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IB Mathematics: Applications and Interpretation HL · AHL 2: Further Derivatives · AHL 2.11

Derivatives: exam-style practice Problem set A Answers

Skills

    1. (a) a = −0.650, b = 44.6, c = −21.4
    2. (b) Yes: R² is close to 1
    3. (c) c = −21.4: f(0) is negative, which is impossible for a number of users
    1. (a) d = 2l − 1
    2. (b) l = 4 m, d = 7 m
    3. (c) 144 m²
    1. (a) h = 350/(πr²)
    2. (b) A = 2πr² + 700/r
    1. (a) \(4\pi r-1200r^{-2}\)
    2. (b) r = 4.57 cm
    3. (c) A = 394 cm²
    1. (a) \(2+\frac{256}{x^3}\)
    2. (b) For x > 0, 256/x³ > 0, so d²A/dx² > 0
    3. (c) Minimum at x = 4; A = 48
    1. (a) \(10-2e^{3x}-6xe^{3x}\)
    2. (b) \(-6(2+3x)e^{3x}\)
    3. (c) a = −2/3

In context

    1. (a) a = −0.0410, b = 6.49, c = −2.90
    2. (b) f′(60) = 1.57: in 2020 the colony is growing by 1.57 hundred seals per year
    3. (c) The data rise by 0.4 hundred per year, so the model overestimates the growth: it is not valid for later years
    1. (a) \(\frac{18e^{-0.3n}}{(1+4e^{-0.3n})^2}\) > 0 for all n
    2. (b) 3 thousand users; 0.72 thousand users per week
    3. (c) P → 15
    4. (d) Curve starts at (0, 3), always rises and levels off at P = 15

Stretch

    1. (a) h = 640/(πr²)
    2. (b) A = 2πr² + 1280/r
    3. (c) \(4\pi r-1280r^{-2}\)
    4. (d) r = 4.67 cm; A = 411 cm²
    5. (e) Cube: side 8.62 cm, area 446 cm². The cylinder (minimum 411 cm²) needs less material
    1. (a) x = e^(ln x), so x^(2x) = e^(2x ln x)
    2. (b) y = 2x − 1
    3. (c) x = 1/e

jmaths.xyz · AI HL · AHL 2.11 Derivatives: exam-style practice