EXAMPLES CHAPTER III
- 1.Given
f(1)=(x—2)(x—3), f(2)=(x—7)(x—5), f(3)=(x—10)(x+1), f(4)=63, obtain a value for x, assuming second differences are constant.
- Find the nth term of the series 2, 12, 36, 98, 270, 768, etc. and the sum of n terms of it.
3. Given thatf(0)=66566, f(1)=65152, f(2)=63677, f(3)=62136, find f (9).6s 4. Find620+29(3x+1) (3x+4) (3x+7) (3.c+10)'
- Find the value of
A" (3x+ 1) (3x+4) (3x+ 7) ... (3x +3,t — 2).
- Prove that
(n—1)"'f(z)+m (n—1)m—if(x +1)+ rn(2 1)(n— 1)m-2f(x+2)+... +f (x+m)=nm f (x) +mn'"-1 of (x)m(m—1){+ 2nm-2 A2 (x)+... +Amf (x)•Hence find the sum of the series(m— 1)12+nb. 22+ - .),32+...+(m+1)2.CHAPTER IV1. Givenf(0)=70795, f(1)=72444, f(2)=74131, and •f (6) = 81283, find f (3), f (4), and f (5).2. Find f (35) givenf (20) = .01313, f (30) = •01727, f (40) = .02392, f (50) = '03493.3. Supply the missing term in the following table: f (0) = 72795,f(1)=71651,f (2) = 70458,f (4)=67919,
- = 66566,
= 65152.4. Givenf(0)=11, f(3)=18, f(6)=74, and f(9)=522,find the intermediate terms.
140 EXAMPLES
5. Given
f(0)=98203, f(1)=97843, f(2)=97459, f(3)=97034, find f (2.25).
- 6.Given
f (0) = 98023, .f(10) = 97651, f (20) =97246, f (30)=96802, find f (15).
- 7.Given
f (0)=58.842, f (2)=55.257, .f(4) = 51.368, f(10) =37.977, complete the series f (0), f (1), ... f (10).
- If you were asked at very short notice to obtain approximate values for the complete series f (0), f (1), f (2), ... f (20), being given that f (0)=.013, f(10)=•248, f(15)=•578, and f(20)=•983, what methods would you adopt, and what value would you obtain for f (9) l
9.f(1)=1; f(2)+f(3)=5.41; f(4)+f(5)+f(6)=18.47; f (7)+.f (8)+.1(9)+f(10)+f (11)+f(12)=90.36. Find the value of f (x) for all values of x from 1 to 12 inclusive.
- Apply Lagrange's formula to find f (5) and f (6), given that
f(1)=2, f(2)=4, f(3)=8, f(4)=16 and f(7)=128;and explain why the results differ from those obtained by completing the series of powers of 2.
- Find the simplest algebraic expression in x which has the values 5, 3, 9, 47 and 165 when x has the values 0, 1, 2, 3 and 4 respectively.
- Prove the following formulas for approximate interpolation :
.f(1)=.f(3)-.3[.f(5)-.f(-3)]+.2[f(-3)-f (- 5)] (1),f(0)=I[f(1)+f(-1)]- [ {.f(3)-.f(1)}- {f(-1)-f(-3)}] ...(2), and apply them to find the logs of 45, 46, 47, 48, 49, being givenlog 42 = 162325, log 50 =1.69897,log 44=1.64345, log 52 =1.71600.
- Given E f (x)=500426, 4 f (x)=329240, r f (x)=175212 and f (10)=40365, find f (1).
CHAPTER V1. Use Gauss' interpolation formula to obtain the value of f (41) given f (30) = 36782, f (35) = 2995.1, f (40) = 2400.1,f(45)=1876.2, f(50)=1416.3.Verify your result by using Lagrange's formula over the same figures.
EXAMPLES 141
2. Given the following table find f (28) using Stirling's formula: f (20) = 98450, f (25) = 96632, f (30)=94472,
f (35)=91852, f (40)=88613.
J (x)=i[f(-i)8.._f( )~+xAf(- +x2i A 2f J (-) 2AJ (-+...,
and apply the formula to find f (32) given
f(25)='2707, f(30)='3027, f(35)='3386, _f(40)='3794.
- From the table of annual net premiums given below find the annual net premium at age 25 by means of Bessel's formula :
AgeAnnual Net Premiums20'0142724'0158128.0177232'01996
- Use Everett's interpolation formula to complete the series f (25) to f (35),
given thatf(15)=305, f(20)=457, f(25)=568, f(30)=671,f(35)=897, f (40) = 1190, f (45) =1481.CHAPTER VI1. Given the following table of f (x) :f(0)=217, f(1)=140, f(2)=23, ,f(3)=—6,show how to find approximately the value of x for which the function is zero.2. Given that, whenx=0, f(x)=0,
- f(x)=100,
f(x)=2000,find x when _f (x)=1900 by Lagrange's formula of interpolation (applied inversely) and explain why the result does not agree with that found by using the formula f (.x)=(1+0)x f (0) and solving the quadratic.3. The following values off (x) are given :f(10)=1754, f(15)=2648, f(20)=3564.Find, correct to one decimal place, the value of x for which f (x) =3000.
4. f(30)=-30, .f(34)=—13, f(38)=3, f(42)=18.
Apply Lagrange's formula of interpolation inversely to find x, where f (x)=0.
142 EXAMPLES
CHAPTER VII
- 1.Find the value of
11n (n+l) (n+2) ... (n+r— 1) ' and use the result to find the value of 1
2n (n + 1) (n+2)'
2. Prove thatn(x2+1).x!=n.(n+1)!.
- 3.Show thatr!
lax~(x)=aak1(x)—aal A~(x)+( a$1 )2A2~(x)—... , where q (x) is any rational integral algebraic function of x.
- Prove that if an diminishes as n increases and converges to the limit zero, the sum to infinity of the series al — a, + a3 ... is the same as the sum to infinity of the series ia1—fAct' +NA1a1—etc.
Find the sum to infinity of the series+iE••• true to four decimal places.
- Prove that if the fourth and higher differences are ignored the sum of n successive terms of a function of which f (0) is the central term is
nf(0)+24Af(—1),where n is an odd number.CHAPTER VIII1. Given the following values off (x):f(1.41)= 7092, f(1.49)=•6711,f (1.52)=•6579, f'(1.53) = -6536, find f (1.45).2. Use Divided Differences to find f (80) to the nearest integer, given f(70)=235, f(71)=256, f(79)=436, f(81)=484.
- Given
f(20)=•342, f(23)=•391, f(31'=•515, f(34)=•559,
find f (30) by means of Divided Differences, and check the result by applying Lagrange's interpolation formula.
EXAMPLES 143 CHAPTER IX
|
1. |
The following values of f (x, y) are given : |
|
|
|
f(35, |
55)=10.020, |
f(35, |
50)=11.196, |
f(35, |
45) =12.019, |
|
f (40, |
55) = 9.796, |
f (40, |
50)=10.894, |
f (40, |
45) =11.641, |
|
1(45, |
55) = 9.583, |
f(45, |
50)=10.591, |
f(45, |
45)=11.243. |
|
|
|
|
|
|
|
- (i)Using only six of the above values, find f (42, 52).
Making use of all the data calculate f (44, 51).
2. Prove that if f (0, 1)=f(1, 0) and f(0, 2)=f(2, 0), then
f(x, y)=.f(0,0)+(x+y)LA.+x+2—1 A2x]f(0, 0)+xy[f(1, 1)-f(2, 0)1
Find f (39, 33), given
f(35, 35)=3.151, .f (35, 45)=3.912, f (35, 40)=3.471, f (40, 40)=3.766.
CHAPTER XI
1. Find the differential coefficients with regard to x of log x= and ,/— 3
(x — 1) A/x2 — 7
- Differentiate with respect to x
2 a3+a2x+axe+x3lo".a'2+ax+x23. Find the differential coefficient, with regard to x2 of /x2 — 2
- Differentiate
ax(1) ax a) log-, with respect to log-, a (2) log x- , with respect to x,(3) x (logx)', with respect to x.
- If (1 —x)v+(1 -y)x=0, find fix.
6. Obtain the differential coefficients of(1) log fix, (2) el', with respect to x.
144 EXAMPLES
CHAPTER XII
- Find (1) the nth differential coefficient of(i)2 1)with respect to x, and (2) the second differential coefficient of logo (1 +x) with respect to log, x.
- Find the nth differential coefficient of e "x (x - 2)2.
Having given thatz2 y+' c/x+y=0, dx2 dda+2d' +1doprove that2dx.+2+(2n+1)x. dx" y+(n+1)n=0.
- Obtain the second differential coefficient of log (a+bx+cx2) with respect to m.
5. Find the nth differential coefficient with respect to .v of x+ 1 2x2 - 5x+3CHAPTER XIII2x41. Prove thatexx 1=1 2 + 12 720 ...and show that no odd power of .v beyond the first can occur.
log/ 1z)4 2. Prove that(~ i +x2 x -x 3+3. 5"'"'3. Prove that the first three terms of the expansion of (1+x)r in powers of exx are e - 2 +.',}ext.
- Expand log (x+a2+x2) in ascending powers of x.
log (1+x2)
- Expand _ 1in ascending powers of x as far as the term in
volving x'.CHAPTER XIV
- EvaluateLt [log (1 +i x)]
x-,-oex
- Find the maximum value of x (x - 1) (x - 2) between the limits 0 and 1.
Find the values of x at the points where the graph of the function (1+x2) el-x' has its greatest slope.
EXA MPLES 145
- A man in a boat at sea, 5 miles distant from the nearest point of a straight shore, wishes to reach a place 12 miles distant along the shore, measuring from this nearest point. At what point should he land to reach this place in the minimum time, if he can row at 3 miles an hour and walk at 4 miles an hour?
Given f (0) = 1876, f (1) = 777, f (3) =19, and ,f (6) = - 218, interpolate the values of f (2), .f(4), and f (5) and find the values of x for which f (x) is a maximum or minimum.6. Find the minimum and maximum values of x4-4x3-8x'-+48x-48.
- A window is in shape a rectangle with a semicircle covering the top. If the perimeter of the window be a fixed length ?, find what is its maximum area.
8. Find(1) Lt x-a(2) Lt (1 +.r3)'.x-0CHAPTER XV
- Show thatd f ~) = A3f (x - ) approximately.
By considering the function f (x)=a+bx+ex and using the above relation prove that loge c=cl-c-1 approximately, where c is a small quantity.
2
- 2.Show that o0'"-% +A3 0"'...=0, when m> 1.
Prove that, if I (x) be a function the fourth differences of which are constant,df(x)=dx '=[f(x-2)-8f(x-1)+8.f(x+1)-f(x+2)]and hence find an approximate value for[log f (x)] where
f (x - 2) = 42.699, f(x-1)=40.365, f(.c)=37'977, f (x + 1) = 35.543, f(x+2)=33.075.
CHAPTER XVIII
- 1.Find the integralsr
(1) f ,x'dx. (2) I — dx (3) flog (log ~) dx..x-1x6 -a6
- 2.Evaluatef x4axdx.
H. T. B. I. 10
146 EXAMPLES
- 5.Find the value ofx2i l ex" dr.
0
CHAPTER XX
1. From the tablef(0)=217, f(1)=140, f(2)=23, f(3)=-6,3find an approximate value for J f (x) dx, and explain why the result differsfrom .f (0)+f (1)+f (2)+f (3). o
- 2.Prove that approximately
f + f(x) dx=h4 (f(-1)+22f (0)+f (1)),
and find thereby an approximate value for I
~ 10,000 x 4-x x 2;'dx.
- 3.Discuss the error in assuming that
Jof (x) dx=f (I)=Il [f(0)+f(1)] if (1) f(x)=a+bx+cx2, (2) f(x)=100x4'.
- 4.Given f ( - 2), f (0), f (2), find an approximate value for
f _3 f (x) dx.
EXAMPLES 147
- 5.Prove that
f 0 1 f (x) dx = ~ [5.f (1) + 8f (0) - f (–1)] approximately. If the speed of a train on a non-stop run is as shown in the table below, findthe approximate mileage travelled between 12.0 and 12.30, using the above formula.
|
Time |
Speed inmiles per hour |
| 11.50 |
24.2 |
| 12.0 |
35.0 |
| 12.10 |
41.3 |
| 12.20 |
42.8 |
| 1 2.30 |
39 2 |
- 6.If.f(1)=4157, f(2)=4527, f(4)=5435, find approximately the value of
4
) (x) dx.
CHAPTER XXI
- n persons are sitting at a round table, and from them three are selected at random ; show that the chance that no two of those selected are sitting next one another is (n–4)(n–5)
(n – 1)(n – 2)'
- A heap of playing cards contains 6 hearts, 5 spades and 4 clubs. A card is chosen at random 9 times in succession and is not replaced. Find the chance (1) that there are no hearts left, and (2) only hearts are left.
A man has two sovereigns and four shillings in his pocket; from these he selects at random 3 coins, and again chooses from these 3 coins at random, the first coin for d, the second for B. and the third for C.Find the values of the expectations of A, B, and C.
- Find the probability that out of 5 persons aged 45 exactly 3 will die in a year.
Probability that a person aged 45 will die in a year= .01224.
- A die with six faces is thrown three times and the sum of the throws is twelve: find the chance (1) that the first throw was a four: (2) that four was thrown each time.
Two men, d and B, each draw a card from a well-shuffled pack of playing cards, find that they are of the same value, and replace the cards; they do this four times in succession. Find the chance that this would happenand show that it is approximately1 84,000
148 EXAMPLES
- The probability that a man aged 50 will survive one year is •98428. Show that the probability that, out of 5 men aged 50, 3 at least will die within a year is •0000385.
The 26 letters of the alphabet are placed in a bag. A and B alternately draw a letter from the bag, the letters drawn not being replaced. The winner is the one who draws most vowels. A starts and draws a vowel with his first draw. What is his chance of winning?If a number of five figures containing any five of the ten digits once only is written down at random, what is the probability that it is divisible by 9?Given the following table find the probability that one at least of three persons aged respectively 20, 30, 40 will die between the 10th and 20th year from now :ProbabilityProbabilityAgeof survivingof surviving10 years20 years20.93363•8565130•91740•8116740•88476•7151711. Two persons, A and B, play for a stake, each throwing alternately two dice, A commencing. A wins if he throws six, B if he throws seven, the game ceasing as soon as either event happens. What ratio will A's chance of winning bear to B's?12. The sum of two positive integers (excluding zero) is 100 ; find the chance that their product exceeds 1200.13. The following table shows the probability that a woman of the age specified will marry in a year:AgeProbability of marriage20•066525•103330.061940•0183Find the probability that, out of 4 women aged 20, 25, 30, 40 respectively, only one marries within a year.14. A bag contains 8 counters, numbered 1 to 8. Four are drawn at random. Find the chances that
- The sum of the numbers on the four counters amounts to at least 17.
The counters numbered 2 and 3 are among the four.The four counters contain at least two of the three counters numbered 3, 5 and 7.15. A penny is tossed six times. Find the chance that neither heads nor tails have occurred three times in succession.
EXAMPLES 149
16. An urn contains counters marked with the digits 6, 7, 8 and 9 ; and the number of times each digit occurs is equal to the value of the digit. If counters are drawn one at a time, each counter being replaced when drawn, what is the probability
- (1)that the digit 6 is drawn before the digit 9;
that the sum of the first three digits drawn is exactly 20?
17. A and B play a set of games, to be won by the player who first wins four games, with the condition that if they each win three they are to play the best of three to decide the set. A's chance of winning a single game is to B's as 2 to 1. Find their respective chances of winning the set.
18. The probability of any one of 10 men each aged 30 surviving a year is •99229.Show that the probability that exactly 5 men out of the 10 survive a year is6.6x10-s.Find also the probability that of the 10 men one particular man will die first and another particular man last.19. A point is taken at random within the area bounded by the curve y=,r. lcgx, the x axis, and the ordinates at the points x=1, and x=4.Find the probability that the distance of the point from the y axis is less than 2.
20. In a game of whist the dealer found that on turning up the last card he had the Ace, King, Queen, Knave, Ten, and 3 other trumps in his hand. Find the chance that this would occur.21. A and B cut a pack of cards, the player who wins the cut six times to be the winner. ;l, having won four times to B's once, cuts a five. Find the chance that A will be the winner.
22. In a line AB of length 3a, a point P is taken at random and then in AP a point Q is taken at random. What is the probability that PQ exceeds a?