MATHEMATICS and CALCULATOR TECHNIQUES
ENGR. REYNILAN L. DIMAL
THE MEMORY VARIABLES MEMORY A B C D E (ES PLUS only) F (ES PLUS only) X Y M
CALCULATOR BUTTONS ALPHA (-) ALPHA O ‘ “ ALPHA hyp ALPHA sin ALPHA cos ALPHA tan ALPHA ) ALPHA S D ALPHA M+
HOW TO CLEAR MEMORY • SHIFT 9 1 = – This means you will automatically go to MODE 1
• SHIFT 9 2 = – All values stored in the memory variables will be erased
• SHIFT 9 3 = – This means you will automatically go to MODE 1 and all values stored in the memory variables will be erased.
MODE 1 : GENERAL CALCULATIONS
HOW TO CONVERT BETWEEN DEGREES, RADIANS AND GRADIANS Convert237.6150 to DMS (DegreeMin Sec)
DISPLAY: 237.615O 237036'54"
BASICS
HOW TO CONVERT BETWEEN DEGREES, RADIANS AND GRADIANS Convert210 47'12"todecimaldegrees.
DISPLAY: 210 470120 21.7866666 7
BASICS
HOW TO CONVERT BETWEEN DEGREES, RADIANS AND GRADIANS Convert1200 to radians.
DISPLAY: 1200 2 3
BASICS
HOW TO CONVERT BETWEEN DEGREES, RADIANS AND GRADIANS π Convert radians to degrees. 2
DISPLAY:
r
2 90 BASICS
PAST CE BOARD EXAM Whatis 1200 in centesimalsystem? ENTER
DISPLAY: 0
120 400 3 BASICS
HOW TO GET THE POLAR AND RECTANGULAR COORDINATE OF A POINT IN THE CARTESIAN PLANE
PAST CE BOARD EXAM Find thepolarcoordinateof thepoint(4, - 6).
DISPLAY: Pol(4,6) r 7.211102551 , 56.30993247 BASICS
HOW TO GET THE POLAR AND RECTANGULAR COORDINATE OF A POINT IN THE CARTESIAN PLANE
PAST ECE BOARD EXAM Find the value of cos if the terminalside containsthepointP(-3,-4) Solution :
BASICS
HOW TO GET THE POLAR AND RECTANGULAR COORDINATE OF A POINT IN THE CARTESIAN PLANE
PAST ECE BOARD EXAM DISPLAY: Pol(3,4) r 5, 126.8698976 NOT E: r is storedautomatically toX and to Y.
DISPLAY : cos(Y ) 3 5
BASICS
PAST ECE BOARD EXAM Find therectangular coordinateof a point whose polarcoordinateis (3,1200 ).
DISPLAY: Rec(3,120) X 1.5, Y 2.59807621
BASICS
HOW TO SOLVE COMBINATION AND PERMUTATION PROBLEMS. PAST ECE BOARD EXAM How many triangles are formedby 10 distinct pointsno threeof which are collinear? Solution :
T henumber of trianglesthatcan be formedfrom 10 non collinearpointsis 10C3. DISP LAY: 10C 3 120
BASICS
PAST ECE BOARD EXAM In how manydifferentways can thejudges choose the winner and thefirst runner up fromamongthe 10 finalistsin a student essay contest? Solution : T hereare10 finaliststaken2 at a time. Note: order is importanthere DISP LAY: 10P 2 90 BASICS
HOW TO EVALUATE FACTORIAL NUMBERS
18. Find the value of 10!
DISP LAY: 10! 3628800
BASICS
PAST EE BOARD EXAM How manydifferentpermutations can be made from thelettersMISSISSIP PI?
Solution : Number of M 1; I's 4; P 's 2; S' s 4; Number of Lett ers 11. Note: T henumber of differentpermutations is : 11! 1!4!2!2! BASICS
PAST EE BOARD EXAM
DISP LAY: 11! 1! x 4! x2!x4! 34650
BASICS
HOW TO EVALUATE FUNCTIONS 11. Evaluatef( 6 ) if f(x) 3x 4 3x 2-5x 6
BASICS
HOW TO EVALUATE FUNCTIONS 12. Evaluatef( 4,3 ) if f(x, y) 4x3 y 2 3x 2 y-2xy2 y 3
BASICS
PAST ME BOARD EXAM Find theremainderwhen 3x4 2x3 - 4x2 x 4 is divided by x 2. Solution: f(x) 3x 4 2 x3-4 x 2 x 4 , remainder f(-2 )
BASICS
HOW TO EVALUATE FUNCTIONS 13. Is (x 3 ) a factorof x6 6x5 8x 4-6x3-9x 2?
Conclusion:Since f(-3) 0, then x 3 is a factorof x 6 6 x 5 8 x 4 -6 x 3-9 x 2 BASICS
PAST ECE BOARD EXAM Find theremainderwhen 4 y3 18y 2 8 y 4 is divided by 2 y 3. Concept: Set thedivisor tozero and solvefor y. ENTER : DISP LAY: 2Y 3 0, Y - 1.5
ENTER : DISP LAY: 4Y 3 18Y 2 8Y - 4 11 Answer : T heremainderis 11 ADVANCE
HOW TO USE THE ∑ SIGN Find the sum. 1 2 3 ... 20
DISP LAY: 20
x
x1
210 BASICS
HOW TO SOLVE LINEAR EQUATIONS SOLVE 4(3 x) 5(4 x)
BASICS
HOW TO SOLVE LINEAR EQUATIONS x 3 x 1 x2 SOLVE 1 12 6 9
BASICS
HOW TO SOLVE A SPECIFIC VARIABLE D ( 2 X 2Y ), X 4, D 2, and A 9, 7 what is the valueof Y ? If A
BASICS
PAST ECE BOARD EXAM When 3 is m ultipliedby 5 less thana num ber, the result is 9 less than5 tim esthe num ber. Find 7 less than5 tim esthe num ber.
BASICS
PAST ECE BOARD EXAM
BASICS
HOW TO USE MULTILINE FUNCTION
PAST EE BOARD EXAM Find thearea of a trianglewhose sides are 6m,8m,12m.
Solution : Using Heron's Formula: A s(s - a)(s- b)(s - c) abc s 2
ENTER :
BASICS
HOW TO USE MULTILINE FUNCTION
PAST EE BOARD EXAM DISP LAY: ABC X : X(X - A)(X- B)(X - C) 2
ENTER : DISP LAY: X
ABC 2 13
DISP LAY: X ( X A)( X B)( X C ) 455 BASICS
HOW TO SOLVE TRIGONOMETRIC EQUATIONS Solve 5tan x - 3 2tan x : 0 x 360 ENTER :
DISPLAY:
ENTER :
5 tan x- 3 2 tan x X 30 L-R
0
Answer : X 300 and 2100
DISPLAY: 5 tan x- 3 2 tan x X 210 L-R
0
BASICS
HOW TO USE LOGARITHMIC EQUATIONS
PAST ME BOARD EXAM Solve for xin log2 x log2 (x 5) 10
ENTER :
DISPLAY: log2 x log2 (x 5) 10 X L-R
29.5975076 9 0 BASICS
HOW TO USE LOGARITHMIC EQUATIONS
PAST ECE BOARD EXAM 3log x
Solve for xin x
100x
DISPLAY: x 3log x 100x X L-R
10 0 BASICS
HOW TO GET THE DERIVATIVE AT A POINT Find thederivativeof x3 3x 2 when x 3.
ENTER : DISP LAY: d ( X 3 3X 2 ) dx x 3 45 BASICS
PAST ECE BOARD EXAM x2 Differentiate theequation y x 1 x2 2x x a. b. c.2 x 2 ( x 1) ( x 1)
2x2 d. ( x 1)
T echnique: Differentiate y at any valueof x, say x 2 and comparethis value to the value of thechoiceswhen same value of x is substituted.
ENTER : DISP LAY: d x2 dx x 1 x 2 0.8888888889 ADVANCE
PAST ECE BOARD EXAM Note: Compareit tothechoicesas thevalueof x is being substituted.
x2 2x a) Substitute x 2 2 ( x 1)
ENTER : DISP LAY:
T he valuesof therest of thechoiceswhen x 2 are summarizedas follows:
x2 2x ( x 1) 2 x 2 b. 0.8888888889 ( x 1) x 2 3
c.2 xx2 4
x2 4 d . ( x 1) x 2 3
x2 2x Answer : a. ( x 1) 2
ADVANCE
HOW TO INTEGRATE 2
Evaluate ( x 3x 1)dx 5
1
ENTER : DISPLAY: 2
x
5
3x 1 dx
1
16 BASICS
MODE 2 : COMPLEX NUMBER CALCULATIONS
HOW TO SOLVE COMPLEX NUMBERS For thecomplexnumber z 3 - 4i a. Find theabsolute value. b. Find theargument.
DISPLAY: 3 4i r 5 53.13010235 0
Answer : T heabsolute valueis 5 and theargumentis 53.13 BASICS
HOW TO SOLVE COMPLEX NUMBERS Given : (2 - 3i)(5 2i), find theproduct.
ENTER :
DISP LAY: (2 3i )(5 2i ) 16 11i
BASICS
HOW TO SOLVE COMPLEX NUMBERS 4 2i Simplify : 5 - 2i ENTER :
DISPLAY: 4 3i 5 2i 14 23 i 29 29 BASICS
HOW TO GET THE COMPONENT OF A FORCE AND RESULTANT OF FORCES Find thex and y componentsof theforceF 300N370 ENTER :
DISPLAY: 300370 239.590635 180.5445069i Answer : T hex componentis 239.5N and they componentis 180.54N.
BASICS
HOW TO GET THE COMPONENT OF A FORCE AND RESULTANT OF FORCES Find the x and y componentsof theforce F 800lbs with angle400 at III Quadrant. Solution: Expresstheforceas F 800(180 40) 8002200
ENTER : DISPLAY: 8002200 612.8355545 5142300877 i Answer : T he x componentis - 612.84lbsand the y componentis - 514.23lbs.
BASICS
HOW TO GET THE COMPONENT OF A FORCE AND RESULTANT OF FORCES Find theresultuntof theforces,F1 350Nat 600 and F2 400Nwith an of angle1400. Solution : T heresultantis thesum of thecomplex numbers35060 400140
ENTER :
DISPLAY:
Answer : T hemagnitudeof the
resultantis 575.43Nwith 103.200 350600 4001400 575.4315683103.2017875 with the x - axis(counterclockwise) BASICS
PAST EE/ECE BOARD EXAM Simplify the expression i1997 i1999 wherei is an imaginarynumber. T echnique: Divide theexponentsto 4 and get theremainder.
ENTER : DISP LAY:
ENTER :
1997 4 DISP LAY: 1 499 4
1999 4 499
Not e:1/4 corresponds t o i (i1 i) 2/4 corresponds t o - 1 (i2 i) 3/4 corresponds t o - i (i3 - i) whole number (0 remainder)
3 4
ENTER : DISPLAY: i i 0 (Answer)
corresponds t o1 (i4 1) ADVANCE
PAST CE/ECE BOARD EXAM Find thevalueof (1 i)5 wherei is an imaginarynumber? T echnique: Rewriteas (1 i)3 (1 i)2
ENTER : ENTER : DISPLAY: (1 i)3 (1 i) 2 4 4i
ADVANCE
MODE 3 : STATISTICAL AND REGRESSION CALCULATIONS
HOW TO FIND THE MEAN AND STANDARD DEVIATION Five light bulbs burned out afterlasting867, 859,840,852,and 888hrs. Find themean. DISP LAY:
DISP LAY: x 861.2
1 2 3 4 5
x 867 859 840 852 888 BASICS
PAST ME BOARD EXAM Given thefollowingstatistical data,determine thestandarddeviation. Data:112 132 144 156 164 176 183 197 ENTER DISP LAY: x 1
112
2 3 4
132 144 156
5 6 7
164 176 183
8
197
DISPLAY:
x 26.21545346 BASICS
HOW TO GET THE MEAN, VARIANCE AND STANDARD DEVIATION OF GROUPED DATA DISP LAY: x 1 62 2 3 4 5 6 7
FREQ 2
65 68 71
5 12 15
74 77 80
8 5 3
BASICS
HOW TO GET THE MEAN, VARIANCE AND STANDARD DEVIATION OF GROUPED DATA DISP LAY: n 50 DISPLAY: x 70.94 DISPLAY: sx 4.391132065
DISPLAY: Ans2 19.28204082 BASICS
HOW TO FIND AREAS IN THE NORMAL CURVE
P(a)meansarea fromz - to z a R(a) meansarea fromz a to z Q(a) meansarea fromz 0 to z a
BASICS
HOW TO FIND AREAS IN THE NORMAL CURVE Find thearea under thenormalcurve to theleft of z 1.64.
ENTER :
DISP LAY: P (1.64) 0.9495 BASICS
HOW TO FIND AREAS IN THE NORMAL CURVE Find thearea beneatha standardnormalcurve between z 0 and thepoint- 1.58.
ENTER :
DISPLAY: Q(-1.58) 0.44295 BASICS
HOW TO FIND AREAS IN THE NORMAL CURVE Find theprobability thata normaldistribution random variable will be within z 1 standarddeviationof themean.
Solution : We are lookingfor thearea fromz -1 to z 1.
ENTER :
DISPLAY: Q(-1) Q(1) 0.68268 BASICS
HOW TO FIND AREAS IN THE NORMAL CURVE Find theprobability thata normallydistributed random variable will lie more than1.5 standarddeviationabove themean.
Solution : Weare lookingfor thearea fromz 1.5 to z .
ENTER :
DISP LAY: R(1.5) 0.066807 BASICS
HOW TO SOLVE LINEAR REGRESSION PROBLEMS
BASICS
HOW TO SOLVE LINEAR REGRESSION PROBLEMS ENTER :
DISP LAY:
DISP LAY: A 3.1359045 DISP LAY: B 0.40449955 409
1 2 3 4
x 20 18 16 14
y 12 10 11 6
5 6 7 8
10 8 6 4
7 8 4 6
9 10
2 0
5 2
BASICS
HOW TO SOLVE LINEAR REGRESSION PROBLEMS T herefore: T heregressionequationsis Y A BX Y 3.1359045 0.40449954 09X b. To determine the correlatio n coefficien t : DISPLAY: r 0.8854825905
c. T o predict the valueof Y when X 23: DISPLAY: 23y 12.43939394
BASICS
HOW TO GET THE EQUATION OF A LINE GIVEN 2 POINTS
PAST ECE BOARD EXAM Find theequationof theline thatpasses through (2,5)and (-3,8). DISPLAY: ENTER : 1 2
DISPLAY: A 31 5
x
y
2 3
5 8
DISP LAY: B 3
5
BASICS
HOW TO GET THE EQUATION OF A LINE GIVEN 2 POINTS
PAST ECE BOARD EXAM T herefore: theequat ionof theline is : Y A BX 31 3 Y X 5 5 or : 5Y 31 3 X 3 X 5Y 31
BASICS
HOW TO GET A POINT ON THE LINE GIVEN TWO POINTS If a line passes through(4,1)and (3,-7)and (x, y) is on theline,what is the value of x in (x,4)and the value of y in (-5,y)?
DISP LAY: 4 xˆ
Answer : When x 4, y
35 8
35 8 BASICS
HOW TO GET A POINT ON THE LINE GIVEN TWO POINTS
DISP LAY: Answer : Wheny - 5, x - 71 5 yˆ 71
BASICS
PAST ME BOARD EXAM T heequationof theline thatinterceptsthe x - axis at x 4 and the y - axis at y - 6 is :
ENTER : DISP LAY:
ENTER : x y 1 4 0 2 0 6
DISP LAY: A -6
ENTER : DISP LAY:
Answer : B 1.5 or 3/2
Y A BX Y - 6 3/2X which can be rewrit t enas : 3X - 2Y - 12 0 ADVANCE
PAST CE/ECE BOARD EXAM Find the30th termof thearithmeticprogression 4, 7,10...
ENTER : DISPLAY: x
y
1 1 4 2 2 7
ENTER : DISPLAY: ˆ 30 Y 91
ADVANCE
PAST CE/ECE BOARD EXAM Whatis thesum of theprogression 4, 9,14...up to the20th term?
ENTER : DISPLAY:
ENTER : x
y
DISPLAY:
1 1 4 2 2 9
A DISPLAY: 1 Ans A 1
ENTER : DISPLAY:
ENTER :
ENTER : B DISP LAY: 5 Ans B 5 ADVANCE
PAST CE/ECE BOARD EXAM Whatis thesum of theprogression 4, 9,14...up to the20th term?
ENTER :
DISP LAY: 20
A BX x 1
1030
ADVANCE
PAST CE BOARD EXAM
T he4th termof theGP is 216and the6th termis 1944.Find the8th term.
ENTER : DISPLAY: x 1 4 2 6
y 216 1944
ENTER : DISPLAY: ˆ 8Y 17496 ADVANCE
MODE 4 : SPECIFIC NUMBER SYSTEMS CALCULATIONS
HOW TO DO BASE NUMBER CALCULATIONS Convert23410 to binary(base 2). ENTER : DISP LAY: 234 Dec 234
ENTER : Thus: 23410 111010102
DISPLAY: 234 Bin 0000000011 101010
BASICS
HOW TO DO BASE NUMBER CALCULATIONS Convert123410 to HEXADECIMAL system. ENTER : ENTER : DISPLAY: 1234 Hex 000004D2
Thus:123410 000004D216 BASICS
HOW TO DO BASE NUMBER CALCULATIONS ConvertABC1216 toOCTALsystem. ENTER : ENTER : DISP LAY: ABC12 Oct 0000253602 2
Thus: ABC1216 00002536022 8 BASICS
HOW TO DO BASE NUMBER CALCULATIONS Evaluate(AB16 )(3F16 ). ENTER : ENTER : DISPLAY: AB x 3F Hex 00002A15
BASICS
HOW TO DO BASE NUMBER CALCULATIONS Evaluate112 4510 AB216 778. (in base 10)
Solution : Convert all values to base 10. For 112
Result : 3
For AB216
Result : 2738
For 778
Result : 63
Add : 3 45 2738 63 Answer : 2849 BASICS
HOW TO DO BASE NUMBER CALCULATIONS Find thelogical AND ( 1012 and1102 ) ENTER :
DISP LAY: 101and110 Bin 0000000000 000100
BASICS
HOW TO DO BASE NUMBER CALCULATIONS Find thelogical XOR ( 1012 and1102 ) ENTER :
DISP LAY: 101xor11 Bin 0000000000 000110
BASICS
MODE 5 : EQUATION SOLUTION
HOW TO SOLVE EQUATIONS IN 2 UNKNOWNS SOLVE 2 x 7 y 4 x 2y 1
BASICS
PAST ME BOARD EXAM In 5 years, Ana' s agewill betwice as the ageof her friend Jun. Five years ago, she was three tim esas old as his friend. Find their presentages.
BASICS
PAST ME BOARD EXAM
BASICS
PAST EE BOARD EXAM A man has 2 investments one paying3% annualinterest and theotherat 4% interest.T he totalincomefrom theinvestments is P1700.If theinterestrates were interchanged, the totalincomewould be P1800. Find theamountof each investment.
BASICS
PAST EE BOARD EXAM
BASICS
PAST ECE BOARD EXAM 2000kg of st eelcont aining8% nickelis t o be made by mixingst eelcont aining14% nickel wit h anot hercont aining6% nickel.How much of each is needed? Solution: Let : X amountof steelcontaining14% nickel Y amountof steelcontaining6% nickel
BASICS
PAST ECE BOARD EXAM
BASICS
HOW TO SOLVE QUADRATIC EQUATIONS Solve thequadraticequation6x 7 x 5 0 2
BASICS
HOW TO SOLVE QUADRATIC EQUATIONS Find the values of xin x2 2x 5 0
BASICS
HOW TO SOLVE QUADRATIC EQUATIONS Solve thequadraticequation5x 2 2x 9 0
NOT E: T hisis theadvantageof CASIO ES PLUS over theOLD ES - It can give irrationalroots
BASICS
HOW TO SOLVE EQUATIONS IN 3 UNKNOWNS Find the values of x, y and zif: 3x - 3 y - z 4 x 9 y 2 z 16 x - y 6 z 14
BASICS
HOW TO SOLVE EQUATIONS IN 3 UNKNOWNS
BASICS
HOW TO SOLVE CUBIC EQUATIONS Solve thecubic equation x3 2x 2 -5x - 6 0
BASICS
HOW TO SOLVE CUBIC EQUATIONS Solve x - 1 0 3
BASICS
MODE 6 : MATRIX CALCULATIONS
HOW TO SOLVE PROBLEMS INVOLVING MATRICES
PAST ECE BOARD EXAM 3 5 9 1 Simplify 37 1 2 7 1 4 9 8 9
3 5 Solution : Store 7 1 to MAT A 4 9
ENTER : 9 1 Solution : Store 7 1 to MAT B 8 9
ENTER :
BASICS
HOW TO SOLVE PROBLEMS INVOLVING MATRICES
PAST ECE BOARD EXAM ENTER : DISP LAY: 3 MAT A 2 MAT B
ENTER :
DISP LAY: Ans 27 17 35 5 28 45 BASICS
HOW TO SOLVE PROBLEMS INVOLVING MATRICES 3 2 - 1 Find the transposeof mat rixA if A 3 7 8 - 1 3 2 3 2 - 1 St ore 3 7 8 t o mat rixA. - 1 3 2
ENTER :
ENTER : DISP LAY: T rn (Mat A) Ent er: Ans 3 3 - 1 2 7 3 - 1 8 2
BASICS
HOW TO SOLVE PROBLEMS INVOLVING MATRICES 2 1 3 Find t heinverseof mat rixA if A 6 1 4 3 7 2 2 1 St ore 6 1 3 7
3 4 t o mat rixA. 2
ENTER :
ENTER : DISP LAY: Mat A-1 Ans - 0.4 0.2923 0.0153 0 0.076 0.1538 0.6 - 0.169 - 0.061
BASICS
HOW TO COMPUTE THE DETERMINANT OF A 3X3 MATRIX Find thedeterminant :
2 4 -5 2 1 7 8 1 2
ENTER :
DISP LAY: det(Mat A) 228
BASICS
PAST CE BOARD EXAM In a Cartesiancoordinates, the verticesof a triangleare defined by thefollowingpoints(-2,0),(4,0)and (3,3).Whatis thearea? Concept: T hearea of any t riang le wit h vertices (x1 , y1 ), (x 2 , y 2 ) and (x3 , y 3 ) is : x1
A
1 x2 2 x3
y1
2 0 1
1
y2 1 y3 1
A
1 det 4 2 3
0 1 3 1
ENTER : DISP LAY:
ENTER :
-2 0 1 4 0 1 3 3 1 DISP LAY: 0.5det (MatA) 9
ADVANCE
MODE 7 : GENERATING TABLE FROM A FUNCTION
HOW TO TABULATE VALUES OF A FUNCTION T abulatevaluesof f(x) x 3 2x2 3 from x 0 to x 10 everyunit step. ENTER :
DISP LAY: 1
X 0
2
1
0
3 4
2 3
13 42
5
4
93
6 7
5 6
172 285
8
7
438
9 10
8 9
637 888
11
10
1197
F(X) 3
BASICS
MODE 8 : VECTOR CALCULATIONS
HOW TO DO VECTOR CALCULATIONS Given the2 vectors: A 4i - j 7k and B 3i 5j 9k. a. Find themagnitudeof theresultantof vectorsA and B. b. Find thedot product of vectorA and B. c. Find thecross product of vectorsA and B.
ENTER : DISP LAY: A [ 4
-1
7 ]
ENTER : DISP LAY: B [ 3
5
9 ]
BASICS
HOW TO DO VECTOR CALCULATIONS ENTER : DISPLAY: Abs(VctA VctB) 17.9164728 7
b. ENTER : DISP LAY: Vct A Vct B 70
c. ENTER : DISPLAY: VctA VctB Ans [ - 44
- 15
23]
BASICS
HOW TO DO VECTOR CALCULATIONS
PAST ME/CE BOARD EXAM Whatis themagnitudeof the vectorA 4i 2j 7k and give its directioncosine vector. ENTER : ENTER :
T o get themagnitude: ENTER :
BASICS
HOW TO DO VECTOR CALCULATIONS
PAST ME/CE BOARD EXAM DISP LAY: Abs (Vct A) 8.30662386 3 Not e: (T hisis storedin Ans) T o get thedirectioncosine:
DISPLAY: Ans [0.4815 0.2407 0.8427] BASICS
THANK YOU VERY MUCH AND GOD BLESS!!!
