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

Further Integration: exam review Exam review Answers

Section A: short questions

    1. (a) \(4x^{\frac{3}{2}}-20x^{\frac{1}{2}}+c\)
    2. (b) \(y=6e^{x}-3\sin x-2\)
    1. (a) \(- \frac{5}{2}\ln|3-2x|+c\)
    2. (b) \(\frac{1}{3}\cos(4-3x)+c\)
    3. (c) \(2\ln\left(x^{3}+4\right)+c\)
    1. (a) \(16\)
    2. (b) \(-6\ln 4\) = \(-8.32\)
    1. (a) \(x=0,\ x=2,\ x=5\)
    2. (b) \(\int_{0}^{2}y\,dx-\int_{2}^{5}y\,dx\) (or \(\int_{0}^{5}|y|\,dx\))
    3. (c) \(21.1\)
    1. (a) AG
    2. (b) \(\ln|1+\sin x|+c\)
    1. (a) \(60\)
    2. (b) \(24\)
    3. (c) \(a=3\)
    1. (a) \(\frac{8}{3}\) = 2.67
    2. (b) \(-12\ln 2\) = -8.32
    3. (c) \(-6\ln 5 + 6 \ln 2\) = -5.5
    1. (a) \(2 + 6 \ln 3\) = 8.59
    2. (b) \(a=2\)
    1. (a) Horizontal stretch, scale factor 3; vertical stretch, scale factor 2; translation of 1 unit up in the \(y\)-direction. Image \((12,-3)\)
    2. (b) \(42\)

Section B: long questions

    1. (a) \(6\ln 3\) = \(6.59\)
    2. (b) \(12\pi\) = \(37.7\)
    3. (c) \(6\ln 3-4\) = \(2.59\)
    4. (d) \(\pi\left(32-24\ln 3\right)\) = \(17.7\)
    1. (a) \(4\)
    2. (b) \(8\)
    3. (c) \(16\)
    4. (d) \(8\)
    5. (e) \(k=4\)
    6. (f) \(4\)
    1. (a) \((0,0)\) and \((2,4)\)
    2. (b) \(\frac{4}{3}\) = \(1.33\)
    3. (c) \(\frac{64 \pi}{15}\) = \(13.4\)
    4. (d) \(\frac{8 \pi}{3}\) = \(8.38\)
    5. (e) \(a=3\)
    1. (a) \(C(t)=3\ln\left(\frac{t^{2}+4}{4}\right)+2\)
    2. (b) \(8.91\)
    3. (c) \(3\ln 5\) = \(4.83\)
    4. (d) \(t=7.32\)
    5. (e) The rate approaches 0, but \(C\) still increases without limit (slowly).
    1. (a) Make \(x^2\) the subject of \(y=\sqrt{x^2+1}\)
    2. (b) \(\frac{20 \pi}{3}\) = 20.9
    3. (c) 20.9 litres
    4. (d) Depth = 2.46 dm
    5. (e) \(3.36\) dm\(^2\)

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