Higher Engineering Mathematics
62 NUMBER AND ALGEBRA = 1 +(−3)(2x)+ (−3)(−4) 2! (2x)^2 + (−3)(−4)(−5) 3! (2x)^3 +··· = 1 − 6 x+ 24 x^2 − 80 x^3 +··· (b) The ex ...
THE BINOMIAL SERIES 63 A Problem 14. Simplify √ (^3) (1− 3 x)√(1+x) ( 1 + x 2 ) 3 given that powers ofxabove the first may be ne ...
64 NUMBER AND ALGEBRA 5. 1 √ 1 + 3 x ⎡ ⎢ ⎢ ⎣ ( 1 − 3 2 x+ 27 8 x^2 − 135 16 x^3 +··· ) |x|< 1 3 ⎤ ⎥ ⎥ ⎦ Expand (2+ 3 x)−^6 t ...
THE BINOMIAL SERIES 65 A New values of b and l are (1+ 0 .035)b and (1− 0 .025)lrespectively. New second moment of area = 1 12 [ ...
66 NUMBER AND ALGEBRA The radius of a cone is increased by 2.7% and its height reduced by 0.9%. Determine the approximate perce ...
A Number and Algebra 8 Maclaurin’s series 8.1 Introduction Some mathematical functions may be represented as power series, conta ...
68 NUMBER AND ALGEBRA on; thus cosxmeets this condition. However, iff(x)=lnx,f′(0)=^10 =∞, thus lnxdoes not meet this condition. ...
MACLAURIN’S SERIES 69 A fiv(x)= − 6 (1+x)^4 fiv(0)= − 6 (1+0)^4 =− 6 fv(x)= 24 (1+x)^5 fv(0)= 24 (1+0)^5 = 24 Substituting these ...
70 NUMBER AND ALGEBRA Problem 9. Develop a series for sinhxusing Maclaurin’s series. f(x)=sinhxf(0)=sinh 0= e^0 −e−^0 2 = 0 f′(x ...
MACLAURIN’S SERIES 71 A Use Maclaurin’s series to determine the expansion of (3+ 2 t)^4. [ 81 + 216 t+ 216 t^2 + 96 t^3 + 16 t^ ...
72 NUMBER AND ALGEBRA = [ θ− θ^3 18 + θ^5 600 − θ^7 7(5040) +··· ] 1 0 = 1 − 1 18 + 1 600 − 1 7(5040) +··· =0.946, correct to 3 ...
MACLAURIN’S SERIES 73 A L’Hopital’s rulewill enable us to determine such limits when the differential coefficients of the numer- ...
74 NUMBER AND ALGEBRA Hence lim x→ 0 { x−sinx x−tanx } =− 1 2 Now try the following exercise. Exercise 38 Further problems on li ...
Assign-02-H8152.tex 23/6/2006 15: 6 Page 75 A Number and Algebra Assignment 2 This assignment covers the material contained in C ...
Number and Algebra 9 Solving equations by iterative methods 9.1 Introduction to iterative methods Many equations can only be sol ...
SOLVING EQUATIONS BY ITERATIVE METHODS 77 A − 3 − 2 − 1 1 2 20 f(x) f(x) = 5 x^2 + 11 x− 17 10 − 10 − (^17) − 20 0 Figure 9.2 Th ...
78 NUMBER AND ALGEBRA Sincef(1) is positive andf(2) is negative, a root lies betweenx=1 andx=2. A sketch off(x) = x+ 3 −ex, i.e. ...
SOLVING EQUATIONS BY ITERATIVE METHODS 79 A Bisecting this interval gives 1. 50390625 + 1. 5078125 2 i.e. 1. 505859375 Hence f(1 ...
80 NUMBER AND ALGEBRA As shown in Problem 2, a table of values is produced to reduce space. x 1 x 2 x 3 = x 1 +x 2 2 f(x 3 ) 0.1 ...
SOLVING EQUATIONS BY ITERATIVE METHODS 81 A These results show that the negative root lies between 0 and −1, since the value off ...
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