Higher Engineering Mathematics

342 DIFFERENTIAL CALCULUS (a) sech−^1 (x−1) (b) tanh−^1 (tanhx) [ (a) − 1 (x−1) √ [x(2−x)] (b) 1 ] (a) cosh−^1 ( t t− 1 ) (b ...

G Differential calculus 34 Partial differentiation 34.1 Introduction to partial derivatives In engineering, it sometimes happens ...

344 DIFFERENTIAL CALCULUS Hence ∂z ∂y = 4 x^3 y− 3. Problem 2. Giveny=4 sin 3xcos 2t, find ∂y ∂x and ∂y ∂t . To find ∂y ∂x ,tis ...

PARTIAL DIFFERENTIATION 345 G t= 2 π √ l g = ( 2 π √ g )√ l= ( 2 π √ g ) l 1 2 Hence ∂t ∂l = ( 2 π √ g ) d dl (l 1 (^2) )= ( 2 π ...

346 DIFFERENTIAL CALCULUS ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (a) ∂k ∂T = AH RT^2 e TS−S RT (b) ∂A ∂T =− kH RT^2 e H−TS RT (c) ∂ ...

PARTIAL DIFFERENTIATION 347 G [ It is noted that ∂^2 z ∂x∂y = ∂^2 z ∂y∂x ] Problem 8. Show that when z=e−tsinθ, (a) ∂^2 z ∂t^2 = ...

348 DIFFERENTIAL CALCULUS z=2lnxy ⎡ ⎣ (a) − 2 x^2 (b) − 2 y^2 (c) 0 (d) 0 ⎤ ⎦ z= (x−y) (x+y) ⎡ ⎢ ⎢ ⎢ ⎣ (a) − 4 y (x+y)^3 (b) ...

G Differential calculus 35 Total differential, rates of change and small changes 35.1 Total differential In Chapter 34, partial ...

350 DIFFERENTIAL CALCULUS Thus dT= V k dp+ p k dV and substituting k= pV T gives: dT= V ( pV T )dp+ p ( pV T )dV i.e. dT= T p dp ...

TOTAL DIFFERENTIAL, RATES OF CHANGE AND SMALL CHANGES 351 G Since the height is increasing at 3 mm/s, i.e. 0.3 cm/s, then dh dt ...

352 DIFFERENTIAL CALCULUS Similarly, ∂b ∂y = y √ (x^2 +y^2 +z^2 ) and ∂b ∂z = z √ (x^2 +y^2 +z^2 ) dx dt =6 mm/s= 0 .6 cm/s, dy ...

TOTAL DIFFERENTIAL, RATES OF CHANGE AND SMALL CHANGES 353 G Using equation (3), the approximate error ink, δk≈ ∂k ∂p δp+ ∂k ∂V δ ...

354 DIFFERENTIAL CALCULUS and ∂t ∂g =−π √ l g^3 (from Problem 6, Chapter 34) δl= 0. 2 100 l= 0. 002 land δg=− 0. 001 g henceδt≈ ...

G Differential calculus 36 Maxima, minima and saddle points for functions of two variables 36.1 Functions of two independent var ...

356 DIFFERENTIAL CALCULUS a minimum if less than at all points in the imme- diate vicinity. Figure 36.3 shows geometrically a ma ...

MAXIMA, MINIMA AND SADDLE POINTS FOR FUNCTIONS OF TWO VARIABLES 357 G (iii) solve the simultaneous equations ∂z ∂x =0 and ∂z ∂y ...

358 DIFFERENTIAL CALCULUS y 1 (^2) z = 4 z = 9 z = 16 12 x z = 1 Figure 36.8 Problem 2. Find the stationary points of the surfac ...

MAXIMA, MINIMA AND SADDLE POINTS FOR FUNCTIONS OF TWO VARIABLES 359 G (iv) ∂^2 z ∂x^2 = 6 x, ∂^2 z ∂y^2 = 6 yand ∂^2 z ∂x∂y = ∂ ...

360 DIFFERENTIAL CALCULUS Wheny=0 in equation (1), 4x^3 − 16 x= 0 i.e. 4 x(x^2 −4)= 0 from which,x=0orx=± 2 The co-ordinates of ...

MAXIMA, MINIMA AND SADDLE POINTS FOR FUNCTIONS OF TWO VARIABLES 361 G h − 2 j − 4 z =^0 c d 2 a e S g f b − 2 x y 4 i 2 z =^9 z ...