Higher Engineering Mathematics

42 NUMBER AND ALGEBRA Hyperbolic functions may be evaluated easiest using a calculator. Many scientific notation calcul- ators a ...

HYPERBOLIC FUNCTIONS 43 A A telegraph wire hangs so that its shape is described byy=50 ch x 50 . Evaluate, correct to 4 signif ...

44 NUMBER AND ALGEBRA − 3 − 2 − 1 01 2 3 1 y = tanh x x y − 3 − 2 − 10 123 y = coth x y = coth x 2 3 y x − 1 − 2 − 3 (a) (b) 1 − ...

HYPERBOLIC FUNCTIONS 45 A Table 5.1 shows some trigonometric identities and their corresponding hyperbolic identities. Problem 6 ...

46 NUMBER AND ALGEBRA Left hand side (L.H.S.) = 1 +2sh^2 x= 1 + 2 ( ex−e−x 2 ) 2 = 1 + 2 ( e^2 x−2exe−x+e−^2 x 4 ) = 1 + e^2 x− ...

HYPERBOLIC FUNCTIONS 47 A 5.4 Solving equations involving hyperbolic functions Equations of the formachx+bshx=c, wherea, bandcar ...

48 NUMBER AND ALGEBRA (b) Wheny= 54 .30, 54. 30 =40 ch x 40 , from which ch x 40 = 54. 30 40 = 1. 3575 Following the above proce ...

HYPERBOLIC FUNCTIONS 49 A Let x=1, then ch 1= 1 + 12 2 × 1 + 14 4 × 3 × 2 × 1 + 16 6 × 5 × 4 × 3 × 2 × 1 + ··· = 1 + 0. 5 + 0. 0 ...

Assign-01-H8152.tex 23/6/2006 15: 5 Page 50 Number and Algebra Assignment 1 This assignment covers the material contained in Cha ...

A Number and Algebra 6 Arithmetic and geometric progressions 6.1 Arithmetic progressions When a sequence has a constant differen ...

52 NUMBER AND ALGEBRA Hence if then’th term is 22 then:a+(n−1)d= 22 i.e. 2^12 +(n−1) ( (^112) ) = 22 (n−1) ( (^112) ) = 22 − 212 ...

ARITHMETIC AND GEOMETRIC PROGRESSIONS 53 A 125 − 80 = 5 d^2 45 = 5 d^2 from which,d^2 = 45 5 =9. Henced= √ 9 =±3. The three numb ...

54 NUMBER AND ALGEBRA Insert four terms between 5 and 22.5 to form an arithmetic progression. [8.5, 12, 15.5, 19] The first, te ...

ARITHMETIC AND GEOMETRIC PROGRESSIONS 55 A 6.5 Worked problems on geometric progressions Problem 10. Determine the tenth term of ...

56 NUMBER AND ALGEBRA Find the sum of the first 7 terms of the series 2, 5, 12^12 ,...(correct to 4 significant figures) [812.5 ...

ARITHMETIC AND GEOMETRIC PROGRESSIONS 57 A Let the GP ofnterms be given bya,ar,ar^2 ,..., arn−^1. The first terma=50 rev/min The ...

Number and Algebra 7 The binomial series 7.1 Pascal’s triangle Abinomial expressionis one which contains two terms connected by ...

THE BINOMIAL SERIES 59 A Using Pascal’s triangle method: (a+x)^5 =a^5 + 5 a^4 x+ 10 a^3 x^2 + 10 a^2 x^3 +··· Hence (2p− 3 q)^5 ...

60 NUMBER AND ALGEBRA Problem 4. Expand ( c− 1 c ) 5 using the bino- mial series. ( c− 1 c ) 5 =c^5 + 5 c^4 ( − 1 c ) + (5)(4) ( ...

THE BINOMIAL SERIES 61 A i.e. (0.97)^6 =0.8330, correct to 4 significant figures Problem 9. Determine the value of (3.039)^4 , c ...