8/19/2019 First Derivative and Graph

1/24

First Derivative

and GraphName/s: Kim Shekinah B. Marasigan &

Pearl Ann LaroaC & S: BS Accountancy-A1Pro: Sir. Renante Tejano

8/19/2019 First Derivative and Graph

2/24

Deinitions o !ncreasin" and Decreasin"Functions

8/19/2019 First Derivative and Graph

3/24

A unction is  increasin"  #hen its"raph rises as it "oes rom $et to

ri"ht% A unction is decreasin" #hen its "raph a$$s  as it "oesrom $et to ri"ht% 

     i    n   c   i   n   c

d     e   c   

8/19/2019 First Derivative and Graph

4/24

he increasin"/decreasin" conceptcan 'e associated #ith the s$ope o

the tan"ent $ine% he s$ope o thetan"ent $ine is positive  #hen  theunction is increasin"  and  ne"ative 

#hen decreasin"

8/19/2019 First Derivative and Graph

5/24

est or !ncreasin" and

Decreasin" Functions

8/19/2019 First Derivative and Graph

6/24

Find the (pen !nterva$s on #hich f  is !ncreasin" or Decreasin"

( )

( )   ( )

( ) ( )

numbers.critical by thesedeterminedintervalsthreetheof testingthe

 summerizesslidenexton thetableThe numbers.criticalonlytheare

 are1and0concludecanyous,'allfordefinedisSince

 1,0 01x3 

033 

0.toequalset,of numberscriticalthedetermineTo

domain.entireitsoncontinuousisthat ote !

3"#$et

 umbers%ritical

&actor 

0$et!

!3

==′

=

=−

=−=′

′

−=

=′

 x x x  f  

 x x

 x x x  f  

 x  f    f  

  f   x x x  f  

 x  f  

'# "'# 1" 0 '#.(" .)( 1 '#!"

sin sin sin

# "0 1

  f x  f f f  

 Increa g Decrea g Increa g 

  f x

+ − +< −∞ − − − − − − − − − − − − − − − − − − − − − − − +∞ >

− = = − =

< −∞ − − − − − − − − − − − − − − − − − − − − − +∞ >

8/19/2019 First Derivative and Graph

7/24

Find the (pen !nterva$s on #hich f  is !ncreasin" or Decreasin"

( )

( )   ( )

( )( )

numbers.critical by thesedeterminedintervalsthreetheof testingthe

 summerizesslidenexton thetableThe numbers.criticalonlytheare

 are1and0concludecanyous,'allfordefinedisSince

 1,0 

01x3 

033 

0.toequalset,of numberscriticalthedetermineTo

domain.entireitsoncontinuousisthat ote !

3"#$et

 umbers%ritical

&actor 

0$et!

!3

==′

=

=−

=−=′

′

−=

=′

 x x x  f  

 x

 x

 x x x  f  

 x  f    f  

  f   x x x  f  

 x  f  

'# "'# 1" 0 '#.(" .)( 1 '#!"

sin sin sin

# "0 1

 f x f f f  

 Increa g Decrea g Increa g 

 f x

+ − +< −∞ − − − − − − − − − − − − − − − − − − − − − − − +∞ >

− = = − =

< −∞ − − − − − − − − − − − − − − − − − − − − − +∞ >

8/19/2019 First Derivative and Graph

8/24

Find the (pen !nterva$s on #hich f  is !ncreasin" or Decreasin"

( )

( )   ( )

( )( )

numbers.critical by thesedeterminedintervalsthreetheof testingthe

 summerizesslidenexton thetableThe numbers.criticalonlytheare

 are1and0concludecanyous,'allfordefinedisSince

 1,0 

01x3 

033 

0.toequalset,of numberscriticalthedetermineTo

domain.entireitsoncontinuousisthat ote !

3"#$et

 umbers%ritical

&actor 

0$et!

!3

==′

=

=−

=−=′

′

−=

=′

 x x x  f  

 x

 x

 x x x  f  

 x  f    f  

  f   x x x  f  

 x  f  

'# "

'# 1" 0 '#.(" .)( 1 '#!"

sin sin sin

# "0 1

 f x f f f  

 Increa g Decrea g Increa g 

 f x

+ − +< −∞ − − − − − − − − − − − − − − − − − − − − − − − +∞ >

− = = − =

< −∞ − − − − − − − − − − − − − − − − − − − − − +∞ >

8/19/2019 First Derivative and Graph

9/24

Find the (pen !nterva$s on #hich f  is !ncreasin" or Decreasin"

( )

( )   ( )

( ) ( )

numbers.critical by thesedeterminedintervalsthreetheof testingthe

 summerizesslidenexton thetableThe numbers.criticalonlytheare

 are1and0concludecanyous,'allfordefinedisSince

 1,0 01x3 

033 

0.toequalset,of numberscriticalthedetermineTo

domain.entireitsoncontinuousisthat ote !

3"#$et

 umbers%ritical

&actor 

0$et!

!3

==′

==−

=−=′

′

−=

=′

 x x x  f  

 x x

 x x x  f  

 x  f    f  

  f   x x x  f  

 x  f  

'# "

'# 1" 0 '#.(" .)( 1 '#!"

sin sin sin# "0 1

 f x f f f  

 Increa g Decrea g Increa g  f x

+ − +< −∞ − − − − − − − − − − − − − − − − − − − − − − − +∞ >

− = = − =

< −∞ − − − − − − − − − − − − − − − − − − − − − +∞ >

8/19/2019 First Derivative and Graph

10/24

Find the (pen !nterva$s on #hich f  is !ncreasin" or Decreasin"

te$$s us #here the unction is increasin" and decreasin"%

( )

( )   ( )

( )( )

numbers.critical by thesedeterminedintervalsthreetheof testingthe

 summerizesslidenexton thetableThe numbers.criticalonlytheare

 are1and0concludecanyous,'allfordefinedisSince

 1,0 

01x3 

033 

0.toequalset,of numberscriticalthedetermineTo

domain.entireitsoncontinuousisthat ote !

3"#$et

 umbers%ritical

&actor 

0$et!

!3

==′

=

=−

=−=′

′

−=

=′

 x x x  f  

 x

 x

 x x x  f  

 x  f    f  

  f   x x x  f  

 x  f  

'# "'# 1" 0 '#.(" .)( 1 '#!"

sin sin sin # "0 1

  f x  f f f  

 Increa g Decrea g Increa g    f x

+ − +< −∞ − − − − − − − − − − − − − − − − − − − − − − − +∞ >

− = = − =

< −∞ − − − − − − − − − − − − − − − − − − − − − +∞ >

8/19/2019 First Derivative and Graph

11/24

Guide$ines or Findin" !nterva$son )hich a Function !s

!ncreasin" or Decreasin"

8/19/2019 First Derivative and Graph

12/24

heorem *%+ he FirstDerivative est

8/19/2019 First Derivative and Graph

13/24

,amp$e 1:  Graph the function f  given by

and find the relative extremes. Suppose that we are trying to graph this function but

do not know any calculus. What can we do? We can

plot a few points to determine in which direction the

graph seems to be turning. Let’s pick some x 

values and see what happens.

.sin" First Derivatives to Find aimum and inimum 0a$ues and SetchGraphs

 f ( x) =  2 x3 − 3 x2 − 12 x + 12.

8/19/2019 First Derivative and Graph

14/24

.sin" First Derivatives to Findaimum and inimum 0a$ues and

Setch Graphs,amp$e 1 2continued3:

8/19/2019 First Derivative and Graph

15/24

.sin" First Derivatives to Findaimum and inimum 0a$ues and

Setch Graphs Example 1 (continued): 

We can see some features of the graph from thesketch.

Now we will calculate the coordinates of thesefeatures

precisel.

!st "nd a general e#pression for the deri$ati$e.

%nd determine where f ′ x ' does not e#ist orwhere

 f ′ x ' ( ). Since f ′ x ' is a polnomial* there is

no′

′ f   ( x) = 6 x2 − 6 x − 12

8/19/2019 First Derivative and Graph

16/24

.sin" First Derivatives to Findaimum and inimum 0a$ues and

Setch Graphs,amp$e 1 2continued3!

"hese two critical values partition the number lineinto

# intervals! $ %& '( &)*( + %&)( ,*( and - %,( '*.

CBA

2-1

6 x 2

− 6 x  −12   =   0

 x 2

−   x  − 2   =   0

( x  − 2)( x  +1)   =   0

 x   =  2   or x   = −1

8/19/2019 First Derivative and Graph

17/24

8/19/2019 First Derivative and Graph

18/24

.sin" First Derivatives to Find aimum

and inimum 0a$ues and Setch Graphs ,amp$e 1 2conc$uded3!

"herefore( by the /irst0erivative "est(

f   has a relative maximum at  x  1 &) given

by

"hus( %&)( )2* is a relative maximum

$nd f   has a relative minimum at  x   1 , 

given by

"hus( %,( &3* is a relative minimum.

 f (−1) =  2(−1)3

− 3(−1)2

− 12(−1) + 12 = 19

 f (2) =  2(2)

3

− 3(2)

2

− 12(2) + 12 = −8

8/19/2019 First Derivative and Graph

19/24

.sin" First Derivatives to Find

aimum and inimum 0a$ues

and Setch the Graph

,amp$e *: /ind the relative extremesfor the

/unction f % x * given by

"hen sketch the graph.

)st find f ′% x *.

 f ( x) = ( x − 2)2 3

+ 1

′ f   ( x) = 2

3

 x − 2( )−1 3

′ f   ( x) =  2

3   x − 23

8/19/2019 First Derivative and Graph

20/24

.sin" First Derivatives to Findaimum and inimum 0a$ues and

Setch Graphs ,amp$e * 2continued3! 

,nd find where f ′% x * does not exist or where f ′% x * 14.

5ote that f ′% x * does not exist where thedenominator

e6uals 4. Since the denominator e6uals 4 when x  1,(

 x  1 , is a critical value.

 f ′% x * 1 4 where the numerator e6uals 4. Since , 74(

 f ′% x * 1 4 has no solution.

"hus( x  1 , is the only critical value.

8/19/2019 First Derivative and Graph

21/24

.sin" First Derivatives to Find aimum andinimum 0a$ues and Setch Graphs

,amp$e * 2continued3! 

#rd  x  1 , partitions the number line into , intervals!

$ %& '( ,* and + %,( '*. So( analye the signs off ′% x * in both intervals.

IntervalBA

2

Test Value  x  = 0  x  = !

Sign of f ′( x )

 – +

Result  f  is decreasing on (–

∞, 2]

f  is increasing on 2, ∞)

8/19/2019 First Derivative and Graph

22/24

.sin" First Derivatives to Findaimum and inimum 0a$ues and

Setch Graphs ,amp$e * 2continued3!

"herefore( by the /irst0erivative "est(

f   has a relative minimum at x  1 ,given by

"hus( %,( )* is a relative minimum.

 f  (2) = (2 − 2)2 3

+1 = 1

8/19/2019 First Derivative and Graph

23/24

 .sin" First Derivatives to Find aimum

and inimum 0a$ues and Setch Graphs ,amp$e * 2conc$uded3!

We use the information obtainedto sketch the graph below( plotting

other function values as needed.

8/19/2019 First Derivative and Graph

24/24

he ,nd4