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2024 TMJC H1 Q4

2024 TMJC H1 Q4

9 marks

The curve \(C\) has equation \(y = \frac{3}{2}x^3 + \frac{1}{2} + \mathrm{e}^{1-x^2}\).

  1. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\). Hence, find the equation of the tangent to \(C\) at the point \(P\) where \(x = 1\). Give your answer in the form \(y = mx + c\), where \(m\) and \(c\) are exact real values.[5]
  2. The tangent to \(C\) at \(P\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the length of \(AB\).[4]

Solution:

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Answer:(i) \(\frac{\mathrm dy}{\mathrm dx}=\frac92x^2-2xe^{1-x^2}\); tangent \(y=\frac52x+\frac12\). (ii) \(AB=\frac{\sqrt{29}}{10}\approx0.539\).

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