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A purpose clause explains why an action is done and so can be considered the motivation behind the verb. It is constructed with 'ut' + a subjunctive verb and is translated in English as '...

Knowing that 1-2sin²(x) = cos(2x), substitute sin²(x) as: sin²(x) = (1-cos(2x))/2Use the linearity of the integral to split it into: Int(1/2) - Int(cos(2x)/2)dxWhich is in...

To begin you must substitute the value of y provided into the first equation. The would change the equation into the following form:x^2+8(x+4)=20. Now expand the bracket so that the equation becomes:x^2+8...

y=x^x, taking natural log of both sides
ln(y)=ln(x^x), using laws of logs
ln(y)=xln(x), using product rule and implicit differentiation
dy/dx 1/y=ln(x)+1
dy/dx=(ln(x)+1)y
dy/dx=(l...

Internal economies of scale are centred around the relationship of a firm increasing its factors of production ie. its inputs and how this affects the average cost of producing goods. The general concept ...