CALCULATOR TECHNIQUES

Transcription

1 CALCULATOR TECHNIQUES Session 1 - Part 1 1. What are the roots of the polynomial: -3, -4 6, 2 2, 3 3,4 Getting familiar with the functions Set Calculator to equation mode: MODE>5>3 for quadratic equation Mode 3 because our polynomial is a 2nd degree equation.. Input the equation coefficients a,b,c: 1 = -7 = 1 2 = = 2. What are the roots of the polynomial: 1,2,3 1,2,5 1,2,4 2,3,5 Set Calculator to equation mode: MODE>5>4 for cubic equation Mode 4 because our polynomial is a 3rd degree equation.. Input the equation coefficients a,b,c: 1 = -7 = 1 4 = -8 = = 3. Which of the following is a possible root of the polynomial: NOTE: A root is any value that, when substituted to the variable(ie x), will satisfy the equation.(in our equation 0=0 ) Set the calculator to computation mode: MODE>1 Input ONLY the left side of the equation. Trial and error, Use the CALCU function.. Pag nagtanong ang calculator X? iinput ang mga choices.. X? 3 = output is repeat the step until you get an output of 0..

2 4. Find the value of x and y in the following equations: -12/19, 16/19-24/19, -22/19 12/19, -16/19 24/19, 22/19 Set calculator to equation mode: MODE>5>1 for two-variable equation. Input the coefficients of a,b and constant c of the first equation: 2 = 3 = 6 = Input the coefficients of a,b and constant c of the second equation: -3 = 5 = 2 = Press the = to get the value of x and press again to get the value of y.. 5. The equation has two rational roots, both of which are positive. Find the larger of these two roots Same method with number 3..We will be using CALCU function..the calculator should display What is the remainder of the polynomial when divided by Set the calculator to computation mode: MODE>1 Input the left side of the equation Apply the Remainder theorem Use CALCU function and substitute x=5 7. Solve the values of y in the system of equations: Set the calculator to equation mode: MODE>5>2 Input the coefficients a,b and constant c of the first, second and third equation.. Press the equal = button twice to get the value of y..

3 8. Solve for the value of 2 2i -2-2i Set the calculator to computation mode: MODE>1 Input the equation in the calculator(as is).. and you will get the answer.. 9. Solve for the value of Set the calculator to computation mode Input the equation in the calculator(as is) 10. Solve for the value of x in the equation: Set the calculator to computation mode Input the equation and use the SOLVED function to get the value of x.. To get the SOLVED function, SHIFT>CALCU>= It should display, X= answer L-R= 0 (important, must be 0)

4 Additional Exercises - PART Evaluate Set the calculator to computation mode: MODE>1 Input the equation as is but replace k with X.. Then press = Find x, if Set calculator to computation mode.. Type the equation as is..use the "SOLVE" function and press = Factor First, type the equation and assign an arbitrary constant to substitute to x and use the CALC function.. for example X=8 The calculator will display "2197".. Second, substitute 8 to the choices and find which one will be equal to "2197".. Start with the first choice until you get the value of "2197".. What we do as well in the previous examples like this is called "REVERSE ENGINEERING" Find the 15th term in the arithmetic sequence: 1,3,5,

5 1. Set the calculator to STAT mode..mode 3>2 2: A+BX is for Arithmetic Progression( P.).. May makikita kayo X and Y column..and X column natin ay ang nth term while sa Y column natin ay ang P.. 2. Input sa X column ang 1,2,3(1st term,2nd term, 3rd term)..sa Y column naman, iinput ang 1,3,5 (kahit 3 number lang, kahit wag na yung 7).. Which mean the 1st term of the P. is 1, 2nd term is 3 and 3rd term is After niyo mainput, press AC(don't worry hindi mabubura ang data niyo) Sa problem ang hinahanap ay ang pang 15th term..anu ba ang value ng ika-15th term?..meaning ang hinahanap ang value sa Y column..gets?.. 4. Therefore, type 15Y-hat.. Ang Y-hat ay makikita when you press SHIFT>1(STAT)>7(Reg)>5(Y-hat)..while ang X-hat ay makikita when you press SHIFT>1(STAT)>7(Reg)>4(X-hat)..take note of this dahil gagamitin natin ito in the next problems.. 5. Then press =.. Kung may tanong kayo dito at di niyo makuha pakipost na lang Find the 12th term in the geometric series: 3, 9, , , ,141 1,594, Set the calculator to STAT mode..mode 3>5 or MODE3>6 (it's the same for Geometric Progression, G.P.) May makikita kayo X and Y column..and X column natin ay ang nth term while sa Y column natin ay ang G.P.. 2. Input sa X column ang 1,2,3(1st term,2nd term, 3rd term)..sa Y column naman, iinput ang 3,9,27.. Which mean the 1st term of the G.P. is 3, 2nd term is 9 and 3rd term is After niyo mainput, press AC(don't worry hindi mabubura ang data niyo) Sa problem ang hinahanap ay ang pang 12th term..anu ba ang value ng ika-12th term?..meaning ang hinahanap ang value sa Y column..gets?.. 4. Therefore, type 12Y-hat.. 5. Then press =.. Kung may tanong kayo dito at di niyo makuha pakipost na lang What is the sum of the first 20 terms of the sequence: 1, 3, 5, Set the calculator to STAT mode: MODE 3>2 (since it is P.) 2. In the X column, input 1,2,3..In the Y column, input 1,3, After you input the data, press A.Don't worry your data will not be deleted (unless you change your mode) 4. Press SHIFT>LOG(found below the ON KEY)..we will use the SUMMATION function..(summation symbol is like E) 5. Press ALPHA>)(for X)>SHIFT>1>7>5(for Y-hat)>SHIFT>)(for comma)>1>shift>)(for comma)>21>) 6. You will noticed that our summation is from 1 to 21..Why 21 and not 20?..Because we start our progression in 1..If we start the progression from 0, then that's the time we use 0 to 20.. And it will be like this..walang Y-hat na symbol e..

6 17. What is the sum of the infinite series: 1, -1/5, 1/25,... 5/6 4/5 5/7 6/7 1. Set the calculator to Computation Mode..MODE>1 Since this is an infinite series, with a common ratio of less than 1, then the sum is a finite number..(note: For an infinite series with a common ratio of greater than 1, the sum is infinite).. 2. Input the equation under SUMMATION function and get the sum from 0 to a large value, say, 50.So it will be like this..then press = 18. What is the sum of ? Set the Calculator to STAT mode..mode3>2 (I hope you already know why MODE 3 2) 2. In the X column, input 1,2,3...In the Y column, input 6,9,12 then press A. 3. We should get the sum but we don't know how many term does it have..so let's determine how many terms it has..earlier in this tutorial I said X is for nth term and Y is for progression..therefore, our code is 171X-hat..Since 171 is our last term..and the calculator will display "56" 4. now use the SUMMATION function like in number 26 but this time, its 1, What is the 6th term of the infinite series: 1, -1/5, 1/25,... 1/3125 1/ /3125-1/ Set the calculator to Computation mode.. 2. Input the general equation of the infinite series and substitute 5 to x using the CALC function to display the 6th term..and press = 20. Rationalize the given equation: 2-5i (5+10i)/6 1+2i

7 (2+i)/3 1. Set the calculator to COMPLEX mode..mode>2 2. Input the equation as it is in the calculator..(note: The imaginary i is found by pressing the ENG key.. 3. Then press = SESSION 2 PART 1 1. Subtract Siguro naman kayang kaya niyo na ito..:d 2. During a rain, 2mm of water fell. Find how many gallons of water fell on a level 10 acre park. 21,381 25,381 27,381 30,381 Conversion only.. V=10 acre (convert acre to m^2)shift>8(for CONV)>11(for acre>m^2)..it will give an answer Then multiply it to.002 m(given in the problem) and it will display in m^3.. Convert this to Liters..1 m^3 = 1000L..And it will display in L.. Finally convert this to gal..press CONV#14 L>gal(US).. And you will get the answer in gal.. 3. The function is discontinous at x=? 2-2 either a or b neither a nor b Pag sinabing discontinous, mageerror siya sa value na isusubstitute mo or yung tinatawag na MATH ERROR..Pag may lumalabas parin na value positive or negative or even zero, continous parin yun..dapat mageerror siya..gets?.. Set the calculator to COMP MODE.. type na equation as it is..press the CALC function and substitute the numbers in the choices and find which one will get an error..the one error is the answer..

8 Alternative solution..we use the TABLE function..mode 7.. Type the equation as it is and press = START? (your number depends on the choices..we start from the lowest which is -2..So enter -2 then press = END? (the largest in the choices)..so enter 2 and press = STEP? (The variation in the choices)..important po ito..kailangan madaanan nung function yung mga choices..in this case 2 and -2..So enter 2 and press = Kung saan may error yun na yun.. 4. Find the possible factors of. (x-2) (2x-1) (2x+3) all of the above Para masabing factor siya, kailangan ang y=0 pag isusubstitute natin yung root of x.. We use CALC function..set the calculator to MODE 1.. Type the equation as it is and press CAL.Get the root of the choices and substitute.. X? enter 2 if y=0, then possible root siya..try the next one.. X? enter 1/2, if y=0, then possible root siya..try the last one.. X? enter -3/2 if y=0, then possible root siya.. Therefore you know the answer.. 5. Determine the area of the triangle bounded by the straight lines x+2y-7=0, 3x-4y-1=0, and 2xy+6= sq.units 15 sq.units 20 sq.units 25 sq.units First get the point of intersection..use the EQN function..mode 5>1 For the first two equation, type: then press = =..Therefore the point of intersection of this is x=3, and y=2.. Then try the next equation Then press = =...x=-5, and y=-4 try the last combination of equation Then press = =...x=-1, and y=4 Then we use the determinant function to get the area of the triangle.. Set the calculator to MATRIX..MODE 6 and press A. What important in this is kailangan natin madaan yung DIM or dimension ng matrix..so press SHIFT>4>1>1>1 and enter then press A. To get the determinant press SHIFT>7(det)>SHIFT>4>3(since we enter our data in Mat.Then press =.. Remember the area of a triangle given the vertices is 0.5 times the determinant..therefore multiply it to 0.5 and get its Abs value.. Then you will get the final answer..

9 6. Find the center of the circle that is circumscribed about the triangle whose vertices are (-3,1), (3,1) and (5,3) (-3,9) (3,-9) (-3,-9) (3,9) Substitute x and y in the general equation of circle.. In the first point you will get the equation -3D-E+F=-10 The second point will give you 3D+E+F=-10 The third point will give you 5D+3E+F=-34.. Therefore we have 3 equations, 3 unknown.. Use the EQN 2 and enter And you will get D=6, E=-18, and F=-10 And write it in the general form.. Ito ang shortcut para makuha ang center.. CENTER OF CIRCLE, ELLIPSE or PARABOLA: (h) (k) Therefore our center (h,k) is at (-3,9).. 7. Find the value of y of the parabola whose axis is vertical and passes through (-1,0), (5,0), (1,8) and (4,y) We use STAT 3( for parabola) in this case.. In the x column enter -1,5 and 1.. In the y column enter 0,0, and 8 then press AC Find the value of y when x = 4.. therefore 4(y-hat) or 8. Find the sum of an P.: 2,5,8...where n=

10 Use STAT 2(for P.) In the x column enter 1,2 In the y column enter 2,5 then press A. Use the SUMMATION FUNCTION.. 9. Find the sum of a G.P.: 2,3,4.5,...where n= Same method with #9 BUT use STAT 6(for G.P.) Then use the SUMMATION function to get the sum In how many ways can a picture be painted by using three(3) or more of the seven(7) different colors? Set the calculator to COMP MODE.. We use the COMBINATION function..it will be found by pressing SHIFT > ( ncr ) And type the equation just like this: SESSION 2 PART Find the decimal equivalent of Convert niyo lang sa decimal.. MODE 4..Set to BIN then type the number.. Press DEC and you got the answer..

11 Additional TIP: Kung hindi na kasya sa calculator at sobrang haba ng BIN mo, iconvert mo muna sa OCT.. 101=5 110=6 001=1 111=7 101=5 Set to OCT and type the OCT equivalent, And press DEC and you got the answer Find the hexadecimal equivalent of x Type niyo lang sa calcu..wag kalimutan base 8 and base 10 yan.. Press HEX for the final answer Evaluate A x MODE 6>A...Tandaan daanan lagi si DIM.. SHIFT>4>1 Itype ang matrix and that's it.. You got the answer..

12 Additional TIP: If A/B, DON'T use divide, instead use the reverse function..and it will be like this 14. Find dy/dx if and when Pag Trigonometric/Inverse, i-set ang calcu sa RADIAN MODE.. I-type sa calcu like this.. Yung po natin diyan, pi po yun If y = 4tanh x + sinh 2x, what is the slope of the curve when x = 2? TANDAAN: Slope = first derivative = To get the slope, simply get the first derivative of the equation Find the slope of the equation when Gagawa tayo ng equation from the figure, since may given tayo na r and theta..then we must come up with Using Pythagorean theorem: And derive with respect to theta.. Substituting the given and using the d/dx function of the calcu, it will be like this:

13 Next we must get From the figure, we have and equating it to x, we have: Substituting the given and getting the derivative of x with respect to theta, we have: Dividing the results and you got the answer Find the equation of the nomal line of the curve at (1,2) 4x-y=2 4x-y=-2 x+4y=9 x+4y=-9 When you are getting the slope of the line using the first derivative, what you are getting is the slope of the tangent line..to get the slope of the Normal Line, you get the negative reciprocal of this.. So, get the first derivate(tangent slope) of the given equation with x=1(given in the problem) And getting the negative reciprocal of this we have -1/4.. Then get the equation using STAT 2.. And press A.Get A and. SHIFT>1>7> and SHIFT>1>7>2 Remember, in the calculator, the equation of STAT 2 is y=a+bx.. Substitute a and b in the equation and you will get the answer The charge in coulombs that passes through a wire after t seconds is given by the function. Determine the current at the end of 2 seconds.. 6 A 8 A 9 A 10 A Simple get the first derivative of this with x=2 (2 seconds) and you will get the answer A 1800-gallon tank of water drains from the bottm in 30 min. According to Torricelli's law, the volume of water remaining in the tant after t minutes is where

14 How fast is the water draining from the tant after 20 minutes? -40 gpm -30 gpm -20 gpm -10 gpm Simply get the derivative of the equation with x=20 (20 minutes).. And you will get the answer in gpm Find the value of x when the function y=lnx/x is at maximum value. 0 e 1 1/e For maxima/minima, the first derivative should be equal to 0.. Try or test all the choices using CALC function if dy/dx=0 with x = x to use CALC function while getting the derivative.. and press CALC x? 0 x? e x? 1 x? 1/e Kung anu sa choices ang nag equal to 0, yun ang sagot.. SESSION 2 PART Find the Remember, in solving Trigonometric or inverse trigonometric, use RADIAN MODE.. Lagyan ng limits from 0 to 1 then press = Makakakuha ka ng sagot..i-note ito.. i-substitute ang limits sa choices..wag intindihin ang constant "C"..remember, upper limit minus lower limit..lets say for example sa choice.

15 Pag pumarehas ang sagot mo kanina(yung ni-note mo) yun ang sagot.. Happy integrating..;) 22. Find the centroid of the area bounded by the parabolas and (1.8, 1.8) (1.9, 1.9) (2,2) (2.1, 2.1) Shorcuts lang ang ibibigay ko since ito ay calculator technique.. Given: Pag nakita niyo na parehas yan, i mean 4x,4x...6x,6y...8x,8y...9x,9y and so on, ang shortcut niyan is 0.9x(center): therefore, 0.9 x (2,2) = answer 23. Find the area of the region bounded by, the x-axis, and the vertical lines x=0 and x=4.. 5 sq. units 16/3 sq. units 17/3 sq. units 6sq. units First get the limits from the equation it self..use EQN 3.. X=3 and X=2 Pasensiya na po kayo sa drawing ko..hirap e..ayun bale alternating po yan.. Then integrate using the limits.. Kaya po negative yung isa kasi dumaan po sa negative half cycle Determine the area of the region bounded by the curve and the x-axis, 2.25 sq. units 2.5 sq. units 3 sq. units

16 37/12 sq. units Get the limits from the equation..use EQN 4.. Ang you will get the answer Find the lenght of arc of the curve y = ln cosx from x=0 to x=pi/ Kung alam niyo formula gamit ang long method, go..di ko na siya ibibigay..shortcuts lang lahat ito.. Ang shortcut dito ay distance formula(two points).. First get the functions of y=f(x) When x=0, x=pi/4 Then y=0, Then use the distance formula The speed of the particle is given by. What distance does it travel while its speed increases from 7 to 99 ft/s? 83.3 ft 90 ft 96.7 ft 100 ft Get the limits from the equation when d=7..use EQN Get the real number and that is your lower limit.. Get the limits from the equation when d=99..use EQN Get the real number and that is your upper limit.. Then integrate the equation using your lower and upper limit Solve for the general solution of the differential equation:

17 Get the roots..use EQN You will get one real root and two complex roots.. KEY: Pag real root = Pag complex root = sin and cos 28. Solved for the particular solution of the differential equation: x + y dy/dx = 2 when x=1 and y=1 Choice one from the choices and Convert the equation to y=f(x).. Differentiate using your calculator with x=1(as given) and you will get the value of dy/dx Substitute x, y and dy/dx to the equation in the problem and see if it is equal to =2 If yes, then your choice is the right answer Solve for the general solution of the differential equation y'-y = 2.. Choose one in the choices.. Choose your favorite number for C and x and substitute it to your choice.. And solved for y..i-note.. i-differentiate ang iyong napiling equation with x=(kung ano ni-substitute mo sa una, be consistent) You will get the y' Then susbtitute in the given equation y'-y=2...if 2=2 Therefore, napili mo ang tamang answer.. SESSION 2 PART 4

18 31. A radioactive substance decreases from 10 grams to 9 grams in two hours. Find its half life hr hr hr hr Dahil sa ito ay radioactive, nagiincrease ito exponentially. Which means kung exponentially ito, gagamitin natin ang STAT 5.. Ang x column natin ay ang time(hr) and y column natin ay quantity.. X Y since time in its half life ang hinahanap, press AC then find 32. A thermometer reading is brought into a room where the temperature is ; 1 min later, the thermometer reading is. Find the temperature equation as a function of time. Pag temperature ang pinaguusapan use STAT 5. Sa x column natin is the time(min) and in y column it's either "Ts-T" or "T-Ts" (temperature). It depends kung sino malaki kung si T or si Ts.. T = Object temperature Ts = surrounding/environment temperature gets?. Let's proceed.. Since malaki ang Ts natin which is, therefore ang gagamitin natin ay Ts-T. Gets?.. X Y(Ts-T) (70-18) (70-31) After natin ma-input ang data, press A Since ang kinukuha natin dito ay equation, therefore kukunin natin ang value ng A and I hope and assume na you know it already kung paano at saan ito kukunin since naturo ko na ito sa naunang session/part.. After makuha ang A and B, alam natin na ang equation sa STAT 5 ay I-substitute ang A and B, x=t and y=ts-t since yun ang ginamit natin(y=70-t) 33. How fast does light travel in glass of refractive index 1.5?

19 Refractive index formula 34. At the surface of the earth,. Assume the earth to be a sphere of radius 6,371 km, compute the mass of the earth. BUT F=mg, assume m = Therefore, cancels' out And you will get the answer. 35. Calculate the form factor of a periodic voltage having the following values for equal time intervals changing one value to the next: 0, 5, 10, 20, 50, 60, 50, 20, 10, 5, etc Use STAT 1 and follow the formula, and you will get the answer. 36. Find the value of 8-8i 8 + 8i 4-4i

20 4 + 4i Ang pwede lang po sa ating calcu ay cube at square ng complex. Pwede pong ganito: Or pwede din ito: 37. Determine the argument of the result of I-type lang po natin sa calcu yung given then press = Convert po natin sa polar form then booomm..yun na I've reached the image limit. Sa next part na lang. Marami pa naman ito. 39. The resistance of a wire is ohms at 100 degrees C and 100 ohms at 30 degrees Determine the temperature when the resistance is 115 ohms degrees C degrees C degrees C

21 75.45 degrees C STAT 2 X column = temp. Y column = Resistance ---x Y What is the present worth of two 10,000 pesos payment at the end of the 3rd and 4th year if the annual interest is 8% compounded quarterly? P 15, 169 P 15, 288 P 16, 721 P 17, 264 Ang formula po talaga nito ay where: F = future worth P = present worth i = interest rate m = number of compounding(quarterl, semi-anually, annually etc) t = time(years,month, days, etc) Kung alam niyo kung paano computin sa long method, then go.. :D Then ito po yung sa CALCU TECHNIQUE: First kukunin natin ang P ng 3rd and 4th year. So unahin natin si 3rd year. gets?.. STAT 5 or 6 X column = mt(ito yung naka-raised sa formula) Y column = X Y , (10,000) bakit po 12? kasi mt=(4)(3) bakit po 13? kasi next year. +1 year gets?.. bakit 10,000 lang, akala ko yung formula? ginagamit lang natin yung formula on the next year. Then get the present worth: Second, kukunin natin ang present worth sa 4th year. Same lang din ito sa 3rd year. Ang pinagkaiba lang yung mt X Y , (10,000) Then get the present worth:

22 Finally get the sum of the present worth in the 3rd year and 4th year. And you will get the answer. SESSION 3 PART 1 Differential Calculus 1. Find indeterminate infinity I-type lang ang equation Press CAL.....x? Wag po natin i-substitute as x=0 dahil magmamath error po yan. Mag-isip po ng number na malapit sa 0. For example Then press '='.. And you will get the answer. 2. Find indeterminate infinity Same method lang po sa #1. So, napaisip ka ngayon kung paano infinity?. Ano po ba ang infinity? Ito ay isang malaking number. So press CAL...x? infinity = 1000 or kahit anong malaking number basta wag lang magmamath error. And you will get the answer. 3. Find Same method as #1.

23 4. Find 0 1/3 1 3 Same method lang din sa #1. Pero wag kalimutan na ilagay sa rad mode ang calcu. 5. Find 0 1/3 1 3 Same method as #4. 6. Find 0 infinity Same method lang din as #4. Palitan ang x ng malapit sa 1. either or Find the derivative of This time I assume na marunong na ang lahat at pamilyar na kayo sa mga steps at function na ginagawa natin. Don't forget: Always use rad mode in trigonometric and inverse trigonometric equations. Gamitin ang d/dx function. SHIFT>INTEGRAL then i-type ang equation. At maglagay ng value ng x. Youre favorite number. I preferred 5. Set x=5

24 Makakakuha tayo ng answer. Say A1(answer #1). Then mula sa choices, isubstitute lang ang 5 sa x at makakauha tayo ng sagot. Say A2 (answer #2) Kung A1 = A2 then tama ang napili niyo sa choices. 8. Find the derivative of y if Same lang sa #7. I hope you get it. That's what you called reverse engineering. 9. Find the derivative of y if 1 + cos x 1 - cos x Same method with # Find the derivative of y if tan y = x. sec x tan x Ang technique lang lagi dito guys is that you must convert the equation into y into a function of x. That is,

25 Then yun, same method na lang as #7. LAPLACE TRANSFORMS 1. Find the laplace transform of f(t) = cosh 5t a. (5 / s^2-25) b. (5s / s^2-25) c. (s / s^2 + 25) d. (s / s^2-25) Steps: By calculator technique (FX991ES+): Let t = x a. Press the Integral sign sa calculator (yung nasa baba ng alpha button) b. Type mo (e^-9x) (cosh 5x) c. Then yung limit po, ilagay nyo po 0 to 9 d. Answer is Take note of the answer. Para po malaman mo kung anu yung laplace transform nya sa choices, trial and error from the choices, just change "s" to 9. Like for this example, letter d yung may kaparehas na sagot. 2. Find the inverse laplace of [3(s^2-2)^2] / 2s^5 a. (3/2) + 3t^2 + (t^4/4) b. (3/2) - 3t^2 - (t^4/4) c. - (3/2) - 3t^2 + (t^4/4) d. (3/2) - 3t^2 + (t^4/4) Steps: By calculator technique (FX991ES+): Let t = x & s = 9 a. From this example, reverse process po ang gagawin. Just input sa calculator yung inverse laplace equation. Ang sagot po dapat sa equation na [3(s^2-2)^2] / 2s^5 ay b. From the given choices, trial and error po uli ang gagawin. Sa question, letter "d" po yung sagot c. Press the Integral sign sa calculator (yung nasa baba ng alpha button) d. Type mo (e^-9x) ([3(s^2-2)^2] / 2s^5). e. Limit is the same from 0 to 9, then press =. Medjo matagal nga lang po lumabas yung sagot. f. Ang sagot po dapat ay