2012 Applications of Differentiation Tutorial Solutions Barely Passed

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  Page 1 of  2 CJC MATHEMATICS DEPARTMENT2012 JC1 H2 MATHEMATICSTOPIC:APPLICATIONS OF DIFFERENTIATION QUESTIONS 1. [2008/SRJC/Promo/7](i) 27)(3)2( 22 =−++  y x y x   0dd1)(3 dd21)2( 0dd1)(6 dd21)2(2 =      −−+      ++=      −−+      ++  x y y x x y y x x y y x x y y x   0dd3dd333dd4dd22 =+−−++++  x y y x y x y x x y y x y x y x    x y x y x y x y x y  x y x y y x y x y x −−=−=−=+−− 74dd4)7( dd0dd7dd4  (ii)(a) Parallel to the  x -axis, 0dd =  x y    x y 4 = ⇒   214127278127)9(3)8( 22222 ±===+=++  x x x x x x x   2214 ±=      ±= ⇒  y   (b)   Parallel to the  y -axis, ∞→  x y dd   x y = ⇒ 771712718927)36(381 27)6(3)9( 222222 ±====+=+  y y y y y y y  7717 ±=      ±= ⇒  x   ________________________________________________ 2.   [2008/IJC/Promo/6](i) Stationary point occurs when   f  ′ (  x ) = 0,  x = −  a and  x = a.    x  ( − a ) −   − a ( − a ) +  f  ′ (  x )+ 0 −   ∴    x = − a (max. point)  x   a −   a   a +  f  ′ (  x ) −  0 + ∴    x = a (min. point) (ii)(a) For  y = f(  x ) to be strictly increasing, f  ′ (  x ) > 0. ⇒    x < − a or  x > a (b) For  y = f(  x ) to concave downwards, f  ′′ (  x ) < 0 ⇒    x < 0____________________________________________   3.   [2009/DHS/Promo/4i] 3 34 r V  π   =   scmt r r t  Ar  Ascmt r t r t r r t V   / 841)4(8 dd8dd4 / 41dd dd)4(416 dd4dd 2222 === ⇒ == ⇒ = ⇒ = π  π  π  π  π  π  π    ___________________________________________ 4.   [2008/IJC/I/6](i) ( ) 21030150021050255  yx yx yx + × =+ == −   ( )( ) ( ) 2222 166430261230180360180(255)3604500540(Shown) Vxyxx xyx xyx xxx xx   = + ×    = + ×= += − += −   k  − a ab  Page 2 of  2 (ii)   45001080 dV  xdx = −  For maximum volume, 0 dV dx =   450010800254.1676  x xor  − ==  
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