Calculator Techniques

(1)

SESSION 1

PART 1

PART 1 1. What are the roots of

1. What are the roots of

the polynomial:

A. -3, -

!. ", #

$. #, 3

%. 3,

$o&e:

Set

Set CalculatorCalculator to equation mode: MODE>5>3 for quadratic equation to equation mode: MODE>5>3 for quadratic equation

Mode 3 because our polynomial is a

Mode 3 because our polynomial is a 2nd degree equation2nd degree equation

!nput t"e equation coefficients a#b#c:

$ % &' % $ 2 % %

(nd you )ill get t"e

(nd you )ill get t"e ans)erans)er

#. What are the roots of

the polynomial:

A. 1,#,3

!. 1,#,'

$. 1,#,

%. #,3,'

$o&e:

Set

Set CalculatorCalculator to equation mode: MODE>5>* for cubic to equation mode: MODE>5>* for cubic equationequation

Mode * because our polynomial is a

Mode * because our polynomial is a 3rd degree equation3rd degree equation

!nput t"e equation coefficients a#b#c:

$ % &' % $ * % &+ % %

3. Whi(h of the follo)in* is a possi+le root of the

polynomial:

A. 3

!. 

$. -#

%. 

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,O-E: ( root is any

,O-E: ( root is any .alue t"at# )"en substituted to t"e .ariable/ie 01# )ill .alue t"at# )"en substituted to t"e .ariable/ie 01# )ill satisfysatisfy

t"e equation/!n our equation % 1

Set t"e

Set t"e calculatorcalculator to computation mode: MODE>$ to computation mode: MODE>$

!nput O,4 t"e left side of

!nput O,4 t"e left side of t"e equationt"e equation

-rial and error# se t"e C(C function

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(3)

6ag nagtanong ang

6ag nagtanong ang calculatorcalculator 78 iinput ang mga c"oices 78 iinput ang mga c"oices

78 3 %

output is 2*

repeat t"e step until you get an output of 

. in& the al/e of 0 an& y in the follo)in* e/ations:

A. -1#21, 1"21

!. -#21, -##21

$. 1#21, -1"21

%. #21, ##21

$o&e:

Set

Set calculatorcalculator to equation mode: MODE>5>$ for t)o&.ariable equation to equation mode: MODE>5>$ for t)o&.ariable equation

!nput t"e coefficients of a#b and constant c

!nput t"e coefficients of a#b and constant c of t"e first equation:of t"e first equation:

2 % 3 % 9 %

!nput t"e coefficients of a#b and constant c

!nput t"e coefficients of a#b and constant c of t"e second equation:of t"e second equation:

&3 % 5 % 2 %

6ress t"e % to get t"e .alue of 0 and press again to get t"e .alue of y

'.

The

e/ation

has

t)o

rational

roots,

+oth

of )hi(h are positie. in& the lar*er of these t)o roots.

A. 1

!. #

$. 3

%. 

$o&e:

Same met"od )it" number 3e )ill be

Same met"od )it" number 3e )ill be using C(C function-"eusing C(C function-"e calculatorcalculator s"ould s"ould

display 

".

What

is

the

remain&

er

of

the

polynomial

)hen

&ii&e& +y

A. #4

!. #14

$. #43

(4)

%. 3'

$o&e:

Set t"e

!nput t"e left side of t"e

!nput t"e left side of t"e equationequation

(pply t"e ;emainder t"eorem

se C(C function and substitute 0%5

5. Sole the al/es of y in the system of

e/ations:

$o&e:

Set t"e

Set t"e calculatorcalculator to equation mode: MODE>5>2 to equation mode: MODE>5>2

!nput t"e coefficients a#b and constant c of

!nput t"e coefficients a#b and constant c of t"e first# second and t"ird equationt"e first# second and t"ird equation

6ress t"e equal % button t)ice to get t"e .alue of y

4. Sole for the al/e of

A. #

!. #i

$. -#

%. -#i

$o&e:

Set t"e

!nput t"e equation in t"e

!nput t"e equation in t"e calculatorcalculator/as is1/as is1

and you )ill get t"e

and you )ill get t"e ans)erans)er

. Sole for the al/e of

A. 1.1

!. #.1

$. 1.1

%. #.1

$o&e:

Set t"e

Set t"e calculatorcalculator to computation mode to computation mode

!nput t"e equation in t"e

!nput t"e equation in t"e calculatorcalculator/as is1/as is1

(5)

1. Sole for the al/e of 0 in the e/ation:

A. 1.1

!. 1.1

$. 1."

%. #.13

$o&e:

Set t"e calculator to computation mode

!nput t"e equation and use t"e SO<ED function to get t"e .alue of 0 -o get t"e SO<ED function# S=!->C(C>%

!t s"ould display# 7% ans)er

&;%  /important# must be 1 (nd you )ill get t"e ans)er

PART #

11. Eal/ate

A. #

!. ##

$. #

%. ##

$o&e:

Set t"e calculator to computation mode: MODE>$ !nput t"e equation as is but replace ? )it" 7 -"en press %

(nd you )ill get t"e ans)er

1#. in& 0, if

A. !.

$.

%.

$o&e:

Set calculator to computation mode

-ype t"e equation as isse t"e @SO<E@ function and press % (nd you )ill get t"e ans)er

13. a(tor

A.

(6)

$.

%.

$o&e:

irst# type t"e equation and assign an arbitrary constant to substitute to 0 and use t"e C(C function

for e0ample 7%+

-"e calculator )ill display @2$A'@

Second# substitute + to t"e c"oices and find )"ic" one )ill be equal to @2$A'@ Start )it" t"e first c"oice until you get t"e .alue of @2$A'@

"at )e do as )ell in t"e pre.ious e0amples li?e t"is is called @;E<E;SE E,B!,EE;!,B@

1. in& the 1'th term in the arithmeti( se/en(e: 1,3,',5 . . .

A. #'

!. #5

$. #

%. 31

$o&e:

$ Set t"e calculator to S-(- modeMODE 3>2 2: (7 is for (rit"metic 6rogression/ (61

May ma?i?ita ?ayo 7 and 4 column(nd 7 column natin ay ang nt" term )"ile sa 4 column natin ay ang (6

2 !nput sa 7 column ang $#2#3/$st term#2nd term# 3rd term1Sa 4 column naman# iinput ang $#3#5 /?a"it 3 number lang# ?a"it )ag na yung '1

"ic" mean t"e $st term of t"e (6 is $# 2nd term is 3 and 3rd term is ' 3 (fter niyo mainput# press (C/dont )orry "indi mabubura ang data niyo1 Sa problem ang "ina"anap ay ang pang $5t" termanu ba ang .alue ng i?a&$5t" term8meaning ang "ina"anap ang .alue sa 4 columngets8

* -"erefore# type $54&"at

(ng 4&"at ay ma?i?ita )"en you press S=!->$/S-(-1>'/;eg1>5/4&"at1)"ile ang 7&"at ay ma?i?ita )"en you press S=!->$/S-(-1>'/;eg1>*/7&"at1ta?e note of t"is da"il

gagamitin natin ito in t"e ne0t problems 5 -"en press %

Fung may tanong ?ayo dito at di niyo ma?u"a pa?ipost na lang

1'. in& the 1#th term in the *eometri( series: 3, , #5 . . .

A. ',

!. 155,15

$. '31,11

%. 1,',3#3

$o&e:

$ Set t"e calculator to S-(- modeMODE 3>5 or MODE3>9 /its t"e same for Beometric 6rogression# B61

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May ma?i?ita ?ayo 7 and 4 column(nd 7 column natin ay ang nt" term )"ile sa 4 column natin ay ang B6

2 !nput sa 7 column ang $#2#3/$st term#2nd term# 3rd term1Sa 4 column naman# iinput ang 3#A#2'

"ic" mean t"e $st term of t"e B6 is 3# 2nd term is A and 3rd term is 2' 3 (fter niyo mainput# press (C/dont )orry "indi mabubura ang data niyo1 Sa problem ang "ina"anap ay ang pang $2t" termanu ba ang .alue ng i?a&$2t" term8meaning ang "ina"anap ang .alue sa 4 columngets8

* -"erefore# type $24&"at 5 -"en press %

Fung may tanong ?ayo dito at di niyo ma?u"a pa?ipost na lang

1". What is the s/m of the 6rst # terms of the se/en(e: 1, 3, ',

5...

A. 3"1

!. 31

$. 

%. 1

$o&e:

$ Set t"e calculator to S-(- mode: MODE 3>2 /since it is (61 2 !n t"e 7 column# input $#2#3!n t"e 4 column# input $#3#5

3 (fter you input t"e data# press (CDont )orry your data )ill not be deleted /unless you c"ange your mode1

* 6ress S=!->OB/found belo) t"e O, FE41)e )ill use t"e SMM(-!O, function /SMM(-!O, symbol is li?e E1

5 6ress (6=(>1/for 71>S=!->$>'>5/for 4&"at1>S=!->1/for comma1>$>S=!->1/for comma1>2$>1

9 4ou )ill noticed t"at our summation is from $ to 2$"y 2$ and not 28ecause )e start our progression in $!f )e start t"e progression from # t"en t"ats t"e time )e use  to 2

(nd it )ill be li?e t"is)alang 4&"at na symbol e

15. What is the s/m of the in6nite series: 1, -12', 12#', . . .

A. '2"

!. 2'

$. '25

%. "25

$o&e:

$ Set t"e calculator to Computation ModeMODE>$

Since t"is is an infinite series# )it" a common ratio of less t"an $# t"en t"e sum is a finite number/,O-E: or an infinite series )it" a common ratio of greater t"an $# t"e sum is infinite1

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2 !nput t"e equation under SMM(-!O, function and get t"e sum from  to a large .alue# say# 5So it )ill be li?e t"is-"en press %

14. What is the s/m of "771#7. . .71518

A. '"

!. 34

$. '14

%. '"#

$o&e:

$ Set t"e Calculator to S-(- modeMODE3>2 /! "ope you already ?no) )"y MODE 3 21 2 !n t"e 7 column# input $#2#3!n t"e 4 column# input 9#A#$2 t"en press (C 3 e s"ould get t"e sum but )e dont ?no) "o) many term does it "a.eSo lets

determine "o) many terms it "asEarlier in t"is tutorial ! said 7 is for nt" term and 4 is for progression-"erefore# our code is $'$7&"atSince $'$ is our last term(nd t"e calculator )ill display @59@

* no) use t"e SMM(-!O, function li?e in number 29 but t"is time# its $#59

1. What is the "th term of the in6nite series: 1, -12', 12#', ...

A. 1231#'

!. 121'"#'

$. -1231#'

%. -121'"#'

$o&e:

$ Set t"e calculator to Computation mode

2 !nput t"e general equation of t"e infinite series and substitute 5 to 0 using t"e C(C function to display t"e 9t" term(nd press %

#. Rationali9e the *ien e/ation:

A. #-'i

!. '71i;2"

$. 17#i

(9)

$o&e:

$ Set t"e calculator to COM6E7 modeMODE>2

2 !nput t"e equation as it is in t"e calculator/,O-E: -"e imaginary i is found by pressing t"e E,B ?ey

3 -"en press %

SESSION #

Part 1 1. S/+tra(t

A. !.

$.

%.

$o&e:

Siguro naman ?ayang ?aya niyo na ito:D

#. %/rin* a rain, #mm of )ater fell. in& ho) many *allons of )ater

fell on a leel 1 a(re par<.

A. #1,341

!. #',341

$. #5,341

%. 3,341

$o&e:

Con.ersion only

<%$ acre /con.ert acre to mG21S=!->+/for CO,<1>$$/for acre>mG21!t )ill gi.e an ans)er **9+59

-"en multiply it to 2 m/gi.en in t"e problem1 and it )ill display +A3'$2 in mG3 Con.ert t"is to iters$ mG3 % $(nd it )ill display +A3'$2 in 

inally con.ert t"is to gal6ress CO,<H$* >gal/S1 (nd you )ill get t"e ans)er in gal

3. The f/n(tion

is &is(ontino/s at 0=8

A. #

!. -#

(10)

%. neither a nor +

$o&e:

6ag sinabing discontinous# mageerror siya sa .alue na isusubstitute mo or yung

tinata)ag na M(-= E;;O;6ag may lumalabas parin na .alue positi.e or negati.e or e.en Iero# continous parin yundapat mageerror siyagets8

Set t"e calculator to COM6 MODE

type na equation as it ispress t"e C(C function and substitute t"e numbers in t"e c"oices and find )"ic" one )ill get an error-"e one error is t"e ans)er

(lternati.e solution)e use t"e -(E functionMODE ' -ype t"e equation as it is and press %

S-(;-8 /your number depends on t"e c"oices)e start from t"e lo)est )"ic" is &2So enter &2 t"en press %

E,D8 /t"e largest in t"e c"oices1So enter 2 and press %

S-E68 /-"e .ariation in t"e c"oices1!mportant po ito?ailangan madaanan nung function yung mga c"oices!n t"is case 2 and &2So enter 2 and press %

Fung saan may error yun na yun

. in& the possi+le fa(tors of

.

A. 0-#;

!. #0-1;

$. #073;

%. all of the a+oe

$o&e:

6ara masabing factor siya# ?ailangan ang y% pag isusubstitute natin yung root of 0 e use C(C functionSet t"e calculator to MODE $

-ype t"e equation as it is and press C(CBet t"e root of t"e c"oices and substitute

78 enter 2 if y%# t"en possible root siya-ry t"e ne0t one 78 enter $J2# if y%# t"en possible root siya-ry t"e last one 78 enter &3J2 if y%# t"en possible root siya

-"erefore you ?no) t"e ans)er

'. %etermine the area of the trianle +o/n&e& +y the straiht lines

07#y-5=, 30-y-1=, an& #0-y7"=..

A. 1 s./nits

!. 1' s./nits

$. # s./nits

%. #' s./nits

$o&e:

irst get t"e point of intersectionuse t"e EK, functionMODE 5>$ or t"e first t)o equation# type:

(11)

3 &* $

t"en press % %-"erefore t"e point of intersection of t"is is 0%3# and y%2 -"en try t"e ne0t equation

3 &* $ 2 &$ &9

-"en press % %0%&5# and y%&* try t"e last combination of equation $ 2 '

2 &$ &9

-"en press % %0%&$# and y%*

-"en )e use t"e determinant function to get t"e area of t"e triangle Set t"e calculator to M(-;!7MODE 9 and press (C

"at important in t"is is ?ailangan natin madaan yung D!M or dimension ng matri0So press S=!->*>$>$>$ and enter

3 2 $ &5 &* $ &$ * $

t"en press (C

-o get t"e determinant press S=!->'/det1>S=!->*>3/since )e enter our data in Mat(-"en press %

;emember t"e area of a triangle gi.en t"e .ertices is 5 times t"e determinant-"erefore multiply it to 5 and get its (bs .alue -"en you )ill get t"e final ans)er

". in& the (enter of the (ir(le that is (ir(/ms(ri+e& a+o/t the

trian*le )hose erti(es are -3,1;, 3,1; an& ',3;

A. -3,;

!. 3,-;

$. -3,-;

%. 3,;

$o&e:

Substitute 0 and y in t"e general equation of circle !n t"e first point you )ill get t"e equation &3D&E%&$ -"e second point )ill gi.e you 3DE%&$

-"e t"ird point )ill gi.e you 5D3E%&3* -"erefore )e "a.e 3 equations# 3 un?no)n se t"e EK, 2 and enter

&3 &$ $ &$ 3 $ $ &$ 5 3 $ &3*

(nd you )ill get D%9# E%&$+# and %&$ (nd )rite it in t"e general form

!to ang s"ortcut para ma?u"a ang center CE,-E; O C!;CE# E!6SE or 6(;(O(: /"1

(12)

-"erefore our center /"#?1 is at /&3#A1

5. in& the al/e of y of the para+ola )hose a0is is erti(al an&

passes thro/*h -1,;, ',;, 1,4; an& ,y;

A. -'

!. '

$. -"

%. "

$o&e:

e use S-(- 3/ for parabola1 in t"is case !n t"e 0 column enter &$#5 and $

!n t"e y column enter ## and + t"en press (C ind t"e .alue of y )"en 0 % *

t"erefore */y&"at1 or

4. in& the s/m of an A.P.: #,',4...)here n=#'..

A. 

!. #'

$. '

%. 5'

$o&e:

se S-(- 2/for (61 !n t"e 0 column enter $#2

!n t"e y column enter 2#5 t"en press (C se t"e SMM(-!O, ,C-!O,

. in& the s/m of a >.P.: #,3,.',...)here n=1.

A. ##".""

!. #3'.#

$. ##.'#

%. #'1.4

$o&e:

(13)

Same met"od )it" HA - use S-(- 9/for B61 -"en use t"e SMM(-!O, function to get t"e sum (nd you )ill get t"e ans)er

1. In ho) many )ays (an a pi(t/re +e painte& +y /sin* three3; or

more of the seen5; &i?erent (olors8.

A. 4

!. 

$. 1'

%. 1#

$o&e:

Set t"e calculator to COM6 MODE

e use t"e COM!,(-!O, function!t )ill be found by pressing S=!- > L / nCr 1 (nd type t"e equation ust li?e t"is:

Part #

11. in& the &e(imal e/ialent of

A. 3#"55

!. 3"#55

$. #3"55

%. #"355

$o&e:

Con.ert niyo lang sa decimal

MODE *Set to !, t"en type t"e number 6ress DEC and you got t"e ans)er

Additional TIP:

Fung "indi na ?asya sa calculator at sobrang "aba ng !, mo# icon.ert mo muna sa OC- $$%5

$$%9 $%$ $$$%' $$%5

Set to OC- and type t"e OC- equi.alent# 59$'5 (nd press DEC and you got t"e ans)er

1#. in& the he0a&e(imal e/ialent of

0

A.

(14)

$.

%.

$o&e:

-ype niyo lang sa calcuag ?alimutan base + and base $ yan 6ress =E7 for t"e final ans)er

13. Eal/ate A 0 !.

A. !.

$.

%.

$o&e:

MODE 9>(C-andaan daanan lagi si D!M S=!->*>$

!type ang matri0 and t"ats it 4ou got t"e ans)er

Additional TIP:

!f (J# DO,- use di.ide# instead use t"e re.erse function(nd it )ill be li?e t"is

1. in& &y2&0 if

an&

)hen

(15)

!. .4""

$. 1.43

%. 1.53#

$o&e:

6ag -rigonometricJ!n.erse# i&set ang calcu sa ;(D!(, MODE !&type sa calcu li?e t"is

4ung po natin diyan# pi po yun

1'. If y = tanh 0 7 sinh #0, )hat is the slope of the (/re )hen 0 =

#8

A. -'.

!. -#4.#

$. '.

%. #4.#

$o&e:

-(,D((,:

Slope % first deri.ati.e %

-o get t"e slope# simply get t"e first deri.ati.e of t"e equation (nd you )ill get t"e ans)er

1". in& the slope of the e/ation

)hen

A. '.#

!. -'.#

$. .#

%. -.#

$o&e:

Baga)a tayo ng equation from t"e figure# since may gi.en tayo na r and t"eta-"en )e

must come up )it"

sing 6yt"agorean t"eorem:

(nd deri.e )it" respect to t"eta

(16)

,e0t )e must get

rom t"e figure# )e "a.e

and equating it to 0# )e "a.e:

Substituting t"e gi.en and getting t"e deri.ati.e of 0 )it" respect to t"eta# )e "a.e:

Di.iding t"e results and you got t"e ans)er

15. in& the e/ation of the nomal line of the (/re

at 1,#;

A. 0-y=#

!. 0-y=-#

$. 07y=

%. 07y=-

$o&e:

"en you are getting t"e slope of t"e line using t"e first deri.ati.e# )"at you are getting is t"e slope of t"e tangent line-o get t"e slope of t"e ,ormal ine# you get t"e negati.e reciprocal of t"is

So# get t"e first deri.ate/-angent slope1 of t"e gi.en equation )it" 0%$/gi.en in t"e problem1

(nd getting t"e negati.e reciprocal of t"is )e "a.e &$J* -"en get t"e equation using S-(- 2

(nd press (CBet ( and  S=!->$>'> and S=!->$>'>2

;emember# in t"e calculator# t"e equation of S-(- 2 is y%ab0 Substitute a and b in t"e equation and you )ill get t"e ans)er

14. The (hare in (o/lom+s that passes thro/h a )ire after t

se(on&s is *ien +y the f/n(tion.

%etermine the (/rrent at the en& of # se(on&s..

A. " A

(17)

$.  A

%. 1 A

$o&e:

Simple get t"e first deri.ati.e of t"is )it" 0%2 /2 seconds1 and you )ill get t"e ans)er

1. A 14-*allon tan< of )ater &rains from the +ottm in 3 min.

A((or&in* to Torri(elli@s la), the ol/me of )ater remainin* in the

tant after t min/tes is

)here

o) fast is the )ater &rainin* from the tant after # min/tes8

A. - *pm

!. -3 *pm

$. -# *pm

%. -1 *pm

$o&e:

Simply get t"e deri.ati.e of t"e equation )it" 0%2 /2 minutes1 (nd you )ill get t"e ans)er in gpm

#. in& the al/e of 0 )hen the f/n(tion y=ln020 is at ma0im/m

al/e.

A. 

!. e

$. 1

%. 12e

$o&e:

or ma0imaJminima# t"e first deri.ati.e s"ould be equal to 

-ry or test all t"e c"oices using C(C function if dyJd0% )it" 0 % 0 to use C(C function )"ile getting t"e deri.ati.e

and press C(C 08  08 e 08 $ 08 $Je

Part 3

(18)

#1. in& the

A. !.

$.

%.

$o&e:

Remember, in solving Trigonometric or inverse trigonometric, use RADIAN MODE.. agyan ng limits from  to $ t"en press %

Ma?a?a?u"a ?a ng sagot!&note ito

i&substitute ang limits sa c"oicesag intindi"in ang constant @C@remember# upper limit minus lo)er limitlets say for e0ample sa c"oice (

6ag pumare"as ang sagot mo ?anina/yung ni&note mo1 yun ang sagot =appy integratingN1

##. in& the (entroi& of the area +o/n&e& +y the para+olas

an&

A. 1.4, 1.4;

!. 1., 1.;

$. #,#;

%. #.1, #.1;

$o&e:

S"orcuts lang ang ibibigay ?o since ito ay calculator tec"nique Bi.en:

6ag na?ita niyo na pare"as yan# i mean *0#*090#9y+0#+yA0#Ay and so on# ang s"ortcut niyan is A0/center1: t"erefore#

A 0 /2#21 % answer

#3. in& the area of the re*ion +o/n&e& +y

, the 0-a0is,

an& the erti(al lines 0= an& 0=..

A. ' s. /nits

!. 1"23 s. /nits

(19)

$. 1523 s. /nits

%. "s. /nits

$o&e:

irst get t"e limits from t"e equation it selfse EK, 3 7%3 and 7%2

6asensiya na po ?ayo sa dra)ing ?o"irap e(yun bale alternating po yan -"en integrate using t"e limits

Faya po negati.e yung isa ?asi dumaan po sa negati.e "alf cycle (nd you )ill get t"e ans)er

#. %etermine the area of the re*ion +o/n&e& +y the

(/re

an& the 0-a0is,

A. #.#' s. /nits

!. #.' s. /nits

$. 3 s. /nits

%. 3521# s. /nits

$o&e:

Bet t"e limits from t"e equationse EK, *

(ng you )ill get t"e ans)er

#'. in& the len*ht of ar( of the (/re y = ln (os0 from 0= to

0=pi2..

A. ."'

!. .5#

$. .41

%. .44

(20)

$o&e:

Fung alam niyo formula gamit ang long met"od# goDi ?o na siya ibibigayS"ortcuts lang la"at ito

(ng s"ortcut dito ay distance formula/t)o points1 irst get t"e functions of y%f/01

"en 0%# 0%piJ* -"en y%# &3*

-"en use t"e distance formula (nd you )ill get t"e ans)er

#". The spee& of the parti(le is *ien +y

.

What &istan(e &oes it trael )hile its spee& in(reases from 5 to 

ft2s8

A. 43.3 ft

!.  ft

$. ".5 ft

%. 1 ft

$o&e:

Bet t"e limits from t"e equation )"en d%'se EK, * 2 5  &'

Bet t"e real number and t"at is your lo)er limit Bet t"e limits from t"e equation )"en d%AAse EK, * 2 5  &AA

Bet t"e real number and t"at is your upper limit

-"en integrate t"e equation using your lo)er and upper limit (nd you )ill get t"e ans)er

#5. Sole for the *eneral sol/tion of the &i?erential

e/ation:

A. !.

$.

%.

$o&e:

Bet t"e rootsse EK, *$   +

4ou )ill get one real root and t)o comple0 roots E!:

(21)

6ag comple0 root % sin and cos

#4. Sole& for the parti(/lar sol/tion of the &i?erential e/ation: 0 7

y &y2&0 = # )hen 0=1 an& y=1

A. !.

$.

%.

$o&e:

C"oice one from t"e c"oices and Con.ert t"e equation to y%f/01

Differentiate using your calculator )it" 0%$/as gi.en1 and you )ill get t"e .alue of dyJd0

Substitute 0# y and dyJd0 to t"e equation in t"e problem and see if it is equal to 2

2%2

!f yes# t"en your c"oice is t"e rig"t ans)er

#. Sole for the *eneral sol/tion of the &i?erential e/ation y@-y =

#..

A. !.

$.

%.

$o&e:

C"oose one in t"e c"oices

C"oose your fa.orite number for C and 0 and substitute it to your c"oice (nd sol.ed for yi&note

i&differentiate ang iyong napiling equation )it" 0%/?ung ano ni&substitute mo sa una# be consistent1

4ou )ill get t"e y

-"en susbtitute in t"e gi.en equation y&y%2if

2%2

-"erefore# napili mo ang tamang ans)er

Part 

31. A ra&ioa(tie s/+stan(e &e(reases from 1 rams to  rams in

t)o ho/rs. in& its half life.

A. 1.1 hr

!. 11. hr

(22)

$. 1#.4 hr

%. 13. 1" hr

$o&e:

Da"il sa ito ay radioacti.e# nagiincrease ito e0ponentially "ic" means ?ung e0ponentially ito# gagamitin natin ang S-(- 5

(ng 0 column natin ay ang time/"r1 and y column natin ay quantity 7&&&&&&&&&&4

&&&&&&&&&&$ 2&&&&&&&&&&A

since time in its "alf life ang "ina"anap# press (C t"en find

3#. A thermometer rea&in*

is +ro/*ht into a room )here the

temperat/re is

B 1 min later, the thermometer rea&in* is

.

in& the temperat/re e/ation as a f/n(tion of time.

A. !.

$.

%.

$o&e:

6ag temperature ang pinaguusapan use S-(- 5

Sa 0 column natin is t"e time/min1 and in y column its eit"er @-s&-@ or @-&-s@ /temperature1 !t depends ?ung sino mala?i ?ung si - or si -s

- % Obect temperature

-s % surroundingJen.ironment temperature gets8 ets proceed

Since mala?i ang -s natin )"ic" is # t"erefore ang gagamitin natin ay -s&- Bets8

7&&&&&&&&&&4/-s&-1 &&&&&&&&&&/'&$+1 $&&&&&&&&&&/'&3$1

(fter natin ma&input ang data# press (C Since ang ?inu?u"a natin dito ay equation# t"erefore ?u?unin natin ang .alue ng ( and  ! "ope and assume na you ?no) it already ?ung paano at saan ito ?u?unin since naturo ?o na ito sa naunang sessionJpart

(fter ma?u"a ang ( and # alam natin na ang equation sa S-(- 5 ay

!&substitute ang ( and # 0%t and y%-s&- since yun ang ginamit natin/y%'&-1

33. o) fast &oes liht trael in lass of refra(tie in&e0 1.'8

A. !.

$.

(23)

%.

$o&e:

;efracti.e inde0 formula

3. At the s/rfa(e of the earth,

. Ass/me the earth to +e

a sphere of ra&i/s ",351 <m, (omp/te the mass of the earth.

A. !.

$.

%.

$o&e:

- %mg# assume m % -"erefore# cancels out

(nd you )ill get t"e ans)er

3'. $al(/late the form fa(tor of a perio&i( oltae hain the

follo)in* al/es for e/al time interals (hanin one al/e to the

ne0t: , ', 1, #, ', ", ', #, 1, ', et(.

A. 1.11

!. 1.#

$. 1.3'

%. 1.1

$o&e:

(24)

se S-(- $ and follo) t"e formula# and you )ill get t"e ans)er

3". in& the al/e of

A. 4 - 4i

!. 4 7 4i

$.  - i

%.  7 i

$o&e:

(ng p)ede lang po sa ating calcu ay cube at square ng comple0 6)ede pong ganito:

Or p)ede din ito:

35. %etermine the ar*/ment of the res/lt of

A. -5#

!. #5

$. 5".

%. 1'.

$o&e:

(25)

!&type lang po natin sa calcu yung gi.en t"en press % Con.ert po natin sa polar form t"en booomm4un na

34. I've reached the image limit. Sa next part na lang. Marami pa

naman ito.

3. The resistan(e of a )ire is 1#".4 ohms at 1 &e*rees $ an&

1 ohms at 3 &e*rees $. %etermine the temperat/re )hen the

resistan(e is 11' ohms.

A. "'.1' &e*rees $

!. "."' &e*rees $

$. 51.#' &e*rees $

%. 5'.' &e*rees $

$o&e:

S-(- 2 7 column % temp 4 column % ;esistance &&&0&&&  &&&4&&& &&&$  &&&$29*+ &&&3  &&&$

(26)

. What is the present )orth of t)o 1, pesos payment at the

en& of the 3r& an& th year if the ann/al interest is 4C (ompo/n&e&

/arterly8

A. P 1', 1"

!. P 1', #44

$. P 1", 5#1

%. P 15, #"

$o&e:

(ng formula po talaga nito ay )"ere:

 % future )ort" 6 % present )ort" i % interest rate

m % number of compounding/quarterl# semi&anually# annually etc1 t % time/years#mont"# days# etc1

Fung alam niyo ?ung paano computin sa long met"od# t"en go :D -"en ito po yung sa C(C -EC=,!KE:

irst ?u?unin natin ang 6 ng 3rd and *t" year So una"in natin si 3rd year gets8 S-(- 5 or 9

7 column % mt/ito yung na?a&raised sa formula1

4 column %

&&&&& 7 &&&&&  &&&&& 4 &&&&& &&&&$2 &&&&&  $#

&&&&$3 &&&&&  $2/$#1 ba?it po $28 ?asi mt%/*1/31

ba?it po $38 ?asi ne0t year $ year gets8

ba?it $# lang# a?ala ?o yung formula8 ginagamit lang natin yung formula on t"e ne0t year

-"en get t"e present )ort":

Second# ?u?unin natin ang present )ort" sa *t" year Same lang din ito sa 3rd year (ng pinag?aiba lang yung mt

&&&&& 7 &&&&&  &&&&& 4 &&&&& &&&&&$9&&&&&  $#

&&&&&$'&&&&&  $2/$#1 -"en get t"e present )ort":

inally get t"e sum of t"e present )ort" in t"e 3rd year and *t" year (nd you )ill get t"e ans)er

(27)

Part 1 1. in&

.

A. 

!. .3'3

$. in&eterminate

%. in6nity

$o&e:

!&type lang ang equation 6ress C(C  08

ag po natin i&substitute as 0% da"il magmaM(-= error po yan

Mag&isip po ng number na malapit sa  or e0ample $ -"en press %  (nd you )ill get t"e ans)er

#. in&

A. 

!. 1.5'

$. in&eterminate

%. in6nity

$o&e:

Same met"od lang po sa H$ So# napaisip ?a ngayon ?ung paano infinity8 (no po ba ang infinity8 !to ay isang mala?ing number

So press C(C 08

infinity % $ or ?a"it anong mala?ing number basta )ag lang magM(M(-= error (nd you )ill get t"e ans)er

3. in&

A. -1

!. 1

$. -#

%. #

$o&e:

Same met"od as H$

(28)

. in&

A. 

!. 123

$. 1

%. 3

$o&e:

Same met"od lang din sa H$ 6ero )ag ?alimutan na ilagay sa rad mode ang calcu

'. in&

A. 

!. 123

$. 1

%. 3

$o&e:

Same met"od as H*

". in&

A. !.

$. 

%. in6nity

$o&e:

Same met"od lang din as H*

6alitan ang 0 ng malapit sa $ eit"er $$ or AAAA

5. in& the &eriatie of

A. !.

$.

%.

(29)

$o&e:

-"is time ! assume na marunong na ang la"at at pamilyar na ?ayo sa mga steps at function na ginaga)a natin

Dont forget: (l)ays use rad mode in trigonometric and in.erse trigonometric equations

Bamitin ang dJd0 function S=!->!,-EB;( t"en i&type ang equation (t maglagay ng .alue ng 0 4oure fa.orite number ! preferred 5

Set 0%5

Ma?a?a?u"a tayo ng ans)er Say ($/ans)er H$1

-"en mula sa c"oices# isubstitute lang ang 5 sa 0 at ma?a?au"a tayo ng sagot Say (2 /ans)er H21

Fung ($ % (2 t"en tama ang napili niyo sa c"oices

4. in& the &eriatie of y if

A. !.

$.

%.

$o&e:

Same lang sa H' ! "ope you get it -"ats )"at you called re.erse engineering

. in& the &eriatie of y if

A. !.

$. 1 7 (os 0

%. 1 - (os 0

$o&e:

(30)

1. in& the &eriatie of y if tan y = 0 .

A. !. se( 0 tan 0

$.

%.

$o&e:

(ng tec"nique lang lagi dito guys is t"at you must con.ert t"e equation into y into a function of 0 -"at is#

-"en yun# same met"od na lang as H'

Part #

11. in& the 6rst &eriatie of y if

A. !.

$.

%.

Same lang ito sa mga nakaraang examples ko sa diferential calculus. Kung hindi niyo alam

ito, basa muna kayo sa mga naunang example.

NOTE: Pag parehas ang sagot, HUWAG GAMITIN ANG 1. USE 2 or 3.

1#. in& the se(on& &eriatie of y if

A. !.

$.

%.

So, mano-mano ito since alang shortcut pag second derivative. Sa board exam may

ganyan. So let's test kung marunong pa kayo sa mano-mano.

13. in& the thir& &eriatie of y = 0 ln 0.

A. -120

(31)

$. -120D3;

%. -1

So, mano-mano ito since alang shortcut pag third derivative. Sa board exam may ganyan.

So let's test kung marunong pa kayo sa mano-mano.

1. in& the &eriatie of y if

.

A. -02y

!. 02y

$. yD-3

%. -yD-3

change to !unction o! x then use d"dx !unction. Kayo na bahala kung anong value ng x gusto

niyo. #hen compare.

$ctually same lang ito sa %&.

1'. in& the 6rst &eriatie of y if

A. y 2 07y;

!. -y 2 07y;

$. y 2 07#y;

%. -y 2 07#y;

Same lang sa %(.

1". in& the partial &eriaties )ith respe(t to 0 of the

f/n(tion

A. yD# - '

!. yD#

$. 0y-'y

%. #0y

Mag-iisip ka ng value ng y and then kapag dinirivative mo, e)ual siya dun sa value ng y.

*or example, y + (.

Substitute ( in the e)uation. #he e)uation no becomes x-/ or  x-(.

#hen use 0d"dx10x-(1.

$ng x natin dito ay alays .

So kung anong nakuha niyo anser diyan, kailangan mag e)ual siya dun sa anser. *or

example ang anser ay . 2etter 3 and tamang sagot kasi y + ( and y4 + .

15. The f/n(tion

is &is(ontino/s at

A. -1, 3

!. 1,-3

$. -1,#

%. 1,-#

(32)

5se d"dx !unction then x+x and press 6$26.

Isubstitute ang lahat ng choices at kung saan nagerror, yun na.

14. in& the slope of the tanent line to the raph of the

f/n(tion

at the point )here 0=3.

A. "

!. ""

$. 5#

%. 54

7adian mode. 5se d"dx !unction ith x+8

Slope tangent is e)ual to the 9rst derivative0y'1

1. If y=(os 0 7 sin #0, )hat is the slope of the (/re )hen 0 = #

ra&ians8

A. -#.#1

!. -.

$. -3.#1

%. #.#1

same ith %:.

#. in& $ so that the line y = 0 7 3 is tan*ent to the (/re y =

0D# 7 $.

A. 

!. '

$. "

%. 5

Ree!er:

m+m 0tangent"parallel1

m+ -"m 0normal"perpendicular1

$nd y+(x/8 should be tangent to y+x4/6

*irst, get the 9rst derivative or slope o! y+(x/8.

5se d"dx ith x+, there!ore the slope is (.

e)uate y' + y'

0y+(x/81 + 0y+x4/61

( + x

and x + . #hen get the value o! y !rom y+(x/8

y + . #here!ore, our pt is 0,1

Substitute this to ; + x4 / 6 to get the value o! 6.

aplace transformation

Examples:

1. Find the laplace t ransform of f(t) = cosh 5t

a. (5 / s^2 - 25)