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5 marksDifferentiation II

CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2 · Question 2(a)(ii)

Let 4x^2 + 3xy^2 + 7x + 3y = 0.

Show that for f(x, y) = 4x^2 + 3xy^2 + 7x + 3y, 6\frac{\partial f(x, y)}{\partial y} - 10 = -\left[\frac{\partial^2 f(x, y)}{\partial y^2}\right] \left[\frac{\partial^2 f(x, y)}{\partial y \partial x}\right] + \frac{\partial^2 f(x, y)}{\partial x^2}.

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Other parts of this question

  1. 2(a)(i)Use implicit differentiation to show that \frac{dy}{dx} = -\frac{8x + 3y^2 + 7}{3(1 + 2xy)}.[5 marks]
  2. 2(b)(i)Express f(x) in the form a + \frac{b}{9x^2 + 4} where a, b \in \mathbb{R}.[2 marks]
  3. 2(b)(ii)Given that f(x) is symmetric about the y-axis, evaluate \int_{-2}^2 f(x)\,dx.[6 marks]
  4. 2(c)(i)Show that \int h^n \ln h\,dh = \frac{h^{n+1}}{(n+1)^2} [-1 + (n+1)\ln h] + C, where -1 \ne n \in \mathbb{Z} and C \in \mathbb{R}.[5 marks]
  5. 2(c)(ii)Hence, find \int \sin^2 x \cos x \ln(\sin x)\,dx.[2 marks]

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