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IB Mathematics: Applications and Interpretation HL · AHL 2: Further Derivatives · AHL 2.12

Further Derivatives: exam review Exam review Answers

Section A: short questions

    1. (a) \(15x^{\frac{3}{2}} - \frac{2}{\sqrt{x}}\)
    2. (b) \(g'(x)=\frac{20}{x}+3\sec^{2}x\)
    1. (a) \(- \frac{2x}{\sqrt{9 - 2x^{2}}}\)
    2. (b) \(3e^{\sin(3x)} \cos(3x)\)
    3. (c) \(29\)
    1. (a) \(xe^{-3x}(2-3x)\)
    2. (b) \(\frac{e^{x}(2x-3)}{(2x+1)^{3}}\)
    1. (a) \(2e^{2x} - 5\)
    2. (b) \(x=\ln(2)\) (0.693)
    3. (c) \(y=\frac{1}{3}x+1\)
    1. (a) \(f'(x)=(1 - 2x)e^{-2x}\)
    2. (b) \((\frac{1}{2}, \frac{1}{2e})\)
    3. (c) \(f''(x)=(4x - 4)e^{-2x}\)
    4. (d) \(f''(\frac{1}{2})=- \frac{2}{e}<0\); \(x=1\)
    1. (a) \(\frac{dC}{dx}=-\frac{320}{x^{3}}\)
    2. (b) \(-16\) ppm per minute
    3. (c) The sign is negative, so the concentration is decreasing: it falls by 16 ppm each minute as the drone moves away.
    1. (a) \(v=6 t^{2} - 30 t + 24\), \(a=12 t - 30\)
    2. (b) \(t=1\) s and \(t=4\) s
    3. (c) \(v(3)=-12\), \(a(3)=6\): the particle is slowing down
    1. (a) \(v=4e^{-0.5t}\)
    2. (b) \(t=1.96\) s
    3. (c) \(8\) m

Section B: long questions

    1. (a) AG
    2. (b) \(\frac{5}{4 \pi}\) = 0.398 cm/s
    3. (c) \(\frac{5}{3}\) = 1.67 cm²/s
    4. (d) As \(h\) increases the surface is wider, so the same volume per second gives a smaller rise.
    1. (a) \(a=-3,\ b=-9\)
    2. (b) \((-1,10)\) maximum; \((3,-22)\) minimum
    3. (c) \(x<1\)
    4. (d) \(-22<k<10\)
    1. (a) \(h=\frac{330}{\pi r^{2}}\)
    2. (b) AG
    3. (c) \(4 \pi r - \frac{660}{r^{2}}\)
    4. (d) \(r=3.74\) cm
    5. (e) \(\frac{d^{2}A}{dr^{2}}=4 \left(\pi + \frac{330}{r^{3}}\right)>0\)
    6. (f) \(264\) cm²
    1. (a) \(P(0)=100\)
    2. (b) \(1200\)
    3. (c) \(\frac{dP}{dt}=\frac{5280e^{-0.4t}}{\left(1+11e^{-0.4t}\right)^{2}}\)
    4. (d) \(67.2\) deer per year
    5. (e) \(t=5.99\), \(120\) deer per year
    6. (f) \(600\); point of inflexion, concave up before and concave down after
    1. (a) \(a=-6\sin(2t)\)
    2. (b) \(t=0.785\) s
    3. (c) \(3\) m s\(^{-1}\)
    4. (d) \(s=1.5\sin(2t)+1\)
    5. (e) \(1.48\) m
    6. (f) \(7.52\) m

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