Higher Engineering Mathematics, Sixth Edition
282 Higher Engineering Mathematics Hence p×( 2 r× 3 q)=( 4 i+j− 2 k) ×(− 24 i− 42 j− 12 k) = ∣ ∣ ∣ ∣∣ ∣ ∣ ijk 41 − 2 − 24 − 42 − ...
Scalar and vector products 283 Now try the following exercise Exercise 113 Further problemson vector products In problems 1 to 4 ...
284 Higher Engineering Mathematics magnitude, direction and sense), thenAP=λb,whereλ is a scalar quantity. Hence, from above, r= ...
Scalar and vector products 285 isparallel tothevector2i+ 7 j− 4 k. Determine the point on the line corresponding toλ=2in the res ...
Revision Test 8 This Revision Test covers the material contained in Chapters 24 to 26.The marks for each question are shown in b ...
Chapter 27 Methods of differentiation 27.1 Introduction to calculus Calculusis a branch of mathematics involving or lead- ing to ...
288 Higher Engineering Mathematics 0 1 1.5 2 3 2 4 6 8 10 f(x) x A D C B f(x) 5 x 2 Figure 27.3 (iii) the gradient of chordAD = ...
Methods of differentiation 289 Substituting(x+δx)forxgives f(x+δx)=(x+δx)^2 =x^2 + 2 xδx+δx^2 , hence f′(x)=limit δx→ 0 { (x^2 + ...
290 Higher Engineering Mathematics Thestandard derivativessummarized below may be proved theoretically and are true for all real ...
Methods of differentiation 291 Thus dy dx =( 5 )( 4 )x^4 −^1 +( 4 )( 1 )x^1 −^1 − 1 2 (− 2 )x−^2 −^1 +( 1 ) ( − 1 2 ) x− 1 2 −^1 ...
292 Higher Engineering Mathematics (a) 4ln9x (b) ex−e−x 2 (c) 1 − √ x x ⎡ ⎢ ⎢ ⎣ (a) 4 x (b) ex+e−x 2 (c) − 1 x^2 + 1 2 √ x^3 ⎤ ...
Methods of differentiation 293 Hence dy dx =(x^3 cos3x) ( 1 x ) +(lnx)[− 3 x^3 sin3x + 3 x^2 cos3x] =x^2 cos3x+ 3 x^2 lnx(cos3x− ...
294 Higher Engineering Mathematics Note that the differential coefficient isnotobtained by merely differentiating each term in t ...
Methods of differentiation 295 2. 2cos3x x^3 [ − 6 x^4 (xsin3x+cos3x) ] 3. 2 x x^2 + 1 [ 2 ( 1 −x^2 ) (x^2 + 1 )^2 ] 4. √ x cosx ...
296 Higher Engineering Mathematics Using the function of a function rule, dy dx = dy du × du dx = ( 1 2 √ u ) ( 6 x+ 4 )= 3 x+ 2 ...
Methods of differentiation 297 Problem 25. Ify=cosx−sinx,evaluatex,in the range 0≤x≤ π 2 ,when d^2 y dx^2 is zero. Since y=cosx− ...
298 Higher Engineering Mathematics Evaluatef′′(θ )whenθ=0given f(θ )=2sec3θ. [18] Show that the differential equation d^2 y dx^ ...
Chapter 28 Some applications of differentiation 28.1 Rates of change If a quantityydepends on and varies with a quantity xthen t ...
300 Higher Engineering Mathematics The rate of change of temperature is dθ dt Since θ=θ 0 e−kt then dθ dt =(θ 0 )(−k)e−kt=−kθ 0 ...
Some applications of differentiation 301 x t Time Distance Figure 28.1 x t B A Time Distance Figure 28.2 for the distancexis kno ...
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