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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.14

Further Integration: exam review Exam review: second paper Answers

Section A: short questions

    1. (a) \(6x^{\frac{3}{2}}-28x^{\frac{1}{2}}+c\)
    2. (b) \(y=4e^{x}+6\cos x-3\)
    1. (a) \(-3\ln|5-4x|+c\)
    2. (b) \(-\frac{1}{5}\sin(2-5x)+c\)
    3. (c) \(5\ln\left(x^{2}+6\right)+c\)
    1. (a) \(21\)
    2. (b) \(-9\ln 3\) = \(-9.89\)
    1. (a) \(x=0,\ x=4,\ x=5\)
    2. (b) \(\int_{0}^{4}y\,dx-\int_{4}^{5}y\,dx\) (or \(\int_{0}^{5}|y|\,dx\))
    3. (c) \(32.75\)
    1. (a) AG
    2. (b) \(-\ln(2+\cos x)+c\)
    1. (a) \(53\)
    2. (b) \(18\)
    3. (c) \(a=4\)
    1. (a) \(\frac{248}{75}\) = 3.31
    2. (b) \(-9\ln 6 + 9 \ln 2\) = -9.89
    3. (c) \(-5\ln 6 + 5 \ln 3\) = -3.47
    1. (a) \(-8\ln 2 + 1 + 8 \ln 5\) = 8.33
    2. (b) \(a=3\)
    1. (a) Horizontal stretch, scale factor 2; vertical stretch, scale factor 3; translation of 1 unit down in the \(y\)-direction. Image \((12,8)\)
    2. (b) \(12\)

Section B: long questions

    1. (a) \(8\ln 4\) = \(11.1\)
    2. (b) \(48\pi\) = \(151\)
    3. (c) \(8\ln 4-6\) = \(5.09\)
    4. (d) \(\pi\left(30-16\ln 4\right)\) = \(24.6\)
    1. (a) \(20.25\)
    2. (b) \(40.5\)
    3. (c) \(78\)
    4. (d) \(60.75\)
    5. (e) \(k=4\)
    6. (f) \(20.25\)
    1. (a) \((0,0)\) and \((3,9)\)
    2. (b) \(\frac{9}{2}\) = \(4.50\)
    3. (c) \(\frac{162 \pi}{5}\) = \(102\)
    4. (d) \(\frac{27 \pi}{2}\) = \(42.4\)
    5. (e) \(a=2\)
    1. (a) \(N(t)=6\ln\left(\frac{t^{2}+6}{6}\right)+3\)
    2. (b) \(12.9\)
    3. (c) \(6\ln 7\) = \(11.7\)
    4. (d) \(t=9.80\)
    5. (e) The rate approaches 0, but \(N\) still increases without limit (slowly).
    1. (a) Make \(x^2\) the subject of \(y=\sqrt{x^2+4}\)
    2. (b) \(\frac{32 \pi}{3}\) = 33.5
    3. (c) 33.5 litres
    4. (d) Depth = 3.39 dm
    5. (e) \(4.29\) dm\(^2\)

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