Answer only the parts written in blue in the images below Mathematics can smile!3PROBLEM Two(a) Using Newton's forward difference formula, find the sumS, = 1³ +2³ +3³ +4³ + ... +n³.(b) Values of x (in degrees) ans sinx are given in the following table.x (in degrees)sinx150.25881900.34202010.42261832025300.50.57357640.64278763540Determine the value of sin 38º.(c) Find the missing term in the following table:y1129381Explain why the result differs from 3³ = 27.(d) From the following table, find the value of el.17 using Gauss's forward formula.et1.00 2.7183| 1.05 | 2.85771.10 3.00421.15 | 3.15821.20 3.32011.25 3.49031.30 | 3.6693(e) The following table gives the values of e for certain equidistant values of x. Find the valueof e when x= 0.644 by using:(i) Bessel's Formula(ii) Stirling's Formula.(iii) Everett's formula.(f) Derive the Lagrange Interpolation formula.(g) The following table gives the values of the probability integral2y =Mathematics can smile!corresponding to certain values of x.For what value of x is this integral equal to ?0.460.470.480.49y 0.4846555 | 0.4937452 | 0.5027498 | 0.5116683(h) The values of f(x) are given at a,b,c.Show that the maximum is obtained byf(a).(b² – c²) + f(b).(c² – a²) + f(c).(a² – b²)2[f(a).(b – c)+f(b).(c– a)+f(c).(a – b)](i) Given log10654=2.8156, log10658 =2.8182,log10659 =2.8189,log10661 =2.8202, find log1065(j) Derive the general error formula. PROBLEM FOUR(a) Find the value of x when y =0.3 by applying Lagrange's formula inversely.x 0.4y| 0.3683| 0.3332 | 0.28970.60.8(b) The following table gives the value of the elliptical integralF(0) =1- įsin²0for certain values of 0.Find the values of 0 if F(0)=0.3887250230F(0) 0.3706 0.4068| 0.4433210(c) Use Lagrange multipliers to obtain the interpolating polynomial which passes throuthe following points.-11 2f(x) | 0Hence use your polynomial to approximate f'(x), f"(x), f" (x), f'(1), f"(1), f" (1).(d) Using Lagrange's formula, find a polynomial which passes through the points-10 15(0, – 12), (1,0), (3,6), (4, 12)Mathematics can smile!6(e) Using Lagrange's interpolation formula, find the value of y corresponding to x= 10the following table:511y= f(x) | 121314 16(f) Given that u =F, ɛr, ɛy and & denote the errors in x,y and z respectively sucix= y=z=1 and ɛ = ɛ, = & = 0.001, then find the relative maximum error in u(g) Show that the relative error in quotient of two quantities x and y may be expresdifference between the relative errors in x and y.Sxy?(h) Given that u = , &, &, and ɛ denote the errors in x,y and z respectively sucx=y=z=2 and & = ɛy = & = 0.003, then find the relative maximum error in u(i) The number x= 37.46235 is rounded off to 4 significant figures.Compute the percentage error in x.(j) Obtain an interpolating polynomial using Newton Backward Difference Formula for-12f(x)0 -1 | 0| 15PROBLEM FIVE(a) Derive the trapezoidal Rule and the error involved in this method Hence or otheevaluate:(i)(b) Use Romberg integration to compute1-Ldx1+xcorrect to three decimal places.(c) erive the Simpson's third Rule and the error involved in this method Hence or otheevaluate:(i)(d) In the following table, locate and correct the value of y that is in error by using difftable, in which y is a cubic polynomial in x.x|01 2y 25 | 21 | 18 | 184 545 76 | 1233727(e) Use Romberg's method to computedxcorrect to 4 decimal places by taking h=0.5,0.25 and 0.125.(f) Given that1.01.21.41.61.82.02.2

a Given that Sn=13+23+33+...+n3 By using 1-x-2=1+2x+3x2+4x3+.....+n+1xn By multiplying x in…

(a) Using Newton's forward difference formula, find the sum Sn = 13 +2° +3³ +4³ +. +n°. (b) Values of x (in degrees) ans sinx are given in the following table. x (in degrees) sinx 15 | 0.2588190 0.2420001

(a) Using Newton's forward difference formula, find the sum Sn = 13 +2° +3³ +4³ +. +n°. (b) Values of x (in degrees) ans sinx are given in the following table. x (in degrees) sinx 15 | 0.2588190 0.2420001

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Answer only the parts written in blue in the images below

Mathematics can smile!
3
PROBLEM Two
(a) Using Newton's forward difference formula, find the sum
S, = 1³ +2³ +3³ +4³ + ... +n³.
(b) Values of x (in degrees) ans sinx are given in the following table.
x (in degrees)
sinx
15
0.2588190
0.3420201
0.4226183
20
25
30
0.5
0.5735764
0.6427876
35
40
Determine the value of sin 38º.
(c) Find the missing term in the following table:
y
1
1
2
9
3
81
Explain why the result differs from 3³ = 27.
(d) From the following table, find the value of el.17 using Gauss's forward formula.
et
1.00 2.7183
| 1.05 | 2.8577
1.10 3.0042
1.15 | 3.1582
1.20 3.3201
1.25 3.4903
1.30 | 3.6693
(e) The following table gives the values of e for certain equidistant values of x. Find the value
of e when x= 0.644 by using:
(i) Bessel's Formula
(ii) Stirling's Formula.
(iii) Everett's formula.
(f) Derive the Lagrange Interpolation formula.
(g) The following table gives the values of the probability integral
2
y =
Mathematics can smile!
corresponding to certain values of x.For what value of x is this integral equal to ?
0.46
0.47
0.48
0.49
y 0.4846555 | 0.4937452 | 0.5027498 | 0.5116683
(h) The values of f(x) are given at a,b,c.Show that the maximum is obtained by
f(a).(b² – c²) + f(b).(c² – a²) + f(c).(a² – b²)
2[f(a).(b – c)+f(b).(c– a)+f(c).(a – b)]
(i) Given log10654=2.8156, log10658 =2.8182,log10659 =2.8189,log10661 =2.8202, find log1065
(j) Derive the general error formula.

PROBLEM FOUR
(a) Find the value of x when y =0.3 by applying Lagrange's formula inversely.
x 0.4
y| 0.3683| 0.3332 | 0.2897
0.6
0.8
(b) The following table gives the value of the elliptical integral
F(0) =
1- įsin²0
for certain values of 0.Find the values of 0 if F(0)=0.3887
250
230
F(0) 0.3706 0.4068| 0.4433
210
(c) Use Lagrange multipliers to obtain the interpolating polynomial which passes throu
the following points.
-1
1 2
f(x) | 0
Hence use your polynomial to approximate f'(x), f"(x), f" (x), f'(1), f"(1), f" (1).
(d) Using Lagrange's formula, find a polynomial which passes through the points
-10 15
(0, – 12), (1,0), (3,6), (4, 12)
Mathematics can smile!
6
(e) Using Lagrange's interpolation formula, find the value of y corresponding to x= 10
the following table:
5
11
y= f(x) | 12
13
14 16
(f) Given that u =
F, ɛr, ɛy and & denote the errors in x,y and z respectively suci
x= y=z=1 and ɛ = ɛ, = & = 0.001, then find the relative maximum error in u
(g) Show that the relative error in quotient of two quantities x and y may be expres
difference between the relative errors in x and y.
Sxy?
(h) Given that u = , &, &, and ɛ denote the errors in x,y and z respectively suc
x=y=z=2 and & = ɛy = & = 0.003, then find the relative maximum error in u
(i) The number x= 37.46235 is rounded off to 4 significant figures.
Compute the percentage error in x.
(j) Obtain an interpolating polynomial using Newton Backward Difference Formula for
-1
2
f(x)
0 -1 | 0| 15
PROBLEM FIVE
(a) Derive the trapezoidal Rule and the error involved in this method Hence or othe
evaluate:
(i)
(b) Use Romberg integration to compute
1-L
dx
1+x
correct to three decimal places.
(c) erive the Simpson's third Rule and the error involved in this method Hence or othe
evaluate:
(i)
(d) In the following table, locate and correct the value of y that is in error by using diff
table, in which y is a cubic polynomial in x.
x|01 2
y 25 | 21 | 18 | 18
4 5
45 76 | 123
3
7
27
(e) Use Romberg's method to compute
dx
correct to 4 decimal places by taking h=0.5,0.25 and 0.125.
(f) Given that
1.0
1.2
1.4
1.6
1.8
2.0
2.2

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