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Nusret Balcı TMUA Paper 1 · Q14

Paper 1CalculusGeometry

Let mm be a real constant such that 0≤m≤10\le m\le1. We wish to approximate the curve y=x2y=x^2 with the line y=mxy=mx on the interval [0,1][0,1]. Let E(m)E(m) be the total area of the region bounded between the curve y=x2y=x^2 and the line y=mxy=mx, for 0≤x≤10\le x\le1.

E(m)=∫01∣x2−mx∣ dxE(m)=\int_0^1|x^2-mx|\,\mathrm{d}x

Find the value of mm for which E(m)E(m) is a minimum. Diagram for question 14

A12\frac12
B23\frac23
C12\frac1{\sqrt2}
D13\frac1{\sqrt3}
E34\frac34
F32\frac{\sqrt3}{2}

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