Additional Exercises Exercises - PART 2
11. Evaluate A. 9920 B. 2920 C. 9200 D. 2290 Code: Set the calculator to computation mode: MODE>1 Input the equation as is but replace k with X.. Then press =.. And you will get the answer..
12. Find x, if A. B. C. D. Code: Set calculator to computation mode.. Type the equation as is..Use the "SOLVE" function and press =.. And you will get the answer..
13. Factor A. B. C. D. Code: 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"..
14. Find the 15th term in the arithmetic sequence: 1,3,5,7 . . . A. 25 B. 27 C. 29 D. 31 Code: 1. Set the calculator to STAT mode..MODE 3>2 2: A+BX is for Arithmetic Progression( A.P.).. May makikita kayo X and Y column..And X column natin ay ang nth term while sa Y column natin ay ang A.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 A.P. is 1, 2nd term is 3 and 3rd term is 7.. 3. 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 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 =.. And you will get the answer.. Kung may tanong kayo dito at di niyo makuha pakipost na lang..
15. Find the 12th term in the geometric series: 3, 9, 27 . . . A. 59,049 B. 177,147 C. 531,141 D. 1,594,323 Code: 1. 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 27.. 3. 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 =.. And you will get the answer.. Kung may tanong kayo dito at di niyo makuha pakipost na lang..
16. What is the sum of the first 20 terms of the sequence: 1, 3, 5, 7... A. 361 B. 391
C. 400 D. 441 Code: 1. Set the calculator to STAT mode: MODE 3>2 (since it is A.P.) 2. In the X column, input 1,2,3..In the Y column, input 1,3,5.. 3. After you input the data, press AC..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..
And you will get the answer..
17. What is the sum of the infinite series: 1, -1/5, 1/25, . . . A. 5/6 B. 4/5 C. 5/7 D. 6/7 Code: 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: number..(NOTE: For an infinite 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 6+9+12+. . .+171? A. 4956 B. 4389 C. 5198 D. 5462 Code: 1. 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 AC..
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 1,56..
19. What is the 6th term of the infinite series: 1, -1/5, 1/25, ... A. 1/3125 B. 1/15625 C. -1/3125 D. -1/15625 Code: 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: A. 2-5i B. (5+10i)/6 C. 1+2i D. (2+i)/3 Code: 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 = And you will get the answer..
SESSION 2 PART 1
1. Subtract A. B. C. D. Code: 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. A. 21,381 B. 25,381 C. 27,381 D. 30,381 Code: Conversion only.. V=10 acre (convert acre to m^2)SHIFT>8(for CONV)>11(for acre>m^2)..It will give an answer 40468.56.. Then multiply it to .002 m(given in the problem) and it will display 80.93712 in m^3.. Convert this to Liters..1 m^3 = 1000L..And it will display 80937.12 in L.. Finally convert this to gal..Press CONV#14 L>gal(US).. And you will get the answer in gal..
3. The function A. 2 B. -2 C. either a or b D. neither a nor b
is discontinous at x=?
Code: 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.. 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 A. (x-2) B. (2x-1) C. (2x+3) D. all of the above
.
Code: 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 CALC..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-4y1=0, and 2x-y+6=0.. A. 10 sq.units B. 15 sq.units C. 20 sq.units D. 25 sq.units Code: First get the point of intersection..use the EQN function..MODE 5>1 For the first two equation, type: 1 2 7 3 -4 1 then press = =..Therefore the point of intersection of this is x=3, and y=2.. Then try the next equation.. 3 -4 1 2 -1 -6 Then press = =......x=-5, and y=-4 try the last combination of equation.. 1 2 7 2 -1 -6 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 AC.. What important in this is kailangan natin madaan yung DIM or dimension ng matrix..So press SHIFT>4>1>1>1 and enter 3 2 1 -5 -4 1 -1 4 1 then press AC.. To get the determinant press SHIFT>7(det)>SHIFT>4>3(since we enter our data in MatA..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..
6. Find the center of the circle that is circumscribed about the triangle whose vertices are (-3,1), (3,1) and (5,3) A. (-3,9) B. (3,-9) C. (-3,-9) D. (3,9) Code: 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 -3 -1 1 -10 3 1 1 -10 5 3 1 -34 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) A. -5 B. 5 C. -6 D. 6 Code: 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 And you will get the answer..
8. Find the sum of an A.P.: 2,5,8..............where n=25.. A. 900 B. 925 C. 950 D. 975 Code: Use STAT 2(for A.P.) In the x column enter 1,2 In the y column enter 2,5 then press AC.. Use the SUMMATION FUNCTION..
And you will get the answer..
9. Find the sum of a G.P.: 2,3,4.5,.................where n=10. A. 226.66 B. 235.42 C. 242.52 D. 251.84 Code: Same method with #9 BUT use STAT 6(for G.P.) Then use the SUMMATION function to get the sum.. And you will get the answer..
10. In how many ways can a picture be painted by using three(3) or more of the seven(7) different colors?. A. 80 B. 99 C. 105 D. 120 Code: 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:
And you will get the answer..
SESSION 2 PART 2
Starting in this Part, di ko na iisa-isahin kung anu pipindutin niyo para madali tayo..Kung may tanong kayo at di niyo makuha, pakipost na lang..
11. Find the decimal equivalent of A. 32677 B. 36277 C. 23677 D. 26377 Code: Convert niyo lang sa decimal.. MODE 4..Set to BIN then type the number.. Press DEC and you got the answer.. 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, 56175 And press DEC and you got the answer..
12. Find the hexadecimal equivalent of
x
A. B. C. D. Code: Type niyo lang sa calcu..Wag kalimutan base 8 and base 10 yan.. Press HEX for the final answer..
13. Evaluate A x B.
A.
B.
C.
D. Code: MODE 6>AC.....Tandaan daanan lagi si DIM.. SHIFT>4>1 Itype ang matrix and that's it.. You got the answer.. 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
A. 0.433 B. 0.866 C. 1.083 D. 1.732 Code: Pag Trigonometric/Inverse, i-set ang calcu sa RADIAN MODE.. I-type sa calcu like this..
Yung
po natin diyan, pi po yun..
15. If y = 4tanh x + sinh 2x, what is the slope of the curve when x = 2? A. -54.9 B. -28.2
C. 54.9 D. 28.2 Code: TANDAAN: Slope = first derivative = To get the slope, simply get the first derivative of the equation.. And you will get the answer..
16. Find the slope of the equation A. 5.2 B. -5.2 C. 4.2 D. -4.2
when
Code: 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:
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..
17. Find the equation of the nomal line of the curve A. 4x-y=2
at (1,2)
B. 4x-y=-2 C. x+4y=9 D. x+4y=-9 Code: 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 AC..Get A and B.. 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..
18. 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.. A. 6 A B. 8 A C. 9 A D. 10 A Code: Simple get the first derivative of this with x=2 (2 seconds) and you will get the answer..
19. 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 How fast is the water draining from the tant after 20 minutes? A. -40 gpm B. -30 gpm C. -20 gpm D. -10 gpm
Code: Simply get the derivative of the equation with x=20 (20 minutes).. And you will get the answer in gpm..
20. Find the value of x when the function y=lnx/x is at maximum value. A. 0 B. e C. 1 D. 1/e Code: 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? x? x? x?
0 e 1 1/e
Kung anu sa choices ang nag equal to 0, yun ang sagot..
SESSION 2 PART 3
21. Find the A. B. C. D. Code: 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 A..
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 A. (1.8, 1.8) B. (1.9, 1.9) C. (2,2) D. (2.1, 2.1) Code: 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 vertical lines x=0 and x=4.. A. 5 sq. units B. 16/3 sq. units C. 17/3 sq. units D. 6sq. units
, the x-axis, and
Code: 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.. And you will get the answer..
24. Determine the area of the region bounded by the curve and the x-axis, A. 2.25 sq. units B. 2.5 sq. units C. 3 sq. units D. 37/12 sq. units Code: Get the limits from the equation..Use EQN 4..
Ang you will get the answer..
25. Find the lenght of arc of the curve y = ln cosx from x=0 to x=pi/4.. A. 0.65 B. 0.72 C. 0.81
D. 0.88 Code: 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, -0.34 Then use the distance formula.. And you will get the answer..
26. The speed of the particle is given by . What distance does it travel while its speed increases from 7 to 99 ft/s? A. 83.3 ft B. 90 ft C. 96.7 ft D. 100 ft Code: Get the limits from the equation when d=7..Use EQN 4.. 2 5 0 -7 Get the real number and that is your lower limit.. Get the limits from the equation when d=99..Use EQN 4.. 2 5 0 -99 Get the real number and that is your upper limit..
Then integrate the equation using your lower and upper limit.. And you will get the answer..
27. Solve for the general solution of the differential equation: A. B. C. D. Code: Get the roots..Use EQN 4..1 0 0 8 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 A. B. C. D. Code: 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... 2=2 If yes, then your choice is the right answer..
29. Solve for the general solution of the differential equation y'-y = 2.. A. B. C. D. Code: 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
31. A radioactive substance decreases from 10 grams to 9 grams in two hours. Find its half life. A. 10.19 hr B. 11.49 hr C. 12.89 hr D. 13. 16 hr Code: 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 0-----|-----10 2-----|-----9 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. A. B. C. D. Code: 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 Gets?..
, therefore ang gagamitin natin ay Ts-T.
X-----|-----Y(Ts-T) 0-----|-----(70-18) 1-----|-----(70-31) After natin ma-input ang data, press AC. Since ang kinukuha natin dito ay equation, therefore kukunin natin ang value ng A and B. 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? A. B. C. D. Code: 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. A. B. C. D. Code:
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. A. 1.11
B. 1.24 C. 1.35 D. 1.41 Code:
Use STAT 1 and follow the formula, and you will get the answer.
36. Find the value of A. 8 - 8i B. 8 + 8i C. 4 - 4i D. 4 + 4i Code: 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 A. -72 B. 27 C. 76.9 D. 159.4 Code:
I-type lang po natin sa calcu yung given then press = Convert po natin sa polar form then booomm..Yun na..
38. I've reached the image limit. Sa next part na lang. Marami pa naman ito. 39. The resistance of a wire is 126.48 ohms at 100 degrees C and 100 ohms at 30 degrees C. Determine the temperature when the resistance is 115 ohms. A. 65.15 degrees C B. 69.65 degrees C C. 71.25 degrees C D. 75.45 degrees C Code: STAT 2 X column = temp. Y column = Resistance ---x--- | ---Y-----100 | ---126.48 ---30 | ---100
40. 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?
A. P 15, 169 B. P 15, 288 C. P 16, 721 D. P 17, 264 Code: 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 --------12 ----- | 10,000 ----13 ----- | 1.02(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 ---------16----- | 10,000 -----17----- | 1.02(10,000) Then get the present worth: 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 A. 0 B. 0.353 C. indeterminate D. infinity
.
Code: I-type lang ang equation Press CALC.... . .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 0.001. Then press '=' .. And you will get the answer.
2. Find A. 0 B. 1.75 C. indeterminate D. infinity Code: 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 CALC... .x? infinity = 1000 or kahit anong malaking number basta wag lang magMAMATH error. And you will get the answer.
3. Find A. -1 B. 1 C. -2 D. 2
Code: Same method as #1.
4. Find A. 0 B. 1/3 C. 1 D. 3 Code: Same method lang din sa #1. Pero wag kalimutan na ilagay sa rad mode ang calcu.
5. Find A. 0 B. 1/3 C. 1 D. 3 Code: Same method as #4.
6. Find A. B. C. 0 D. infinity Code: Same method lang din as #4. Palitan ang x ng malapit sa 1. either 1.001 or .9999
7. Find the derivative of A. B.
C. D. Code: 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 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 A. B. C. D. Code: Same lang sa #7. I hope you get it. That's what you called reverse engineering.
9. Find the derivative of y if A. B. C. 1 + cos x D. 1 - cos x
Code: Same method with #7.
10. Find the derivative of y if tan y = x . A. B. sec x tan x C. D. Code: Ang technique lang lagi dito guys is that you must convert the equation into y into a function of x. That is,
Then yun, same method na lang as #7.
SESSION 3 PART 2
11. Find the first derivative of y if A. B. C. D. Same lang ito sa mga nakaraang examples ko sa differential 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.
12. Find the second derivative of y if A. B. C. D. So, mano-mano ito since walang shortcut pag second derivative. Sa board exam may ganyan. So let's test kung marunong pa kayo sa mano-mano.
13. Find the third derivative of y = x ln x. A. -1/x B. -1/(x^2) C. -1/(x^3) D. -1 So, mano-mano ito since walang shortcut pag third derivative. Sa board exam may ganyan. So let's test kung marunong pa kayo sa mano-mano.
14. Find the derivative of y if A. -x/y B. x/y C. y^-3 D. -y^-3
.
change to function of x then use d/dx function. Kayo na bahala kung anong value ng x gusto niyo. Then compare. Actually same lang ito sa #7.
15. Find the first derivative of y if A. y / (x+y) B. -y / (x+y) C. y / (x+2y) D. -y / (x+2y) Same lang sa #14.
16. Find the partial derivatives with respect to x of the function A. y^2 - 5 B. y^2 C. xy-5y D. 2xy Mag-iisip ka ng value ng y and then kapag dinirivative mo, equal siya dun sa value ng y. For example, y = 4. Substitute 4 in the equation. The equation now becomes 16x-20+6 or 16x-14. Then use (d/dx)(16x-4). Ang x natin dito ay always 1. So kung anong nakuha niyo answer diyan, kailangan mag equal siya dun sa answer. For example ang answer ay 16. Letter B and tamang sagot kasi y = 4 and y^2 = 16.
17. The function A. -1, 3 B. 1,-3 C. -1,2 D. 1,-2
is discontinous at
Use d/dx function then x=x and press CALC. Isubstitute ang lahat ng choices at kung saan nagerror, yun na.
18. Find the slope of the tangent line to the graph of the function at the point where x=3. A. 60 B. 66 C. 72 D. 78 Radian mode. Use d/dx function with x=3 Slope tangent is equal to the first derivative(y')
19. If y=4cos x + sin 2x, what is the slope of the curve when x = 2 radians? A. -2.21
B. -4.94 C. -3.21 D. 2.21 same with #18.
20. Find C so that the line y = 4x + 3 is tangent to the curve y = x^2 + C. A. 4 B. 5 C. 6 D. 7 Remember: m1=m2 (tangent/parallel) m1= -1/m2 (normal/perpendicular) And y=4x+3 should be tangent to y=x^2+C First, get the first derivative or slope of y=4x+3. Use d/dx with x=1, therefore the slope is 4. equate y' = y' (y=4x+3) = (y=x^2+C) 4 = 2x and x = 2. Then get the value of y from y=4x+3 y = 11. Therefore, our pt is (2,11) Substitute this to Y = x^2 + C to get the value of C.
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)
By calcu technique (FX991ES+): Let t = x 1. Press the Integral sign sa calcu 2. Type mo (e^-9x) (cosh 5x) 3. Then yung limit po, ilagay nyo po 0 to 9 4. Answer is .1607 . 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)
By calcu technique (FX991ES+): Let t = x & s = 9 1. From this example, reverse process po ang gagawin. Just input sa calcu yung inverse laplace equation. Ang sagot po dapat sa equation na [3(s^2-2)^2] / 2s^5 ay .158537 2. From the given choices, trial and error po uli ang gagawin. Sa question, letter "d" po yung sagot 3. Press the Integral sign sa calcu 4. Type mo (e^-9x) ([3(s^2-2)^2] / 2s^5) . 5. Limit is the same from 0 to 9 , then press =. Medjo matagal nga lang po lumabas yung sagot. 6. Ang sagot po dapat ay .158537
Problem: Arithmetic Progression th The 6 term of an arithmetic progression is 12 and the 30 term is 180. 1. What is the common difference of the sequence? 2. Determine the first term? nd 3. Find the 52 term. th 4. If the n term is 250, find n. 5. Calculate the sum of the first 60 terms. 6. Compute for the sum between 12th and 37th terms, inclusive. th
Traditional Solution For a little background about Arithmetic Progression, the traditional way of solving this problem is presented here.
HideClick here to show or hide the solution
→ common difference
→ first term
nd
→ 52 term
th
→ 40 term, a40 = 250 Sum of AP is given by the formula
Sum of the first 60 terms
→ answer Sum between 12th and 37th terms, inclusive.
→ answer
Calculator Technique for Arithmetic Progression Among the many STAT type, why A+BX? The formula a n = am + (n - m)d is linear in n. In calculator, we input n at X column and a n at Y column. Thus our X is linear representing the variable n in the formula. Bring your calculator to Linear Regression in STAT mode: MODE → 3:STAT → 2:A+BX and input the coordinates.
X (for n) 6 30
Y (for an) 12 180
To find the first term: AC → 1 SHIFT → 1[STAT] → 7:Reg → 5:y -caret and calculate 1y-caret, be sure to place 1 in front of y-caret. 1y-caret = -23
→ answer for the first term
nd To find the 52 term, and again AC → 52 SHIFT → 1[STAT] → 7:Reg → 5:y -caret and make sure you place 52 in front of y-caret.
52y-caret = 334
nd
→ answer for the 52 term
To find n for a n = 250, AC → 250 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 250x-caret = 40
→ answer for n
To find the common difference, solve for any term adjacent to a given term, say 7th term because the 6th term is given then do 7y-caret - 12 = 7 for d. For some fun, randomly subtract any two adjacent terms like 18y-caret - 17y-caret, etc. Try it! Sum of Arithmetic Progression by Calculator Bring the your calculator to Quadratic Regression in STAT mode MODE → 3:STAT → 3:_+cX2
2
Why MODE → 3:STAT → 3:_+cX ? The formula S = ½n[ 2a 1 + (n - 1)d ] for sum of arithmetic progression is quadratic in n. In our calculator, we input n in the X column and the sum at the Y column. Note that for the given AP, a 1 = -23, a2 = -16, and a 3 = -9. Input three coordinates
X 1 2 3
Y -23 -23-16 -23-16-9
Sum of the first 60 terms: ( AC → 60 SHIFT → 1[STAT] → 7:Reg → 6:y -caret) 60y-caret = 11010 Sum from 12th to 37th terms, use SHIFT → 1[STAT] → 7:Reg → 6:y-caret twice 37y-caret - 11y-caret = 3679 Another way to solve for the sum is to use the Σ calculation. The concept is to add each term in the progression. Any term in the progression is given by a n = a1 + (n - 1)d. In this problem, a 1 = -23 and d = 7, thus, our equation for a n is an = -23 + (n - 1)(7). Reset your calculator into general calculation mode: MODE → 1:COMP then SHIFT → log. Sum of first 60 terms:
(-23 + (ALPHA X - 1) × 7) = 11010
Or you can do
(-23 + 7 ALPHA X) = 11010 which yield the same result.
Sum from 12th to 37th terms
(-23 + (ALPHA X - 1) × 7) = 3679
Or you may do
(-23 + 7 ALPHA X) = 3679
Calculator Technique for Geometric Progression Problem Given the sequence 2, 6, 18, 54, ... 1. Find the 12th term 2. Find n if a n = 9,565,938. 3. Find the sum of the first ten terms. Traditional Solution
HideClick here to show or hide the solution Common ratio,
answer
→ Line (2)
answer
Or you may use SHIFT → SOLVE of the calculator in Line 2 to find n directly 9565938 ALPHA CALC 2(3ALPHA
X-1
) SHIFT → SOLVE =
wait for a little while... Calculator will output X = 15 L - R = 0
Answer for n is 15
answer
Solution by Calculator Why A·B^X? The nth term formula a n = a1r for geometric progression is exponential in form, the variable n in the formula is the X equivalent in the calculator. n – 1
MODE → 3:STAT → 6:A·B^X
X 1 2 3
Y 2 6 18 To solve for the 12 th term
AC → 12 SHIFT → 1[STAT] → 7:Reg → 5:y -caret 12y-caret = 354294
answer
To solve for n, AC → 9565938 SHIFT → 1[STAT] → 7:Reg → 4:x-caret 9565938x-caret = 15
answer
Sum of the first ten terms ( MODE → 1:COMP then SHIFT → log) n – 1 Each term which is given by a n = a1r .
(2(3ALPHA
X - 1
)) = 59048
answer
Or you may do
(2 × 3ALPHA X) = 59048
Calculator Technique for Harmonic Progression Problem Find the 30th term of the sequence 6, 3, 2, ...
Solution by Calculator MODE → 3:STAT → 8:1/X
X 1 2 3
Y 6 3 2
AC → 30 SHIFT → 1[STAT] → 7:Reg → 5:y -caret 30y-caret = 0.2
answer
I hope you find this post helpful. With some practice, you will get familiar with your calculator and the methods we present here. I encourage you to do some practice, once you grasp it, you can easily solve basic problems in progression. If you have another way of using your calculator for solving progression problems, please share it to us. We will be happy to have variety of ways posted here. You can use the comment form below to do it.
The following models of CASIO calculator may work with these methods: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus. Before we go to Calculator technique, let us first understand the movements of the hands of our continuously driven clock. For simplicity, let "dial" be the unit of one hand movement and there are 60 dials in the complete circle as shown in the figure.
1. When the minute-hand moves 60 dials, the hour hand moves 5 dials. The ratio of the two movements, hour-hand over minute-hand, is 5/60 = 1/12. Thus, if the minute-hand will move x-minutes, the hour-hand moves by x/12 minutes. 2. When the second-hand moves 60 dials, the minutehand moves 1 dial. The ratio of the two movements, minute-hand over second-hand, is 1/60. Thus, if the second-hand will move x-seconds, the minute-hand moves by x/60 seconds, and the hour-hand also moves by 1/12 of x/60 or x/720 seconds. 3. The relationship of hand-movements can also be translated in terms of degree unit which I found handy in calculator technique for board exam problems. We know that a complete circle is equal to 360° and equal to 60 dials. Thus, 1 dial is equivalent to 360°/60 = 6° and five dials is equivalent to 5(6°) = 30°. Note that 1 dial move of the minute-hand is equivalent to 1 minute of time, and five dials move of the hour-hand is equivalent to 1 hour of time.
Knowing all of the above, we can now develop the calculator technique for solving clock-related problem. We will solve some example here in order to apply this time saving technique. Problem What time after 3:00 o'clock will the minute-hand and the hour-hand of the clock be (a) together for the first time, (b) perpendicular for the first time, and (c) in straight line for the first time? Traditional Solution ShowClick here to show or hide the solution
Solution by Calculator The following calculator keys will be used. Name
Key
Operation
Name
Key
Operation
Shift
SHIFT
Stat
SHIFT → 1[STAT]
Mode
MODE
AC
AC
The relationship between the movements of the clock hands is linear. We can therefore use the Linear Regression in STAT mode. Approach No. 1 Take 3:00 pm as reference point. Initially, the minute-hand of the clock is at 0 dial and the hour-hand of the clock is advance by 15 dials, thus, coordinates (0, 15). After 1 hour (4:00 pm), the minutehand advanced by 60 dials leaving the hour-hand 40 dials, thus, coordinates (60, -40). MODE → 3:STAT → 2:A+BX
X
Y
0
15
60
-40
(a) Together for the first time: The distance between the hands of the clock is zero. We will therefore find X when Y is zero in our table. AC → 0 SHIFT → 1[STAT] → 7:Reg → 4:x-caret 0x-caret = 16.36
Thus, time = 3:16.36 pm (b) Perpendicular for the first time: The hour-hand is behind by 15 dials by the minute hand. Let us find X when Y is -15. AC → -15 SHIFT → 1[STAT] → 7:Reg → 4:x-caret -15x-caret = 32.73
Thus, time = 3:32.73 pm
(c) Straight line for the first time: The hour-hand is behind by 30 dials by the minute hand, thus find X when Y is -30. AC → -30 SHIFT → 1[STAT] → 7:Reg → 4:x-caret -30x-caret = 49.09
Thus, time = 3:49.09 pm The above approach works fine but you need to mentally visualize the hands of the clock to get the proper sign (positive or negative) and value of the coordinates. See for example if the given time is 10:00 pm, the hands will be in straight line for the first time with the hour hand advancing the minute hand by 30 dials. Thus Y is +30 and not -30. This mental visualization takes the same effort as the traditional solution; the only difference is the absence of drawing. Without the drawing is good already but we can do better than that. The next approach will be more consistent, the only catch is that you need to memorize the numbers 30 and 330. I think it is not hard to memorize that numbers. Approach No. 2 In the first approach, both X and Y are in dial units. In this second approach, the X coordinate will be in dial and Y coordinates in degrees. Recall that in 1 hour, the hour-hand will move 5 dials equivalent to 30° and the minute-hand moves for 60 dials or 360°. The 1 hour difference is therefore 360° - 30° = 330° for the hour and minute-hands of the clock. At 3:00 pm, the minute-hand is at -90° in reference with the hour-hand, thus coordinates (0, -90). After 1 hour, that is at 4:00 pm, the minute hand advanced the right hand by 330° - 90° = 240°, thus coordinates (60, 240) MODE → 3:STAT → 2:A+BX
X
Y
Explanation
0
-90
←
60
240
← 330 -
-3 × 30 90
(a) Together for the first time: The angle between the hands of the clock is zero. Find X when Y is zero in our table. AC → 0 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 0x-caret = 16.36
Thus, time = 3:16.36 pm (b) Perpendicular for the first time: The angle between the hour-hand and minute-hand is 90°. Let us find X when Y is 90.
AC → 90 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 90x-caret = 32.73
Thus, time = 3:32.73 pm (c) Straight line for the first time: The angle between the hour-hand and minute-hand is 180°, thus find X when Y is 180. AC → 180 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 180x-caret = 49.09
Thus, time = 3:49.09 pm For me, the second approach is more rapid and easy to implement. I recommend you master just one and be good at it. Problem How soon after 5:00 o'clock will the hands of the clock form a (a) 60-degree angle for the first time, (b) 60-degree angle for the second time, and (c) 150-degree angle? Solution by Calculator Technique MODE → 3:STAT → 2:A+BX
X
Y
Explanation
0
-150
←
60
180
← 330 -
-5 × 30 150
(a) 60-degree angle for the first time AC → -60 SHIFT → 1[STAT] → 7:Reg → 4:x-caret -60x-caret = 16.36 minutes
answer
(b) 60-degree angle for the second time AC → 60 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 60x-caret = 38.18 minutes
answer
(c) 150-degree angle AC → 150 SHIFT → 1[STAT] → 7:Reg → 4:x -caret 150x-caret = 54.54 minutes
answer
Flow Rate Problem Water is poured into a conical tank at the rate of 2.15 cubic meters per minute. The tank is 8 meters in diameter across the top and 10 meters high. How fast the water level rising when the water stands 3.5 meters deep. Traditional Solution
Volume of water inside the tank
Differentiate both sides with respect to time
When h = 3.5 m
answer
Solution by Calculator
ShowClick here to show or hide the concept behind this technique MODE → 3:STAT → 3:_+cX2
X 0 10 5
Y 0 π4 π2
AC → 2.15 ÷ 3.5y -caret = 0.3492
answer
To input the 3.5y-caret above, do 3.5 → SHIFT → 1[STAT] → 7:Reg → 6:y -caret
