N2019 P1 Q7

N2019 P1 Q7

Junior College 2
8 marks
Free

A curve \(C\) has equation \(y=x\mathrm{e}^{-x}\).

  1. Find the equations of the tangents to \(C\) at the points where \(x=1\) and \(x=-1\).[6]
  2. Find the acute angle between these tangents.[2]

Default solution

  1. \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{e}^{-x}(1-x)\)
    \(x=1:\quad y=\mathrm{e}^{-1},\quad m=0\Rightarrow y=\mathrm{e}^{-1}\)
    \(x=-1:\quad y=-\mathrm{e},\quad m=2\mathrm{e}\)
    \(y+\mathrm{e}=2\mathrm{e}(x+1)\Rightarrow y=2\mathrm{e}x+\mathrm{e}\)
  2. \(\tan\theta=2\mathrm{e}\) (one tangent is horizontal).
    \(\theta=\tan^{-1}(2\mathrm{e})=79.6^\circ\) (1 d.p.).
Answer:(i) \(y=\mathrm{e}^{-1}\), \(y=2\mathrm{e}x+\mathrm{e}\). (ii) \(79.6^\circ\).

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