o c a l M a x i m a & M i n i m a

C o n t i n u o u s f u n c t i o n s - S e c o n d d e r i v a t i v e t e s t f o r f u n c t i o n s r e p r e s e n t e d p a r a m e t r i c a l l y

y =

f (

x )

i s r e p r e s e n t e d a s y = φ ( t

)a n d x

= ψ ( t )

t h e n ( φ a n d ψ

a r e t w i c e d i e r e n t i a b l e )

d y

d x =

d y /d t

d x /d t =

φ ( t )

ψ ( t ) =0

=⇒ φ (t

) =0

(s a y t

=t

0

)

d

2

y

d x

2

=d

d x φ ( t

)

ψ ( t )=d

d t φ ( t

)

ψ ( t ) ·1

d x /d t

=ψ ( t )φ ( t )−φ (t )ψ (t )

(ψ ( t ))3

∴d

2

y

d x

2

=φ (t )

ψ ( t ) 2

=⇒ s i g n o f

d

2

y

d x

2

i s s a m e a s s i g n o f φ (t )






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P r o b l e m s

F i r s t a n d s e c o n d d e r i v a t i v e t e s t

E x a m p l e s

1

I n v e s t i g a t e t h e f u n c t i o n f ( x ) = x

4

e

−x

2

f o r e x t r e m a

2

I n v e s t i g a t e t h e f u n c t i o n f ( x ) = x

2 +1

x

2

3

D i s c u s s t h e e x t r e m u m o f f (

x ) =

1 0

3 x

4 + 8 x

3 −1 8 x

2 + 6 0

4

B e t w e e n a n y t w o m a x i m a t h e r e i s a m i n i m a a n d b e t w e e n a n y t w o

m i n i m a t h e r e i s a m a x i m a i s t r u e f o r a n y g i v e n f u n c t i o n f : t r u e o r f a l s e .

5

D i s c u s s t h e e x t r e m a f o r f

(x

) =s i n x

(1

+c o s x

)w h e r e x ∈ (

0

,π

2

)

6

T h e f u n c t i o n y = f (x ) i s r e p r e s e n t e d p a r a m e t r i c a l l y

x = φ (t

) =t

5 −5 t

3 −2 0 t

+7 a n d y

= ψ ( t ) =

4 t

3 −3 t

2 −1 8 t

+3 , (|

t |<

2 )

t h e n n d t h e m a x i m u m a n d m i n i m u m v a l u e s f o t h e f u n c t i o n s y = f ( x )






http://find/

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O u t l i n e

3











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C o n t i n u o u s b u t n o t d i e r e n t i a b l e

C o n t i n u o u s b u t n o t d i e r e n t i a b l e ,

c a n b e o f t w o t y p e s

1

d e r i v a t i v e t o l e f t a n d r i g h t o f

x = a a r e b o t h n o n - z e r o - t o p

t w o g u r e s

2

O n e o f l e f t o r r i g h t d e r i v a t i v e

i s z e r o a t x =

a - b o t t o m t w o

g u r e s

T r e a t r s t w i t h r s t d e r i v a t i v e t e s t ,

i . e . c h a n g e o f s i g n i s e x t r e m a

T r e a t s e c o n d w i t h b a s i c d e n i t i o n


D i s c o n t i n u o u s f u n c t i o n s - E x t r e m a

G l o b a l M a x i m a a n d M i n i m a

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C o n t i n u o u s b u t n o t d i e r e n t i a b l e

P r o b l e m s

1

D i s c u s s e x t r e m a i n t h e f o l l o w i n g f u n c t i o n s

1

f

(x

) =2 s i n x

,x

≥0

c o s

−1

x − π

2

,x <

0

2

f (

x ) =

x

2 ,x ≥

0

t a n x ,

x <

0

2

T r y t h e s e p r o b l e m s

1

f (

x ) =

2 x +

3 x

2

/3

, d i s c u s s e x t r e m a , a n d e x p l a i n t h e s t r a t e g y

y o u t o o k !




T y p e s o f D i s c o n t i n u i t y

D e n i t i o n

R e m o v a b l e D i s c o n t i n u i t y

U n r e m o v a b l e d i s c o n t i n u i t y

http://find/

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S u m m a r y

O u t l i n e

4



D e n i t i o n



S u m m a r y

5


O n a C l o s e d i n t e r v a l

O n a n o p e n i n t e r v a l





D e n i t i o n



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S u m m a r y


D i s c o n t i n u o u s f u n c t i o n s

L o c a l M a x i m a , M i n i m a f o r D i s c o n t i n u o u s f u n c t i o n s

D i s c o n t i n o u s f u n c t i o n c a n b e d i v i d e d i n t o

1

R e m o v a b l e d i s c o n t i n u i t y a t x = a

2

N o n - R e m o v a b l e d i s c o n t i n u i t y a t x =

a





D e n i t i o n



http://find/

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S u m m a r y

O u t l i n e

4



D e n i t i o n



S u m m a r y

5








D e n i t i o n



http://find/

http://goback/



S u m m a r y



F o r d i s c o n t i n u o u s w e w i l l g o t o t h e d e n i t i o n o f l o c a l m a x i m a a n d l o c a l m i n i m a ,

D e n i t i o n

F o r l o c a l m a x i m a t o e x i s t a t x = a , f (a − h ) < f ( a ) a n d f ( a + h ) < f (a ) f o r a l l

h >

0 a n d s m a l l

D e n i t i o n

F o r l o c a l m a x i m a t o e x i s t a t x = a , f (a −

h ) > f ( a ) a n d f ( a + h ) > f (a ) f o r a l l

h > 0 a n d s m a l l .

A t a n y p o i n t a s s e e n e a r l i e r , w e h a v e

f i s m o n o t o n e

o r f a t t a i n s a n e x t r e m u m

S o t h i s h e l p s o u r t h o u g h t , a b s e n s e o f m o n o t o n i c i t y i s p r e s e n c e o f e x t r e m u m i n

d i s c o n t i n u o u s c a s e s .





D e n i t i o n



http://find/

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S u m m a r y

O u t l i n e

4



D e n i t i o n



S u m m a r y

5








D e n i t i o n



http://find/

http://goback/



S u m m a r y


E x t r e m a - R e m o v a b l e

D e n i t i o n

I f f i s r e m o v a b l y d i s c o n t i n u o u s a t x =

a , t h e n w e g o b y

m o n o t o n o c i t y o r d e n i t i o n

N o t e

I n t h i s c a s e w e n e v e r a p p l y t h e c h a n g e s i g n o f d e r i v a t i v e , f r o m r s t d e r i v a t i v e

t e s t , s i n c e t h a t m a y n o t w o r k c o r r e c t l y

f ( x ) =

t a n x ,

x >

0

3 , x = 0

c o s

−1

x − π

2

,x <

0

h a s c h a n g e o f s i g n , f r o m n e g a t i v e t o p o s i t i v e a t x = 0 ( p o i n t o f d i s c o n t i n u i t y )

i m p l y i n g t h e r e i s a l o c a l m i n i m a b u t i t a t t a i n s a l o c a l m a x i m a .





D e n i t i o n



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S u m m a r y

G r a p h s - D i s c o n t i n u o u s f u n c t i o n s

E x t r e m a - R e m o v a b l e

A t a p o i n t x =

a , h a v i n g

r e m o v a b l e d i s c o n t i n u i t y w e

a l w a y s a t t a i n e x t r e m a , a s

s h o w n i n t h e a d j o i n i n g c a s e s .

S i n c e a n y f u n c t i o n w i l l s a t i s f y

o n e o f t h e c a s e s o f e x t r e m a

1

f ( a ) > f ( a + h ) a n d

f

(a

) >f

(a − h

)2

f ( a ) < f ( a + h ) a n d

f (

a ) <

f (

a

−h

)

D i s c u s s e x t r e m a i n

f (

x ) =

x

2 , x < 0

5 , x = 0

2 s i n x , x > 0





D e n i t i o n



S u m m a r y

http://find/

http://goback/



O u t l i n e

4



D e n i t i o n



S u m m a r y

5








D e n i t i o n



S u m m a r y

http://find/

http://goback/




E x t r e m a - U n r e m o v a b l e : f ( a ) e q u a l s L H L o r R H L

D e n i t i o n

I f f h a s n o n - r e m o v a b l e

( L H L = R H L a t x = a )d i s c o n t i n u i t y a t x = a t h e y w e

c h e c k e x t r e m a a t a b y

D e n i t i o n o f e x t r e m a o r

m o n o t o n o c i t y

H e r e w e e x e m p l i f y , r s t

c a t e g o r y ,

f (

a ) =

L H L o r f (

a ) =

R H L





D e n i t i o n



S u m m a r y

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E x t r e m a - U n r e m o v a b l e : f ( a ) n o t e q u a l t o b o t h L H L o r R H L

A s d e n e d i n p r e v i o u s s l i d e , w e g o b y d e n i t i o n

h e r e t o o . H e r e ,

f (

a ) =

L H L o r f (

a ) =

R H L

A p o i n t i s n i c e l y e x e m p l i e d t h a t i f

1

f (

a ) >

m a x (

L H L ,

R H L )

t h e n w e h a v e a l o c a l

m a x i m a

2

f (a ) < m i n (L H L , R H L ) t h e n w e h a v e a l o c a l

m i n i m a

3

e l s e x = a i s a m o n o t o n i c p o i n t .

f ( x ) =1

|x | , x

= 0 a n d f (x ) = 0 f o r x = 0 t h e n

d i s c u s s e x t r e m a





D e n i t i o n



S u m m a r y

http://find/

http://goback/



O u t l i n e

4



D e n i t i o n



S u m m a r y

5








D e n i t i o n



S u m m a r y

http://find/

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S u m m a r y

Function

Continuous at x = a Discontinuous at x = a

Differentiable at x = a Not differentiable at x = a Removable Non Removable

1st derivative Test

If twice differentiable2nd Derivative test

Failure to apply 1st and 2nd derivative test

Not Monotone means extremum

Neither of LHD or RHD nonzero

One of LHD or RHD is zero

or Definition





D e n i t i o n



S u m m a r y

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P r o b l e m s

1

I n v e s t i g a t e t h e e x t r e m a o f

1

f ( x ) =

−

2 x , x < 0

3 x + 5 , x ≥ 0

2

f (

x ) = 2 x

2 +3

,x =

0

4

,x

=0

3

I f f (x ) =

7 −

x

2 , x < 2

1 1 −

x , x ≥

2

t h e n f ( x ) h a s ( A ) l o c a l m a x i m a a t x = 0

( B ) l o c a l m a x i m a a t x = 2 ( C ) l o c a l m a x i m a a t x = 1 1 ( D ) N o n e o f

t h e s e

2

T h e p o i n t o f i n e x i o n o f ( x − 5 )5 5

(x − 6 )6 6

i s ( A ) 0 ( B ) 5 ( C ) 6 ( D )

6 6 0

1 2 1

3

D o e s t h e f u n c t i o n f ( x ) =

1 /x

2 , x > 0

3 x

2 , x ≤

0

h a s a m i n o r m a x i m u m a t

x = 0 ? a n d d o e s c h a n g e o f s i g n o f d e r i v a t i v e j u s t i f y t h e o u t c o m e .





D e n i t i o n



S u m m a r y

http://find/

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1

P r o v e t h a t t h e f u n c t i o n f (x ) =x

2

s i n

2 ( 1 /x ) , x = 0

0

,x

=0

h a s a m i n i m u m a t

t h e p o i n t x

0

= 0 ( t h o u g h n o t a s t r i c t m i n i m u m )






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O u t l i n e

4



D e n i t i o n



S u m m a r y

5









http://find/

http://goback/



G l o b a l M a x i m a a n d G l o b a l M i n i m a

C o n t i n u o u s f u n c t i o n s - D e n i t i o n & C l o s e d i n t e r v a l

G l o b a l M a x i m a / M i n i m a o f f : D

f

→ R

G l o b a l M a x i m a / M i n i m a i s d e n e d a s t h e m a x i m u m / m i n i m u m v a l u e

t a k e n b y f (x ) f o r a l l x ∈

D

f

G l o b a l M a x i m a / M i n i m a o f f : [ a , b ] →R

G l o b a l M a x i m a / M i n i m a o f a f u n c t i o n f f r o m [a , b ] ⊂ R→ R

r e p r e s e n t e d a s M ( G l o b a l m a x i m a ) a n d m ( G l o b a l m i n i m a ) i s g i v e n

a s

M = m a x { f ( a ), f ( c

1 ), f ( c

2 ), . . . , f (c

n ), f ( b )}m = m i n

{f ( a ), f ( c

1

), f ( c

2

), . . . , f ( c

n

), f ( b )}w h e r e c

1

,c

2

, . . . ,c

n

a r e t h e c r i t i c a l p o i n t s a n d a a n d b a r e e n d

p o i n t s o f t h e d o m a i n .






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O u t l i n e

4



D e n i t i o n



S u m m a r y

5









http://find/

http://goback/



G l o b a l M a x i m a a n d G l o b a l M i n i m a

C o n t i n u o u s f u n c t i o n s - O p e n i n t e r v a l

G l o b a l M a x i m a / M i n i m a o f f : ( a , b ) → R

G l o b a l M a x i m a / M i n i m a o f a f u n c t i o n f f r o m (a , b ) ⊂R→ R r e p r e s e n t e d a s M

( G l o b a l m a x i m a ) a n d m ( G l o b a l m i n i m a ) i s g i v e n a s

M =

m a x

{f

(c

1

),f

(c

2

), . . . ,f

(c

n

)}

m = m i n {

f (c

1

), f (c

2

), . . . , f ( c

n

)}w h e r e c

1

, c

2

, . . . , c

n

a r e t h e c r i t i c a l p o i n t s a n d

1

M > m a x

l i m

x →a

+f (x ), l i m

x →b

−f (x )

a n d

2

m < m i n

l i m

x →a

+f ( x ), l i m

x →b

−f ( x )

E l s e i f a n y o f c o n d i t i o n s ( 1 ) a n d ( 2 ) f a i l s t h e n t h e r e d o e s n ' t e x i s t

g l o b a l m a x i m a o r m i n i m a f o r f






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P r o b l e m s

A b s o l u t e o r g l o b a l e x t r e m a

1

F i n d t h e g r e a t e s t a n d l e a s t v a l u e s o f t h e f o l l o w i n g f u n c t i o n s o n

t h e i n d i c a t e d i n t e r v a l s

1

f (x ) = 2 x

3 − 3 x

2 − 2 x + 1 o n [−2 , 5 /2 ]2

f (

x ) =

x e

−x

o n [

1 ,

e ]

3

f

(x

) =x

+√

x o n [ 1 , 4 ]

4

f (

x ) =

√4 −

x

2

o n [ - 2 , 2 ]

5

f

(x

) = 2 x

2 +2

x

2

,x

∈[−

2 ,

2 ]−{

0

}1

,x

=0






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P r o b l e m s

A b s o l u t e o r g l o b a l e x t r e m a

1

D o t h e f o l l o w i n g f u n c t i o n s h a v e t h e g r e a t e s t a n d t h e l e a s t

v a l u e s o n t h e i n d i c a t e d i n t e r v a l s ?

1

f (

x ) =

c o s x f o r x ∈ [−π /2 ,π )

2

f

(x

) =s i n −

1

x f o r x ∈ (−1

,1

)

2

F i n d t h e l o c a l , g l o b a l e x t r e m a f o r f (

x ) =

a x +

b

c x +

d

3

S h o w t h a t f

(x

) = 2

−s i n

1

x |x

|, x

= 0

0 , x = 0

h a s a m i n i m u m

a t x = 0 b u t i s n o t m o n o t o n i c e i t h e r o n t h e l e f t o r o n t h e r i g h t

o f x =

0 T a n g e n t s a n d N o r m a l s






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T a n g e n t s a n d N o r m a l

D e n i t i o n

E q u a t i o n o f T a n g e n t a t (x

1

, y

1

) o n t h e c u r v e y = f ( x ) i s

y

−y

1

= d y

d x

(

x

1

,y

1

) (x

−x

1

)

D e n i t i o n

E q u a t i o n o f t h e n o r m a l a t p o i n t P ( x

1

, y

1

) o n t h e c u r v e y = f ( x ) i s

y −

y

1

= −

d y

d x

−1

(x

1

, y

1

)

( x −

x

1

)






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S u b t a n g e n t & S u b n o r m a l

D e n i t i o n

S u b t a n g e n t l e n g t h = N T =y

y

S u b n o r m a l l e n g t h = N G = y y

P G =

y

1

+y

2

a n d

P T =

y

y

1

+y

2

d s

= d x

2

+d y

2






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P r o b l e m s

E x a m p l e s

1

F i n d a l l t h e t a n g e n t s t o t h e c u r v e y =

c o s (

x +

y )

, −

2 π ≤

x ≤

2 π

t h a t a r e p a r a l l e l t o t h e l i n e x +

2 y =

0

2

F i n d t h e e q u a t i o n o f t h e n o r m a l t o t h e c u r v e

y

= (1

+x

)y +s i n

−1 (s i n

2

x

)a t x

=0

3

T h e c u r v e y =

a x

3 +b x

2 +c x

+5 t o u c h e s t h e x − a x i s a t P

(−2 ,

0 )

a n d c u t s t h e y −

a x i s a t a p o i n t Q w h e r e t h e g r a d i e n t i s 3 .

T h e n

n d t h e v a l u e s o f a ,

b ,

c

4

S h o w t h a t t h e s u b t a n g e n t a t a n y p o i n t o n t h e c u r v e x

a

y

b =c

a + b

v a r i e s a s t h e a b s c i s s a






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I n t e r s e c t i o n o f t w o c u r v e s

D e n i t i o n

A n g l e o f i n t e r s e c t i o n o f t w o c u r v e s i s d e n e d a s a n g l e b e t w e e n t a n g e n t s

t o t h e c u r v e s a t t h e p o i n t o f i n t e r s e c t i o n .

E x a m p l e s

1

T h e e q u a t i o n o f t h e t w o c u r v e s a r e y

2 =4 a x a n d x

2 =3 2 a y t h e n

n d t h e a n g l e o f i n t e r s e c t i o n o f t h e c u r v e s

2

E q u a t i o n o f t h e t a n g e n t s t o t h e c u r v e (

1 +

x

2 )y

=1 a t t h e p o i n t s o f

i t s i n t e r s e c t i o n w i t h t h e c u r v e (

x +

1 )

y =

1 a r e g i v e n b y

1

x +

2 y =

1 ;

y =

1

2

x +

2 y =

2 ;

x =

1

3

x +

2 y =

2 ;

y =

1

4

x +

2 y =

1 ;

x =

1






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P r o b l e m s

E x a m p l e s

1

T h e c u r v e y =

a

x

a n d y =

b

x

i n t e r s e c t a t a n a n g l e o f

t a n

−1

l n

(a /

b )

1 +

l n a l n b

a b o v e s t a t e m e n t i s

1

t r u e

2

f a l s e

3

c a n n o t s a y a n y t h i n g

2

I f t h e p a r a m e t r i c e q u a t i o n o f a c u r v e i s g i v e n b y x =

e

t

c o s t a n d

y =

e

t

s i n t t h e n t h e t a n g e n t t o t h e c u r v e a t t h e p o i n t t = π /

4

m a k e s w i t h a x i s o f x t h e a n g l e

( a ) 0 ( b ) π

/ 3 ( c ) π

/ 4 ( d ) π

/ 2

3

T h e l e n g t h o f t h e s u b t a n g e n t t o t h e c u r v e

√x

+√

y =

3 a t t h e

p o i n t (

4 ,

1 )

i s

( a ) 2 ( b )

1

2

( c ) - 3 ( d ) N o n e o f t h e s e






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