1.20: Past-paper question 9

4PM1/1R/June/2025 — Question 9 · 9 marks

Given that f(x)=(1+x2)4\displaystyle f(x) = (1+x^2)^4
(a) show that f′(x)=8x(1+x2)3\displaystyle f'(x) = 8x(1+x^2)^3
(2)
The curve C\displaystyle C has equation y=sin3x(1+x2)4\displaystyle y = \frac{\mathrm{sin} 3x}{(1+x^2)^4}
The point A\displaystyle A on C\displaystyle C has x\displaystyle x coordinate 2π3\displaystyle \frac{2\pi}{3}
(b) Show that the gradient of the normal to C\displaystyle C at A\displaystyle A is
−13(1+(2π3)2)4-\frac{1}{3} \left( 1 + \left( \frac{2\pi}{3} \right)^2 \right)^4
(7)