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7.[10marks]A rectangular box with an open top is to have a volume of 48 m3.The length of the baseis to be twice the width.The material for the base costs $8 per square metre.Material for the sideswill cost $6 per square metre.Find the cost of producing the cheapest such box.5
8.[8marks]Use a Riemann sum with a regular partition to determine(x+ 3)2dx.6R
9.[7marks]LetRdenote the region bounded between the curvesy=x2andy=px.Find thevolume of the solid obtained by rotatingRabout the linex= 4.-2246-2-112346
10.[5marks]Use a substitution to evaluate1xpx+ 3dx.11.[7marks]Use integration by parts to evaluate2R1tan°1px°1dx.6R7
13.[6marks]Show that the two-variable functionu= sin (x°at) + ln (x+at)satis°es thewave equationutt=a2uxx(whereais a constant).14.[8marks]Find all critical points and determine the relative extrema (if any) off(x; y) =x3°3xy+ 3y2°9y.Use the Second Derivative Test to justify your conclusions.9