Rate of Change & Limit

7/25/2019 Rate of Change & Limit

1/19

2.1

Rates of Change

and Limits

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by ic!ie Kelly, 2""#

Grand $eton %ational Par!, Wyoming


2/19

S&''ose yo& dri(e 2"" miles, and it ta!es yo& ) ho&rs.

$hen yo&r a(erage s'eed is*mi

200 mi 4 hr 50hr

=

distance

average speed elapsed time

x

t

= =

+f yo& loo! at yo&r s'eedometer d&ring this tri', it mightread - m'h. $his is yo&r instantaneo&s s'eed.


3/19

roc! falls from a high cliff.

$he 'osition of the roc! is gi(en by* 216y t=

fter 2 seconds* 216 2 64y= =

a(erage s'eed* av 64 ft ft322 sec sec

V = =

What is the instantaneouss'eed at 2 seconds/


4/19

instantaneous

yV

t

for some (ery small

change in t

( ) ( )2 2

16 2 16 2h

h

+ =

0here h some (ery

small change in t

We can &se the $+34 to e(al&ate this e5'ression forsmaller and smaller (al&es of h.


5/19

instantaneous

yV

t

( ) ( )

2 216 2 16 2h

h

+ =

h

y

t

1 3"

".1 -.

."1 ).1

.""1 )."1

."""1 ).""1

.""""1 )."""2

( )( ) { }16 2 ^ 2 64 1,.1,.01,.001,.0001,.00001h h h + =

We can see that the (elocity

a''roaches ) ft6sec as hbecomes(ery small.

We say that the (elocity has a limiting

(al&e of ) as ha''roaches 7ero.

8%ote that hne(er act&ally becomes

7ero.9


6/19

( )2 2

0

16 2 16 2limh

h

h

+ $he limit as h

a''roaches 7ero*

( )2

0

4 4 416 lim

h

h h

h

+ +

2

0

4 4 416 lim

h

h h

h

+ +

( )0

16 lim 4h

h

+0

64=

Since the 1 is

&nchanged as ha''roaches 7ero,

0e can factor 1

o&t.


7/19

Consider*sinx

yx

=

What ha''ens as xa''roaches 7ero/

Gra'hically*

( )sin /y x x=

22

/ 2

W+%:;W


8/19

( )sin /y x x=

Loo!s li!e y1


9/19

( )sin /y x x=

%&merically*

$blSet


10/19

( )sin /y x x=

+t a''ears that the limit of as xa''roaches 7ero is 1sinx

x


11/19

Limit notation* ( )limx c f x L =

?$he limit of fof xasxa''roaches cisL.@

So*0

sinlim 1x

xx

=


12/19

$he limit of a f&nction refers to the (al&e that the

f&nction a''roaches, not the act&al (al&e 8if any9.

( )2lim 2x f x =

not 1


13/19

Pro'erties of Limits*

Limits can be added, s&btracted, m&lti'lied, m&lti'liedby a constant, di(ided, and raised to a 'o0er.

8See yo&r boo! for details.9

Aor a limit to e5ist, the f&nction m&st a''roach the

same (al&e from both sides.

;nesided limits a''roach from either the left or right side only.


14/19

1 2 B )

1

2

t 51* ( )1

lim 0x

f x

=

( )1lim 1x f x+ =( )1 1f =

left hand limit

right hand limit

(al&e of the f&nction

( )1

limx

f x does not e5ist

beca&se the left and

right hand limits do not

match


15/19

t 52* ( )2

lim 1x

f x

=

( )2lim 1x f x+ =( )2 2f =

left hand limit

right hand limit


( )2lim 1x f x =beca&se the left and

right hand limits match.

1 2 B )

1

2


16/19

t 5B* ( )3

lim 2x

f x

=

( )3lim 2x f x+ =( )3 2f =

left hand limit

right hand limit


( )3lim 2x f x =beca&se the left and

right hand limits match.

1 2 B )

1

2


17/19

$he Sand0ich $heorem*

( ) ( ) ( )( ) ( ) ( )

If for all in some interval aout

and lim lim , then lim .x c x c x c

g x f x h x x c c

g x h x L f x L

= = =

Sho0 that* 20

1lim sin 0x x x =

$he ma5im&m (al&e of sine is 1, so2 21sinx x

x

$he minim&m (al&e of sine is 1, so2 21sinx x

x

So*2 2 21

sinx x x

x


18/19

2 2 2

0 0 0

1lim lim sin limx x x

x x xx

2

0

10 lim sin 0x x x

2

0

1lim sin 0x

xx

=

=y the sand0ich theorem*


19/19

Rate of Change & Limit

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