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Mathematic Methods HW Solutions 21

# Mathematic Methods HW Solutions 21 - =-6 ln3 ∂y/∂x = 1...

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Chapter 4 21 9.6 V = 1 / 3 9.7 V = d 3 / (27 abc ) 9.8 (8 / 13 , 12 / 13) 9.9 A = 3 ab 3 / 4 9.10 d = 5 / 2 9.11 d = 6 / 2 9.12 Let legs of right triangle be a and b , height of prism = h ; then a = b , h = ( 2 - 2 ) a . 10.1 d = 1 10.2 4, 2 10.3 2, 14 10.4 d = 1 10.5 d = 1 10.6 d = 2 10.7 1 2 11 10.8 T = 8 10.9 max T = 4 at ( - 1 , 0) 10.10 (a) max T = 1 / 2, min T = - 1 / 2 min T = - 16 5 at ( 1 5 , ± 2 5 6 ) (b) max T = 1, min T = - 1 / 2 (c) max T = 1, min T = - 1 / 2 10.11 max T = 14 at ( - 1 , 0) 10.12 Largest sum = 180 min T = 13 / 2 at (1 / 2 , ± 1) Smallest sum = 3 arc cos 1 3 = 164 . 2 10.13 Largest sum = 3 arc sin(1 / 3) = 105 . 8 , smallest sum = 90 11.1 z = f ( y + 2 x ) + g ( y + 3 x ) 11.2 z = f (5 x - 2 y ) + g (2 x + y ) 11.3 w = ( x 2 - y 2 ) / 4 + F ( x + y ) + G ( x - y ) 11.6 d 2 y dz 2 + dy dz - 5 y = 0 11.10 f = u - Ts h = u + pv g = u + pv - Ts df = - pdv - sdT dh = Tds + vdp dg = vdp - sdT 11.11 H = p ˙ q - L 11.13 (a) ( ∂s/∂v ) T = ( ∂p/∂T ) v (b) ( ∂T/∂p ) s = ( ∂v/∂s ) p (c) ( ∂v/∂T ) p = - ( ∂s/∂p ) T 12.1 sin x 2 x 12.2 ∂s ∂v = 1 - e v v → - 1; ∂s ∂u = e u - 1 u 1 12.3 dz/dx = - sin(cos x ) tan x - sin(sin x ) cot x 12.4 (sin 2) / 2 12.5 ∂u/∂x = - 4 , ∂u/∂y = 2 , ∂y/∂x = 2 12.6 ∂w/∂x
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Unformatted text preview: =-6 / ln3, ∂y/∂x = 1 / 6 12.7 ( ∂u/∂x ) y =-e 4 , ( ∂u/∂y ) x = e 4 / ln 2, ( ∂y/∂x ) u = ln 2 12.8 dx/du = e x 2 12.9 (cos πx + πx sin πx-1) /x 2 12.10 dy/dx = ( e x-1) /x 12.11 3 x 2-2 x 3 + 3 x-6 12.12 (2 x + 1) / ln( x + x 2 )-2 / ln(2 x ) 12.13 0 12.14 π/ (4 y 3 ) 12.16 n = 2, I = 1 4 √ π a-3 / 2 n = 4, I = 1 · 3 8 √ π a-5 / 2 n = 2 m , I = 1 · 3 · 5 ··· (2 m-1) 2 m +1 √ π a-(2 m +1) / 2 13.2 (a) and (b) d = 4 / √ 13 13.3 sec 2 θ 13.4-csc θ cot θ 13.5-6 x , 2 x 2 tan θ sec 2 θ , 4 x tan θ sec 2 θ 13.6 2 r sin 2 θ , 2 r 2 sin θ cos θ , 4 r sin θ cos θ , 0 13.7 5%...
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