Higher Engineering Mathematics

302 DIFFERENTIAL CALCULUS height reached, (d) the velocity with which the missile strikes the ground. [ (a) 100 m/s(b)4s (c) 200 ...

SOME APPLICATIONS OF DIFFERENTIATION 303 G Figure 28.5 (ii) Let dy dx =0 and solve for the values ofx. (iii) Substitute the valu ...

304 DIFFERENTIAL CALCULUS Since the gradient changes from negative to positive,the point (1, 3) is a minimum point. Considering ...

SOME APPLICATIONS OF DIFFERENTIATION 305 G Hence ( 3,− (^1156) ) is a minimum point. Knowing (−2, 9) is a maximum point (i.e. cr ...

306 DIFFERENTIAL CALCULUS y= 3 x^2 − 4 x+ 2 [ Minimum at ( 2 3 , 2 3 )] x=θ(6−θ) [Maximum at (3, 9)] y= 4 x^3 + 3 x^2 − 60 x− ...

SOME APPLICATIONS OF DIFFERENTIATION 307 G dV dx = 240 − 128 x+ 12 x^2 = 0 for a turning point Hence 4(60− 32 x+ 3 x^2 )=0, i.e. ...

308 DIFFERENTIAL CALCULUS From Fig. 28.9, x+ 2 y= 100 (1) Area of rectangle, A=xy (2) Since the maximum area is required, a form ...

SOME APPLICATIONS OF DIFFERENTIATION 309 G Figure 28.11 Substituting into equation (1) gives: V=π ( 144 − h^2 4 ) h= 144 πh− πh^ ...

310 DIFFERENTIAL CALCULUS An electrical voltageEis given by E=(15 sin 50πt+40 cos 50πt) volts, wheretis the time in seconds. De ...

SOME APPLICATIONS OF DIFFERENTIATION 311 G Problem 23. Determine the equations of the tangent and normal to the curvey= x^3 5 at ...

312 DIFFERENTIAL CALCULUS [Obviously, in this case, the exact value of dy may be obtained by evaluatingywhenx= 1 .02, i.e. y=4(1 ...

SOME APPLICATIONS OF DIFFERENTIATION 313 G is the pressure difference between the ends of the tube,ris the radius of the tube,L ...

Differential calculus 29 Differentiation of parametric equations 29.1 Introduction to parametric equations Certain mathematical ...

DIFFERENTIATION OF PARAMETRIC EQUATIONS 315 G differentiation (from Chapter 27): dy dx = dy dθ × dθ dx It may be shown that this ...

316 DIFFERENTIAL CALCULUS (a)x=4(θ−sinθ), hence dx dθ = 4 −4 cosθ=4(1−cosθ) y=4(1−cosθ), hence dy dθ =4 sinθ From equation (1), ...

DIFFERENTIATION OF PARAMETRIC EQUATIONS 317 G From equation (1), dy dx = dy dθ dx dθ = 6 sin^2 θcosθ −6 cos^2 θsinθ =− sinθ cosθ ...

318 DIFFERENTIAL CALCULUS Hence, radius of curvature,ρ= √ √ √ √ [ 1 + ( dy dx ) 2 ] 3 d^2 y dx^2 = √√ √ √ [ 1 + ( 1 t ) 2 ]^3 − ...

G Differential calculus 30 Differentiation of implicit functions 30.1 Implicit functions When an equation can be written in the ...

320 DIFFERENTIAL CALCULUS Now try the following exercise. Exercise 130 Further problems on differen- tiating implicit functions ...

DIFFERENTIATION OF IMPLICIT FUNCTIONS 321 G dz dy = d dy (x^2 )+ d dy (3xcos 3y) = 2 x dx dy + [ (3x)(−3 sin 3y)+( cos 3y) ( 3 d ...