# Xxn xxee

A notion of fibre bundle is described, which makes sense in any category with finite inverse limits.An assumption is that some of the structural maps that occur are descent maps: this is the categorical aspect of the notion of glueing objects together out of local data. 2 ln( 3 ) x Dx x π 2 2 1 (3) 3 x Dx x xx =⋅ π π 2 23 3 x xx = π 4.11 ln( – 9) ln ln – 9 – ln 22 x xx x = – 9–9 ln ln xx x x == 29.Step 1: 3 (–1)yx= 3 –1x y= 3 1x y= Step 2: –1 3 () 1f yy= Step 3: –1 3 () 1f xx= Check: –1 33 (() 1 (–1) 1(–1)f fx x x x= = = –1 3 333 ( ( )) [(1 ) –1] ( )f fx x x x= = = 24.

ln 6 = ln (2 · 3) = ln 2 ln 3 = 0.693 1.099 = 1.792 b. By long division, 2 33 21244(21) xxx = − − so 2 2 33 21 2 4 4(21) 331 44 421 xx x dx dx dx dx x xdx x = −− = − ∫∫∫∫ ∫ Let 21ux= − ; then 2du dx= .

0 32vv t= − 0v = when 0 32vt= , that is, when 0 32 v t = . The ball then reaches a height of 22 000 00 2 ( / 32) 16 32 64 32 vvv Hsv v==−= 2 0 0 64 8 v H v H = = 31. () () 1 2 2 2 11 ()() 1 1cos f A f − π π ′ == = ′ b. ax b y cx d = cxy dy = ax b (cy – a)x = b – dy bdy dyb x cy a cy a − − ==− − − 1 () dy b fy cy a − − =− − 1 () dx b fx cx a − − =− − b. [][2] xy xx De xy D = ()()0 xy xx e x Dy y x Dy y = 0 xy xy xe D y ye x D y y = – – xy xy xx xe Dyx Dy ye y = –(1) – – (1) xy xy x xy xy ye y y e y Dy x xe x x e − == = 22. f has a maximum at 1 (1, ) e f is concave up on (2, )∞ and concave down on (,2)−∞ . 1 ln12 ln 4 –1θ = ln12 –1 ln 4 θ= ln12 1 2.7925 ln 4 θ = ≈ 17. 22 2–3 2–3 2 (3 ) 3 ln 3 (2 – 3 ) xx xx =⋅ 2 2–3 (4 – 3) 3 ln3 xx x=⋅ 19.

() () 1 5 27 5 6 4 6 111 ()() 1cos f B f − π π ′ == = ′ 2 7 = c. If bc – ad = 0, then f(x) is either a constant function or undefined. If 1 f f − = , then for all x in the domain we have: 0 ax b dx b cx d cx a − = − (ax b)(cx – a) (dx – b)(cx d) = 0 22 2 ()acx bcaxabdcx − − 2 () 0dbcxbd −−= 222 ()( )( )0ac dc x d a x ab bd − −−= Setting the coefficients equal to 0 gives three requirements: (1) a = –d or c = 0 (2) a = ±d (3) a = –d or b = 0 If a = d, then 1 f f − = requires b = 0 and c = 0, so () ax f xx d = = . [] [4 ] xy xx De D x y = (1 ) 1 xy x x e Dy Dy = 1 xy xy x x e e Dy Dy = –1– x yxy xx e Dy Dy e = 1– –1 –1 xy x xy e Dy e = = 23. f has a point of inflection at 2 2 (2, ) e −3 5 −5 x y 8 362 Section 6.3 Instructor’s Resource Manual 28. 3 1 log ln 3 x x x x x De De e =⋅ 1 0.9102 ln 3 ln 3 x x e e ==≈ Alternate method: 333 log ( log ) log x xx De Dxe e== ln 1 0.9102 ln 3 ln 3 e ==≈ 368 Section 6.4 Instructor’s Resource Manual 20.

To elucidate the energy-saving effect of pumps along with the heat-transfer performance of terminal units in heat pump systems, a comprehensive analysis of an assumed chilled-water circuit at two supply water temperatures under four variable-flow control modes was carried out from the viewpoint of available energy, i.e., exergy.

Subsequently, based on the operating data, the exergy analysis of a heat pump system was carried out to verify the energy-saving effect after variable-frequency transformation of the chilled water pump.