Higher Engineering Mathematics, Sixth Edition
62 Higher Engineering Mathematics Expand(p+ 2 q)^11 as far as the fifth term. ⎡ ⎣ p^11 + 22 p^10 q+ 220 p^9 q^2 + 1320 p^8 q^3 ...
The binomial series 63 + ( 1 / 2 )(− 1 / 2 )(− 3 / 2 ) 3! (x 4 ) 3 +··· ] = 2 ( 1 + x 8 − x^2 128 + x^3 1024 −··· ) = 2 + x 4 − ...
64 Higher Engineering Mathematics Now try the following exercise Exercise 30 Further problems on the binomial series In problems ...
The binomial series 65 Hence new volume ≈πr^2 h( 1 − 0. 08 )( 1 + 0. 02 ) ≈πr^2 h( 1 − 0. 08 + 0. 02 ),neglecting products of sm ...
66 Higher Engineering Mathematics Now try the following exercise Exercise 31 Further practical problems involving the binomial t ...
Revision Test 2 This Revision Test covers the material contained in Chapters 5 to 7.The marks for each question are shown in bra ...
Chapter 8 Maclaurin’s series 8.1 Introduction Some mathematical functions may be represented as power series, containing terms i ...
Maclaurin’s series 69 Continuing the same procedure gives a 4 = fiv(0) 4! , a 5 = fv(0) 5! , and so on. Substituting fora 0 ,a 1 ...
70 Higher Engineering Mathematics f(x)=sinxf( 0 )=sin0= 0 f′(x)=cosxf′( 0 )=cos0= 1 f′′(x)=−sinxf′′( 0 )=−sin0= 0 f′′′(x)=−cosxf ...
Maclaurin’s series 71 Substituting these values into equation (5) gives: f(x)=ln( 1 +x)= 0 +x( 1 )+ x^2 2! (− 1 ) + x^3 3! ( 2 ) ...
72 Higher Engineering Mathematics Problem 11. Develop a series for sinhxusing Maclaurin’s series. f(x)=sinhxf( 0 )=sinh0= e^0 −e ...
Maclaurin’s series 73 8.5 Numerical integration using Maclaurin’s series The value of many integrals cannot be determined using ...
74 Higher Engineering Mathematics = [ θ− θ^3 18 + θ^5 600 − θ^7 7 ( 5040 ) +··· ] 1 0 = 1 − 1 18 + 1 600 − 1 7 ( 5040 ) +··· =0. ...
Maclaurin’s series 75 However, a knowledge of series does not help with examples such as lim x→ 1 { x^2 + 3 x− 4 x^2 − 7 x+ 6 } ...
76 Higher Engineering Mathematics Hence lim x→ 0 { x−sinx x−tanx } =− 1 2 Now try the following exercise Exercise 34 Further pro ...
Chapter 9 Solving equations by iterative methods 9.1 Introduction to iterative methods Many equations can only be solved graphic ...
78 Higher Engineering Mathematics Problem 1. Use the method of bisection to find the positive root of the equation 5x^2 + 11 x− ...
Solving equations by iterative methods 79 1.05, correct to 3 significant figure. We therefore stop the iterations here. Thus, co ...
80 Higher Engineering Mathematics Sincef( 1. 5078125 )is negative andf( 1. 5 )is positive, a root lies betweenx= 1 .5078125 andx ...
Solving equations by iterative methods 81 f(x) 2 1 (^01234) x f(x)x 2 f(x) 2 In x 1 2 Figure 9.4 As shown in Problem 2, ...
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