Higher Engineering Mathematics

282 MATRICES AND DETERMINANTS DI 3 = ∣ ∣ ∣ ∣ ∣ 23 − 26 1 − 587 − 72 − 12 ∣ ∣ ∣ ∣ ∣ =(2)(60−174)−(3)(− 12 +609) +(−26)(2−35) =− 2 ...

THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 283 F Find the values of F 1 ,F 2 andF 3 using determinants. ...

284 MATRICES AND DETERMINANTS 26.4 Solution of simultaneous equations using the Gaussian elimination method Consider the followi ...

THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 285 F From equation (3′′), I 3 = − 89. 308 − 9. 923 =9mA, ...

Assign-07-H8152.tex 17/7/2006 16: 9 Page 286 Complex numbers and Matrices and Determinants Assignment 7 This assignment covers t ...

Differential calculus G 27 Methods of differentiation 27.1 The gradient of a curve If a tangent is drawn at a point P on a curve ...

288 DIFFERENTIAL CALCULUS (v) ifFis the point on the curve (1.01,f(1.01)) then the gradient of chordAF = f(1.01)−f(1) 1. 01 − 1 ...

METHODS OF DIFFERENTIATION 289 G (or,iff(x)=axnthenf′(x)=anxn−^1 ) and is true for all real values ofaandn. For example, ify= 4 ...

290 DIFFERENTIAL CALCULUS (a) Since y= 12 x^3 , a= 12 and n= 3 thus dy dx =(12)(3)x^3 −^1 = 36 x^2 (b)y= 12 x^3 is rewritten in ...

METHODS OF DIFFERENTIATION 291 G (c) Wheny=6ln2xthen dy dx = 6 ( 1 x ) = 6 x Problem 8. Find the gradient of the curve y= 3 x^4 ...

292 DIFFERENTIAL CALCULUS 27.4 Differentiation of a product Wheny=uv, anduandvare both functions ofx, then dy dx =u dv dx +v du ...

METHODS OF DIFFERENTIATION 293 G Now try the following exercise. Exercise 118 Further problems on differen- tiating products In ...

294 DIFFERENTIAL CALCULUS Problem 16. Find the derivative ofy=secax. y=secax= 1 cosax (i.e. a quotient). Letu=1 and v=cosax dy d ...

METHODS OF DIFFERENTIATION 295 G 27.6 Function of a function It is often easier to make a substitution before differentiating. I ...

296 DIFFERENTIAL CALCULUS y= 2 (2t^3 −5)^4 =2(2t^3 −5)−^4. Let u=(2t^3 −5), theny= 2 u−^4 Hence du dt = 6 t^2 and dy du =− 8 u−^ ...

METHODS OF DIFFERENTIATION 297 G d^2 y dx^2 =[(− 6 x)(−3e−^3 x)+(e−^3 x)(−6)] +(−6e−^3 x) = 18 xe−^3 x−6e−^3 x−6e−^3 x i.e. d^2 ...

Differential calculus 28 Some applications of differentiation 28.1 Rates of change If a quantityydepends on and varies with a qu ...

SOME APPLICATIONS OF DIFFERENTIATION 299 G Problem 4. The displacementscm of the end of a stiff spring at timetseconds is given ...

300 DIFFERENTIAL CALCULUS Figure 28.2 Hence the velocity of the car at any instant is given by the gradient of the distance/time ...

SOME APPLICATIONS OF DIFFERENTIATION 301 G Distance x= 1 2 gt^2 = 1 2 (9.8)t^2 = 4. 9 t^2 m Velocity v= dv dt = 9. 8 tm/s and ac ...