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

Further Derivatives: exam review Exam review: second paper Answers

Section A: short questions

    1. (a) \(35x^{\frac{5}{2}} - 9 \sqrt{x}\)
    2. (b) \(g'(x)=\frac{21}{x}+5\sec^{2}x\)
    1. (a) \(\frac{10x}{3 \left(5x^{2} + 7\right)^{\frac{2}{3}}}\)
    2. (b) \(- 2e^{\cos(2x)} \sin(2x)\)
    3. (c) \(57\)
    1. (a) \(3x^{2}\ln(2x-1)+\frac{2x^{3}}{2x-1}\)
    2. (b) \(\frac{e^{x}(3x-8)}{(3x-2)^{3}}\)
    1. (a) \(3e^{3x} - 7\)
    2. (b) \(x=\frac{2 \ln(2)}{3}\) (0.462)
    3. (c) \(y=\frac{1}{4}x+1\)
    1. (a) \(f'(t)=(10 - 5t)e^{-0.5t}\)
    2. (b) \((2, \frac{20}{e})\)
    3. (c) \(f''(t)=(\frac{5t}{2} - 10)e^{-0.5t}\)
    4. (d) \(f''(2)=- \frac{5}{e}<0\); \(t=4\)
    1. (a) \(\frac{dI}{dx}=-\frac{500}{x^{3}}\)
    2. (b) \(-2.40\) lux per second
    3. (c) The sign is negative, so the light intensity at the moth is decreasing: it falls by 2.4 lux each second as the moth flies away.
    1. (a) \(v=6 t^{2} - 42 t + 60\), \(a=12 t - 42\)
    2. (b) \(t=2\) s and \(t=5\) s
    3. (c) \(v(6)=24\), \(a(6)=30\): the particle is speeding up
    1. (a) \(v=3e^{-0.25t}\)
    2. (b) \(t=4.82\) s
    3. (c) \(12\) m

Section B: long questions

    1. (a) AG
    2. (b) \(\frac{1}{2 \pi}\) = 0.159 m/min
    3. (c) \(\frac{8}{5}\) = 1.60 m²/min
    4. (d) As \(h\) increases the surface is wider, so the same volume per minute gives a smaller rise.
    1. (a) \(a=-6,\ b=-15\)
    2. (b) \((-1,18)\) maximum; \((5,-90)\) minimum
    3. (c) \(x<2\)
    4. (d) \(-90<k<18\)
    1. (a) \(h=\frac{1000}{\pi r^{2}}\)
    2. (b) AG
    3. (c) \(4 \pi r - \frac{2000}{r^{2}}\)
    4. (d) \(r=5.42\) cm
    5. (e) \(\frac{d^{2}A}{dr^{2}}=4 \left(\pi + \frac{1000}{r^{3}}\right)>0\)
    6. (f) \(554\) cm²
    1. (a) \(P(0)=300\)
    2. (b) \(4500\)
    3. (c) \(\frac{dP}{dt}=\frac{44100e^{-0.7t}}{\left(1+14e^{-0.7t}\right)^{2}}\)
    4. (d) \(733\) members per month
    5. (e) \(t=3.77\), \(788\) members per month
    6. (f) \(2250\); point of inflexion, concave up before and concave down after
    1. (a) \(a=6\cos(1.5t)\)
    2. (b) \(t=2.09\) s
    3. (c) \(4\) m s\(^{-1}\)
    4. (d) \(s=\frac{14}{3}-\frac{8}{3}\cos(1.5t)\)
    5. (e) \(1.74\) m
    6. (f) \(12.4\) m

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