Calculator Techniques
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engineering mathematics...
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SESSION 1 PART 1 1. What are the roots of the polynomial: A. -3, - !. ", # $. #, 3 %. 3, $o&e: Set Calculator Calculator to to equation mode: MODE>5>3 for quadratic equation Mode 3 because our polynomial is a 2nd degree equation !nput t"e equation coefficients a#b#c: $ % &' % $ 2 % % (nd you )ill get t"e ans)er
#. What are the roots of the polynomial: A. 1,#,3 !. 1,#,' $. 1,#, %. #,3,' $o&e: Set Calculator Calculator to to equation mode: MODE>5>* for cubic equation Mode * because our polynomial is a 3rd degree equation !nput t"e equation coefficients a#b#c: $ % &' % $ * % &+ % % (nd you )ill get t"e ans)er
3. Whi(h of the follo)in* is a possi+le root of the polynomial: A. 3 !. $. -# %. $o&e: ,O-E: ( root is any .alue t"at# )"en substituted to t"e .ariable/ie 01# )ill satisfy t"e equation/!n our equation % 1 Set t"e calculator calculator to to computation mode: MODE>$ !nput O,4 t"e left side of t"e equation -rial and error# se t"e C(C function
6ag nagtanong ang calculator calculator 78 78 iinput ang mga c"oices 78 3 % output is 2* repeat t"e step until you get an output of (nd you )ill get t"e ans)er
. 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 $o&e: Set calculator calculator to to equation mode: MODE>5>$ for t)o&.ariable equation !nput t"e coefficients of a#b and constant c of t"e first equation: 2 % 3 % 9 % !nput t"e coefficients of a#b and constant c 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 (nd you )ill get t"e ans)er
'. The e/ation has t)o rational roots, +oth of )hi(h are positie. in& the lar*er of these t)o roots. A. 1 !. # $. 3 %. $o&e: Same met"od )it" number 3e )ill be using C(C function-"e calculator calculator s"ould s"ould display (nd you )ill get t"e ans)er
". What is the remain& remain&er er of the polynomial &ii&e& +y A. #4 !. #14 $. #43
)hen
%. 3' $o&e: Set t"e calculator calculator to to computation mode: MODE>$ !nput t"e left side of t"e equation (pply t"e ;emainder t"eorem se C(C function and substitute 0%5 (nd you )ill get t"e ans)er
5. Sole the al/es of y in the system of e/ations:
$o&e: Set t"e calculator calculator to to equation mode: MODE>5>2 !nput t"e coefficients a#b and constant c of t"e first# second and t"ird equation 6ress t"e equal % button t)ice to get t"e .alue of y (nd you )ill get t"e ans)er
4. Sole for the al/e of A. # !. #i $. -# %. -#i $o&e: Set t"e calculator calculator to to computation mode: MODE>$ !nput t"e equation in t"e calculator calculator/as /as is1 and you )ill get t"e ans)er
. Sole for the al/e of A. 1.1 !. #.1 $. 1.1 %. #.1 $o&e: Set t"e calculator calculator to to computation mode !nput t"e equation in t"e calculator calculator/as /as is1 (nd you )ill get t"e ans)er
1. Sole 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% !t s"ould display# 7% ans)er &;% /important# must be 1 (nd you )ill get t"e ans)er
PART # 11. Eal/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. !. $. %. $o&e: Set calculator to computation mode -ype t"e equation as isse t"e @SOS=!->1/for comma1>$>S=!->1/for comma1>2$>1 9 4ou )ill noticed t"at our summation is from $ to 2$"y 2$ and not 28ecause )e start our progression in $!f )e start t"e progression from # t"en t"ats t"e time )e use to 2 (nd it )ill be li?e t"is)alang 4&"at na symbol e
(nd you )ill get t"e ans)er
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 ModeMODE>$ 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
2 !nput t"e equation under SMM(-!O, function and get t"e sum from to a large .alue# say# 5So it )ill be li?e t"is-"en press %
14. What is the s/m of "771#7. . .71518 A. '" !. 34 $. '14 %. '"# $o&e: $ Set t"e Calculator to S-(- modeMODE3>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 dont ?no) "o) many term does it "a.eSo lets determine "o) many terms it "asEarlier in t"is tutorial ! said 7 is for nt" term and 4 is for progression-"erefore# our code is $'$7&"atSince $'$ is our last term(nd t"e calculator )ill display @59@ * no) use t"e SMM(-!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 *ien e/ation: A. #-'i !. '71i;2" $. 17#i %. #7i;23
$o&e: $ Set t"e calculator to COM6E7 modeMODE>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 % (nd you )ill get t"e ans)er
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 leel 1 a(re parS=!->*>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 3DE%&$ -"e t"ird point )ill gi.e you 5D3E%&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!;CE# E!6SE or 6(;(O(: /"1
/?1
-"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 (nd you )ill get t"e ans)er
4. in& the s/m of an A.P.: #,',4..............)here n=#'.. A. !. #' $. ' %. 5' $o&e: se S-(- 2/for (61 !n t"e 0 column enter $#2 !n t"e y column enter 2#5 t"en press (C se t"e SMM(-!O, ,C-!O,
(nd you )ill get t"e ans)er
. in& the s/m of a >.P.: #,3,.',.................)here n=1. A. ##"."" !. #3'.# $. ##.'# %. #'1.4 $o&e:
Same met"od )it" HA - use S-(- 9/for B61 -"en use t"e SMM(-!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* three3; or more of the seen5; &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:
(nd you )ill get t"e ans)er
Part # 11. in& the &e(imal e/ialent 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/ialent of A. !.
0
$. %. $o&e: -ype niyo lang sa calcuag ?alimutan base + and base $ yan 6ress =E7 for t"e final ans)er
13. Eal/ate A 0 !.
A.
!.
$.
%. $o&e: MODE 9>(C-andaan daanan lagi si D!M S=!->*>$ !type ang matri0 and t"ats 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 A. .33
an&
)hen
!. .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 (/re )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 A. '.# !. -'.# $. .# %. -.#
)hen
$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 Substituting t"e gi.en and using t"e dJd0 function of t"e calcu# it )ill be li?e t"is:
,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 (/re at 1,#; A. 0-y=# !. 0-y=-# $. 07y= %. 07y=- $o&e: "en you are getting t"e slope of t"e line using t"e first deri.ati.e# )"at you are getting is t"e slope of t"e tangent line-o get t"e slope of t"e ,ormal ine# you get t"e negati.e reciprocal of t"is So# get t"e first deri.ate/-angent slope1 of t"e gi.en equation )it" 0%$/gi.en in t"e problem1
(nd getting t"e negati.e reciprocal of t"is )e "a.e &$J* -"en get t"e equation using S-(- 2
(nd press (CBet ( and S=!->$>'> and S=!->$>'>2 ;emember# in t"e calculator# t"e equation of S-(- 2 is y%ab0 Substitute a and b in t"e equation and you )ill get t"e ans)er
14. The (har*e in (o/lom+s that passes thro/*h a )ire after t se(on&s is *ien +y the f/n(tion. %etermine the (/rrent at the en& of # se(on&s.. A. " A !. 4 A
$. 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 08 08
e $ $Je
Part 3
#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!¬e ito i&substitute ang limits sa c"oicesag intindi"in ang constant @C@remember# upper limit minus lo)er limitlets say for e0ample sa c"oice (
6ag pumare"as ang sagot mo ?anina/yung ni¬e mo1 yun ang sagot =appy integratingN1
##. 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#*090#9y+0#+yA0#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 an& the erti(al lines 0= an& 0=.. A. ' s. /nits !. 1"23 s. /nits
, the 0-a0is,
$. 1523 s. /nits %. "s. /nits $o&e: irst get t"e limits from t"e equation it selfse 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 (/re an& the 0-a0is, A. #.#' s. /nits !. #.' s. /nits $. 3 s. /nits %. 3521# s. /nits $o&e: Bet t"e limits from t"e equationse EK, *
(ng you )ill get t"e ans)er
#'. in& the len*ht of ar( of the (/re y = ln (os0 from 0= to 0=pi2.. A. ."' !. .5# $. .41 %. .44
$o&e: Fung alam niyo formula gamit ang long met"od# goDi ?o na siya ibibigayS"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 *ien +y . What &istan(e &oes it trael )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%AAse 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. Sole for the *eneral sol/tion of the &i?erential e/ation: A. !. $. %. $o&e: Bet t"e rootsse EK, *$ + 4ou )ill get one real root and t)o comple0 roots E!:
6ag real root %
6ag comple0 root % sin and cos
#4. Sole& 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
#. Sole 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 yi¬e 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%2if 2%2 -"erefore# napili mo ang tamang ans)er
Part 31. A ra&ioa(tie 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
$. 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. !. $. %. $o&e: 6ag temperature ang pinaguusapan use S-(- 5 Sa 0 column natin is t"e time/min1 and in y column its eit"er @-s&-@ or @-&-s@ /temperature1 !t depends ?ung sino mala?i ?ung si - or si -s - % Obect temperature -s % surroundingJen.ironment temperature gets8 ets proceed Since mala?i ang -s natin )"ic" is Bets8
# t"erefore ang gagamitin natin ay -s&-
7&&&&&&&&&&4/-s&-1 &&&&&&&&&&/'&$+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 li*ht trael in *lass of refra(tie in&e0 1.'8 A. !. $.
%. $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
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