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4 marksDifferentiation II

CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2 · Question 1(b)(i)

Given that y = e^{\tan^{-1}(2x)}, where -\frac{1}{2}\pi < \tan^{-1}(2x) < \frac{1}{2}\pi, show that (1 + 4x^2)\frac{\mathrm{d}y}{\mathrm{d}x} = 2y.

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Other parts of this question

  1. 1(a)(i)Determine the initial temperature of the water in the tank.[2 marks]
  2. 1(a)(ii)Determine the temperature at which the water in the tank will eventually stabilize.[2 marks]
  3. 1(a)(iii)Determine the time when the temperature of the water in the tank is 70^\circ\text{ C}.[4 marks]
  4. 1(b)(ii)Hence, show that (1 + 4x^2)^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 4y(1 - 4x).[4 marks]
  5. 1(c)(i)Determine \int \frac{4}{e^x + 1}\,\mathrm{d}x by using the substitution u = e^x.[6 marks]
  6. 1(c)(ii)Determine \int \frac{4}{e^x + 1}\,\mathrm{d}x by first multiplying both the numerator and denominator of the integrand by e^{-x} before integrating.[3 marks]

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