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

Derivatives: exam-style practice Problem set B Answers

Skills

    1. (a) a = −0.856, b = 29.9, c = 84.8
    2. (b) Yes: R² is very close to 1
    3. (c) a = −0.856 < 0, so f becomes negative after about x = 37.6: the model fails for large x
    1. (a) h = w + 4
    2. (b) w = 3 m, h = 7 m
    3. (c) 102 m²
    1. (a) h = 280/(πr²)
    2. (b) A = 2πr² + 560/r
    1. (a) \(4\pi r-720r^{-2}\)
    2. (b) r = 3.86 cm
    3. (c) A = 280 cm²
    1. (a) \(2+\frac{864}{x^3}\)
    2. (b) For x > 0, 864/x³ > 0, so d²A/dx² > 0
    3. (c) Minimum at x = 6; A = 108
    1. (a) \(3-3e^{4x}-12xe^{4x}\)
    2. (b) \(-24(1+2x)e^{4x}\)
    3. (c) a = −1/2

In context

    1. (a) a = −1.41, b = 79.3, c = 295
    2. (b) f′(30) = −5.29: in 2025 the model says membership is falling by 5.29 members per year
    3. (c) The data rise by 2 members per year, but the model says they fall: it is not valid at the end of the data
    1. (a) \(\frac{72e^{-0.5n}}{(1+8e^{-0.5n})^2}\) > 0 for all n
    2. (b) 2 hundred birds; 0.889 hundred birds per year
    3. (c) P → 18
    4. (d) Curve starts at (0, 2), always rises and levels off at P = 18

Stretch

    1. (a) h = 480/(πr²)
    2. (b) A = 2πr² + 960/r
    3. (c) \(4\pi r-960r^{-2}\)
    4. (d) r = 4.24 cm; A = 339 cm²
    5. (e) Cube: side 7.83 cm, area 368 cm². The cylinder (minimum 339 cm²) needs less material
    1. (a) x = e^(ln x), so x^(4x) = e^(4x ln x)
    2. (b) y = 4x − 3
    3. (c) x = 1/e

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