tugasan hbmt 4303

8/9/2019 tugasan hbmt 4303

1/20

HBMT4303

FACULTY OF EDUCATION AND LANGUAGES

SEPTEMBER 2012

HBMT4303

TEACHING MATHEMATICS IN FORM FIVE

MATRICULATION NO: 810610146336002

IDENTITY CARD NO. : 810610146336

TELEPHONE NO. : 0138706454

E-MAIL : n!"#$%&'#n()"#%*+,"

LEARNING CENTRE : IPGM -#"./ Tn)/ A"./#n A&'#n


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HBMT4303

INTRODUCTION

R# & ,$#n) % #n %".!#n ,n,. %n ,#*,/*/+ I #..*%,#%n ,! #!%/ &%* %n !#* *%&

%/#%n+ I #" #! !/%! #!,$ #n !# # *# #!%,* &!" %&&!n /!, n $

#,$%n) & !# & ,$#n) #n /""#!% $ #..!#,$ / %n $ #!%,*+ I #* n .*#%n $

"#n%n) & average rate of change9 instantaneous rate of change9 #n relative rate of change+

In $% #%)n"n I $# .!.#! # & /%#:* #,$%n) #n *#!n%n) "#!%#* &! F!" 5

"#$"#%, ,*# : / %n!/, $ ,n,. & !# & ,$#n)9 # ** # #,$ $

#..*%,#%n & !# & ,$#n) %n #!%/ &%* %n !#* *%& %/#%n+ T$ "#!%#* /* ,n% &

#,$%n) #%9 #n #".* & #,%%% %n $%,$ /n %n )!/. ,#n .#!%,%.# /%n) $

"#!%#*

T$ #* !/%! / .!.#! # *n .*#n &! # %n %n #,$%n) $ .%, & !# & ,$#n)

%n $%,$ I /* / .#! ! #** & $ "#!%#* I $# .!.#!+ I $# $#! "; *n .*#n %$ #n

.!%n, F!" 5 "#$"#%, #,$! ) $ &:#, &!" $"+


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HBMT4303

SUMMARY OF APPROACH IN TEACHING RATE OF CHANGE

C#*,/*/ % $ /: #n $ #!%/ #..*%,#%n & $% ,n,.+ Nn? S,n

L# & "%n % # #"n #:/ # !# & ,$#n)@ % #; $# $ &!, #,%n) n # :;

/#* $ "# & $ :; "/*%.*% :; $ #,,*!#%n & $ :; "%n@ $n $

&!, % nn9 $ *# :," # #"n #:/ #,,*!#%n9 !# & ,$#n) & .

,".#! %"+ T$ %ffi,/*; n .!%%n) # /n n%%n & $# *,#* !# &

,$#n) !#**; #9 "# C#*,/*/ : $/)$ # /n/n :/ )%%n) ,!!, !/*9

.!$#. :,#/ !!! ,""% ! ff %n) #,$ $!F!#,%n9 !#%9 #n

.!.!%n#*%; #! .%, & "#$"#%, $# #! "#$"#%,#**; !%,$9 ,)n%%*;

,".*%,# #n %&&%,/* #,$ ="%$9 20029 .+3>+

In "#$"#%,9 # rate % # !#% :n "#/!"n %$ %&&!n /n%+1 I& $

/n% ! /#n%; %n !., & $%,$ "$%n) % ,$#n)%n) % n .,%&%9 //#**; $ !#

% per unit time+ H!9 # !# & ,$#n) ,#n : .,%&% .! /n% %"9 ! .! /n%

& *n)$ ! "# ! #n$! /#n%;+ T$ " ,""n ;. & !# % .! /n% %"9

/,$ #.9 $#! !# #n &*/+ R# $# $# # nn%" n"%n#! %n,*/ ,$#n)

!#9 *%!#,; !# #n *,!%, &*/+In ,!%:%n) $ /n% & # !#9 $ ! .! % /

.#!# $ /n% & $ "#/!"n / ,#*,/*# $ !# =&! #".* # $#!

!# % .! :# .! "%n/>+ A !# &%n /%n) n/":! & $

#" /n% =/,$ # # !#> ! ,/n =/,$ # *%!#,; !#> %** !/* %n # %"n%n*

/#n%;9 $%,$ ,#n : .! # # .!,n#) =&! #".*9 $ )*:#* *%!#,; !# %n

18 # 80> ! &!#,%n ! # # "/*%.*+

O&n !# % # ;nn;" & !$;$" ! &!/n,;9 # ,/n .! ,n =%++ H!'> +)+ !#%

&!/n,% ! $#! !# ! #".* !#+

http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Rate_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Rate_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Exchange_ratehttp://en.wikipedia.org/wiki/Exchange_ratehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Electric_fluxhttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Units_of_measurementhttp://en.wikipedia.org/wiki/Tax_ratehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Dimensionless_quantityhttp://en.wikipedia.org/wiki/Dimensionless_quantityhttp://en.wikipedia.org/wiki/Percentagehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Fraction_(mathematics)http://en.wikipedia.org/wiki/Multiplehttp://en.wikipedia.org/wiki/Rhythmhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Hertzhttp://en.wikipedia.org/wiki/Radio_frequencieshttp://en.wikipedia.org/wiki/Radio_frequencieshttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Sample_ratehttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Rate_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Exchange_ratehttp://en.wikipedia.org/wiki/Exchange_ratehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Electric_fluxhttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Units_of_measurementhttp://en.wikipedia.org/wiki/Tax_ratehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Dimensionless_quantityhttp://en.wikipedia.org/wiki/Dimensionless_quantityhttp://en.wikipedia.org/wiki/Percentagehttp://en.wikipedia.org/wiki/Literacy_ratehttp://en.wikipedia.org/wiki/Fraction_(mathematics)http://en.wikipedia.org/wiki/Multiplehttp://en.wikipedia.org/wiki/Rhythmhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Hertzhttp://en.wikipedia.org/wiki/Radio_frequencieshttp://en.wikipedia.org/wiki/Radio_frequencieshttp://en.wikipedia.org/wiki/Heart_ratehttp://en.wikipedia.org/wiki/Sample_rate


4/20

HBMT4303

A !# & ,$#n) ,#n : &!"#**; &%n %n #;@

$! f(x) % $ &/n,%n %$ !., x ! $ %n!#* &!" a aJh+ An %n#n#n/ !#

& ,$#n) % /%#*n # !%#%+

An #".* ,n!# $ %&&!n, :n $ average #n instantaneous &%n%%n@

$ . & # ,#! ,#n : ,#*,/*#@

1+ An #!#) !# ,#n : ,#*,/*# /%n) $ #* %#n, !#** :n a #n b9 %% :;

$ !#* %"

2+ An %n#n#n/ !# ,#n : !"%n :; %%n) # ."! +

http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Speedometerhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Speedometer


5/20

HBMT4303

EXPLANATION OF THE MEANINGS OF

INSTANTANEOUS RATE OF CHANGE, AVERAGE RATE OF CHANGE ,

AND RELATIVE RATE OF CHANGE.

T$ . # $%,$ # #!%#:* ,$#n) ! # .,%&%, .!% & %"+ R# & ,$#n) % &n

/ $n .#%n) #:/ ""n/"9 #n % ,#n )n!#**; : .! # # !#% :n # ,$#n) %n

n #!%#:* !*#% # ,!!.n%n) ,$#n) %n #n$!+ G!#.$%,#**;9 $ !# & ,$#n) % !.!n

:; $ *. & # *%n+ R# & ,$#n) % &n %**/!# :; $ G! *! *#+ M#n; !#! .#; ,*

#n%n $ . # $%,$ n #!%#:* ,$#n) !*#% #n$!+ F! #".*9 .%n !#!

/; $ !*#%n$%. :n $ !# & ,$#n) %n $ .!%, & #n .%n !*#% # "#** ,$#n) %n

$ .!%, & $ /n!*;%n) #9 nn # #n .%n *#+

B#%,#**;9 $ !#% & $ ,$#n) %n $ /./ #*/ #n ,$#n) %n $ %n./ #*/ & # &/n,%n

% ,#** # !# & ,$#n)+ R# & ,$#n) % $ !#% $# $ $ !*#%n$%. :n $

#!%#:* %n /#%n+ In )n!#*9 $ ,&&%,%n & x % ,#** # $ !# & ,$#n) & #n /#%n+ F!

#".*9 %n yK3 xJ49 $ !# & ,$#n) % 3+ 3 % $ ,&&%,%n & x+ In $% /#%n9 $ ,n#n

#!%#:* % 4+T$ %.*#,"n %n .%%n % nn # $ !# & ,$#n) %n .%%n+ I& # .#!%,* %

%n%%#**; # A #n #&! %" % % # $ .%n B9 $n R# & ,$#n) %n .%%n K P#!%,* # P%nBP#!%,* # P%n A T%" T#n &! C$#n) %n P%%n+

V*,%; % $ . & # .#!%,* ! :


6/20

HBMT4303

S*. K R% & $ /#%n

R/n & $ /#%n

S*. K y2 y1

x2 x1

T$ &!"/*# &! !# & ,$#n) % )%n # &**@

R# & ,$#n) & x %$ !., y % )%n #

R# & ,$#n) & y K dx/dy

T$% % $ !*#%n$%. :n !# & ,$#n) #n *.+

INSTANTANEOUS RATE OF CHANGE

In#n#n/ !# & ,$#n) % $ *%"% & $ &/n,%n $# ,!%: $ #!#) !# & ,$#n)+ I %/ ,!%: $ #n :K2 x2J x3 # xK4+

S*/%n@

http://math.tutorvista.com/calculus/instantaneous-rate-of-change.htmlhttp://math.tutorvista.com/calculus/instantaneous-rate-of-change.htmlhttp://math.tutorvista.com/calculus/instantaneous-rate-of-change.html


7/20

HBMT4303

F! $ %n#n#n/ !# & ,$#n)9 ,#*,/*# $ &%! !%#% & $ )%n &/n,%n+

f = x>K4 xJ1

T$!&!9 &! %n#n#n/ !# & ,$#n) # xK4

f =4>K4 x4J1K17

AVERAGE RATE OF CHANGE

In )"!%,#* ,n,.9 $ !" !# & ,$#n) % $ *. & $ *%n #n =697> % $ ,$#n) %n $ ;,!%n# ! $

,$#n) %n $ ,!%n#9$%,$ % #* ,#** # $ *. & $ *%n+


8/20

HBMT4303

A!#) !# & ,$#n) &!"/*# % " K y2 y1 x2 x1 $!9 " % ,#** $ *. & $ *%n #n = x29 y2>+ T$ *. & $ *%n %n $ )!#.$ K " K 7364 K 42 K 2

Hn,9 $ #!#) !# & ,$#n) & $ *%n #n =697> % 2+

A!#) R# & C$#n) F!"/*#

F! #n; .%n = x19 y1> #n = x29 y2> %n $ ,!%n# .*#n9 %& δ % $ ,$#n) %n x #n %& δ; % $

,!!.n%n) ,$#n) %n y9 $n $ %&&!n, /%n δyδx K y2 y1 x2 x1 % ,#** $ #!#) !# &

,$#n) & y %$ !., x ! $ %n!#* = x19 y1>+ I % #* ,#** # $ !#% & ,$#n) %n ;

,!%n# ! $ ,$#n) %n ,!%n#+ ,#n #%*; &%n $ #!#) !# & ,$#n) %$ $ $*.

& $ #!#) !# & ,$#n) &!"/*#+

A &/n,%n % n $! $! % # !*#%n$%. :n x #n y % n # yK f = x>9 $! x % #n

%n.nn /#n%; #n y # .nn /#n%;+

RELATIVE RATE OF CHANGE

T$ !*#% !# & ,$#n) & # &/n,%n f = x> % )%n :; $ !*#%n f = x> f = x>9 $! f = x> % $ &%!

!%#% & $ &/n,%n+

S* E#".*

/%n@ I& f = x>K3 xJ1 % $ &/n,%n+ T$n9 &%n $ !*#% !# & ,$#n) # xK1+

S*/%n@

G%n $# f = x>K3 xJ1

f = x>K3

T$ !*#% !# & ,$#n) # xK1 K f = x> f = x>

K 33 xJ1

K 33Q1J1

K 34


9/20

HBMT4303

TEACHING & LEARNING MATERIAL AND LESSON PLAN

INVOLVING THE USE OF PREPARED TEACHING AND LEARNING MATERIAL

Mat L e ar! '!" Goa%(

R D,!%: !#*!* #..*%,#%n & !# & ,$#n) /%n) !:#* #n )!#.$%,#*

!.!n#%n9 +)+9 :/%n9 $#%n)9 ,*%n)9 "%n9 ,/!!n9 #! .!/!9

../*#%n9 n%!n"n9 !#n.!#%n+

R D,!%: ,nn,%n :n #!#) !# & ,$#n) #n %n#n#n/ !# &

,$#n) %n ,n+

Mater'a%(

R ,$#! .#.! #n "#!!

,"./! #n ## .!


10/20


11/20

1.1.1: Revisiting Rates of Change

Co!(o%'*ate

De1r'ef

)o%e G r o $ D'( $ ( ( 'o!

S$#! &%n%n) %$ $ ,*#+ A! #n; "%/n!#n%n)+

G/%%n) /%n@

R D,!%: $ *, .%n n # ,/! $# $ *. ,#n :! !.!n $

%n#n#n/ !# & ,$#n) # #n; .%n %n $ %n!#*+

R H /* ;/ ,$#n) $ %n!#* #!/n #,$ :/n, .!% :! %n&!"#%n #:/ $

#!#) #n %n#n#n/ !# & ,$#n) & $ :#**

Pa'r( Pa'r2Sare: Fra3 er Mo*e%

A ,#,$ B %n ,".*%n) # F!#;! "* &! #!#) !# & ,$#n)+ B ,#,$

A %n ,".*%n) F!#;! "* &! %n#n#n/ !# & ,$#n) =BLM 1+1+4>+


12/20

1.1.1: Revisiting Rates of Change

25-year Walleye Population

Year Walleye Population

0 3000

1 3400

2 3720

3 3976

4 4181

5 4345

6 4476

7 4581

8 4665

9 4732

10 4786

11 4829

12 4863

13 4890

14 4912

15 4930

16 4944

17 4955

18 4964

19 4971

20 4977

21 4982

22 4986

23 4989

24 4991

25 4993

Draining Waterfrom a Hot u!

ime "s# $olume "%#

0 1600

10 1344

20 1111

30 900

40 711

50 544

60 400

70 278

80 178

90 100

100 44

110 11

120 0

& 'ish(tory

A pond was stocked with a type of fish caed !waeye"#The ta$e on the eft %i&es the pop'ation of waeye in thepond fo( the 25 yea(s foowin% the stockin% of the pond"

2"

Do)n the Drain

The p'% is p'ed in a s)a hot t'$" The ta$e onthe (i%ht %i&es the &o')e of wate( in the t'$ f(o)the )o)ent the p'% is p'ed* 'nti it is e)pty"

(our*e:http+,,www"c ips a hoy"co) ,we $ %( a ph ics ,as 0 963"ht)

http://www.clipsahoy.com/webgraphics/as0963.htmhttp://www.clipsahoy.com/webgraphics/as0963.htmhttp://www.clipsahoy.com/webgraphics/as0963.htm


13/20

1.1.1: Revisiting Rates of Change "*ontinue+#

3"

,last -ff

A (ocket isa'nched" The%(aph shows itshei%ht a$o&e the%(o'nd f(o) ti)e of a'nchin% to (et'(nto ea(th"

4"


14/20

1.1.2: hats the Way the ,all ,oun*es/

-e&in d(opped a $a and coected the hei%ht .)/ at &a(io's ti)es .s/" A %(aph of thedata he coected is p(o&ided"

Part &: &verage Rate of Change

-e&in wants to ook at (ate of chan%e of the hei%ht at &a(io's ti)es" He hopes to

dete()ine how 'icky the hei%ht was chan%in% at &a(io's ti)es" -e&in fi(st ooks at thea&e(a%e (ate of chan%e fo( specific ti)e inte(&as" o)pete the ta$e with theinfo()ation in the %(aph"

0nterval Coor+inates of n+ Points &verage

Rate of

AB

B1

12

23

34

45

5H

hat do the &a'es fo( a&e(a%e (ate of chan%e te yo' a$o't the path of the $a andthe speed of the $a


15/20

Part ,: 0nstantaneous Rate of Change

et* -e&in wants to find a point in each inte(&a whose tan%ent has the sa)einstantaneo's (ate of chan%e as each secant"

:ketch a c'(&e of $est fit fo( this data in one coo'( and d(aw the secants fo( theinte(&as in anothe( coo'("

o( each ti)e inte(&a* ocate the point on the %(aph $etween the endpoints whose

tan%ent appea(s to ha&e the sa)e (ate of chan%e as the a&e(a%e (ate of chan%efo( that inte(&a"

1" ;n the inte(&a AB the a&e(a%e (ate of chan%e is " The point whose

tan%ent )atches this (ate of chan%e is . * /" (aw the

instantaneo's (ate of chan%e at this point* if possi$e"

2" ;n the inte(&a B* the a&e(a%e (ate of chan%e is " The point that

)ost cosey )atches this (ate of chan%e is . * /"

3" ;n the inte(&a * the a&e(a%e (ate of chan%e is " The point that

)ost cosey )atches this (ate of chan%e is . * /" (aw the

tan%ent at this point* if possi$e"

4" ;n the inte(&a * the a&e(a%e (ate of chan%e is " The point that

)ost cosey )atches this (ate of chan%e is . * /" (aw the

tan%ent at this point* if possi$e"


16/20


17/20

Ref%et'o! o! a(('"!#e!t

T$ ,n,. & !# & ,$#n) % .!$#. $ ,n!#* ,n,. %n %&&!n%#*

,#*,/*/+ I % ; 3 /n5!03#n5%n) #..!8%"#3%n #n "#8%"%'#3%n "$9 #n

$ &/n5#"n3#* $!" & ,#*,/*/09 #"n) $! $%n)+ Y 03/5n30

,n0%03n3*; $# "%0,n,.3%n0 #:/ !# & ,$#n)+

M#n; 03/5n30 $# # .! )!#. & $ ,n,. & !# & ,$#n)9 #n n /.

%n%&;%n) # !# & ,$#n) %$ # *. & # 3#n)n3 *%n ! # ,$! %3$/3 :%n)

#:* /n5!03#n5 % %n # !#*!* ! #*):!#%, ,n3183+

S3/51n30 n ,".#!3"n3#*%' $%! n*)9 #n "#; n !#*%' 3$#3

.%, /,$ #0 $ ,$#%n !/*9 !*# !#9 %".*%,% 5%&&!n3%#3%n9 ,+9 #! #** #:/

!# & ,$#n)+ T$; $# # %&&%,/* %" !,#**%n) $ ,n,. & !# & ,$#n) %n

$ ,n & $ F/n5#"n3#* T$!" & C#*,/*/0+

T$! #! 5%&&!n3 n%n 3$#3 %n* !# & ,$#n) & &/n,%n & @

1+ A!#) !# & ,$#n) & & :n 37 .%n30+

2+ In03#n3#n/0 !# & ,$#n) & & # # .%n3+

3+ A!#) !# & ,$#n) & & :n # #n 9 # # &/n,%n & 8+

4+ In03#n3#n/0 !# & ,$#n) & & # 9 # # &/n,%n & 8+


18/20

7T /n5!03#n5 $% ,n,. .!.!*;9 03/5n30 &%! n $# # *% )!#. &

$ n%n & ,n03#n3 !# & ,$#n)+ S#;%n) 3$#3 n 6/#n3%3; ,$#n) # #

,n03#n3 !# %$ !., #n$! "#n0 3$#3 #n; &!#,%n & $ #* ,$#n)

& n 6/#n3%3; ,!!.n %$ $ #" &!#,%n & $ 33#* ,$#n) & $ $!

6/#n3%3;+ F! #".*9 %& #n :9

.+ 2329 %**/03!#30 $% .!%n,%.*@

I& # .#!%,* % "%n) # # ,n03#n3 !# =.>9 #n % !#* 140 & %n 4

,n9 $n $ 3%" % # !#* #n; %n!#* & *n)$ 1 W 140 K 35 & %

1W 4 K 1 ,n+ S%"%*#!*;9 $ %" %3

# !#* #n; %n!#* & *n)$ 1 W 35 K 7 & % 1 W 1 K 1 0,n5+

A 31#,$%n) /))%n % / "%n #*n) .#!#*** *%n %n # ,!5%n#35

"#nn!9 :%n) 8.*%,%3 #:/ $ &&! ,!%n# $ "%n+ P$;%,#* "%n9

#n%"#3%n09 #n $! ,"./3!G:#05 * ,#n $*. ,%*%# $% *#!n%n)+


19/20

%$ $% n%n %n .*#,9 $ #!#) !# & ,$#n) & n 6/#n3%3; %$

!., #n$! 5/!%n) #n %n!#* ,#n : %n3!5/,5 # $ ,n03#n3 !# &

,$#n) n,#!; ;%* $ #" n ,$#n) %n $ &/n,%n #*/ ! $

n%! %n3!#*+ F! #".*9 %& # .!n !#* 20 "! %n 4 0,n50 # #

#!%#:* !#9 $n $ #!#) !# & ,$#n) % 5 "! .! ,n9 :,#/ #

.!n 3!#*%n) # # ,n03#n3 . & 5 "! .! ,n /* !#* $

#" 20 "31!0 %n 4 ,n = 3$ &%)/! :*>+ In $% #..!#,$9 $

#!#) !# & ,$#n) % # #; ,".#! # #!%#:* !# & ,$#n) #

,n03#n3 !# & ,$#n) ! #n %n3!#*+

T$ .##) &!" #!#) !# & ,$#n) %n03#n3#n/0 !# & ,$#n) %

.!$#. $ : "3%G #%n &! %n3!5/,%n) $ n%n & *%"% %n ,#*,/*/+


20/20

REFERENCES

Ann9 H#!X B%n9 I!*X D#%9 S.$n =F:!/#!; 29 2005>9 Calculus: Early

Transcendentals Single and ultivariable =8$ +>9 N Y!@ %*;9 ISBN 780471

472445

A.*9 T" M+ =/n 167>9 Calculus! "ol# $: %ne&"ariable Calculus 'ith an

ntroduction to inear *lgebra9 4 =2n +>9 %*;9 ISBN 780471000051

A.*9 T" M+ =/n 16>9 Calculus! "ol# +: ulti&"ariable Calculus and inear

*lgebra 'ith *pplications9 4 =2n +>9 %*;9 ISBN 780471000075

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