Higher Engineering Mathematics, Sixth Edition
342 Higher Engineering Mathematics Problem 13. Determine d dx [ cosh−^1 √ (x^2 + 1 ) ] Ify=cosh−^1 f(x), dy dx = f′(x) √ [f(x)]^ ...
Differentiation of inverse trigonometric and hyperbolic functions 343 Hence d dx [coth−^1 (sinx)]= cosx [1−(sinx)^2 ] = cosx cos ...
344 Higher Engineering Mathematics (a) cosech−^1 x 4 (b) 1 2 cosech−^14 x [ (a) − 4 x √ (x^2 + 16 ) (b) − 1 2 x √ ( 16 x^2 + 1 ...
Chapter 34 Partial differentiation 72.1 Introduction to inequalities derivatives In engineering, it sometimes happens that the v ...
346 Higher Engineering Mathematics (b) To find ∂z ∂y ,xis kept constant. Sincez=( 5 x^4 )+( 2 x^3 )y^2 − 3 y then, ∂z ∂y =( 5 x^ ...
Partial differentiation 347 To find ∂t ∂l ,gis kept constant. t= 2 π √ l g = ( 2 π √ g ) √ l= ( 2 π √ g ) l 1 2 Hence ∂t ∂l = ( ...
348 Higher Engineering Mathematics In a thermodynamic system,k=Ae TS−H RT , whereR,kandAare constants. Find(a) ∂k ∂T (b) ∂A ...
Partial differentiation 349 (c) ∂^2 z ∂x∂y = ∂ ∂x ( ∂z ∂y ) = ∂ ∂x ( 12 x^2 y^2 + 14 y)= 24 xy^2 (d) ∂^2 z ∂y∂x = ∂ ∂y ( ∂z ∂x ) ...
350 Higher Engineering Mathematics z=2lnxy ⎡ ⎢ ⎣ (a) − 2 x^2 (b) − 2 y^2 (c) 0 (d) 0 ⎤ ⎥ ⎦ z= (x−y) (x+y) ⎡ ⎢ ⎢ ⎢ ⎣ (a) − 4 ...
Chapter 35 Total differential, rates of change and small changes 35.1 Total differential In Chapter 34, partial differentiation ...
352 Higher Engineering Mathematics Since pV=kT,k= pV T Hence dp= ( pV T ) V dT− ( pV T ) T V^2 dV i.e. dp= p T dT− p V dV (b) To ...
Total differential, rates of change and small changes 353 Hence the rate of change of z, dz dt =75.14units/s, correct to 4 signi ...
354 Higher Engineering Mathematics Using equation (2), the rate of change of diagonalbis given by: db dt = ∂b ∂x dx dt + ∂b ∂y d ...
Total differential, rates of change and small changes 355 Problem 8. Pressurepand volumeVof a gas are connected by the equationp ...
356 Higher Engineering Mathematics Using equation (3), the approximate change int, δt≈ ∂t ∂l δl+ ∂t ∂g δg Sincet= 2 π √ l g , ∂t ...
Chapter 36 Maxima, minima and saddle points for functions of two variables 36.1 Functions of two independent variables If a rela ...
358 Higher Engineering Mathematics f(x,y),andthenf(x,y)calculated for each, a large number of lines such asPP′can be constructed ...
Maxima, minima and saddle points for functions of two variables 359 36.3 Procedure to determine maxima, minima and saddle points ...
360 Higher Engineering Mathematics z (^12) 1 o x y Figure 36.7 A contour map forz=(x− 1 )^2 +(y− 2 )^2 is shown in Fig.36.8.Thev ...
Maxima, minima and saddle points for functions of two variables 361 and substituting in equation (2) gives: − 6 x+ 3 ( 1 2 x^2 ) ...
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