Approximate calculations and modelling
Answers are for parents and teachers.
1
Approximate 1.004⁵⁰ and explain why (1 + x)ᵐ ≈ 1 + mx may be used.
1 + 50 · 0.004 = 1.2; x = 0.004 is small, so the formula is reasonable (the exact value is ≈ 1.221).
2
Approximate √36.18 using f(x) = √x.
x₀ = 36, f′ = 1/12; 6 + 0.18/12 = 6.015.
3
Approximate cos 62° (x₀ = 60°, 2° = π/90 ≈ 0.0349, sin 60° = √3/2 ≈ 0.866).
(cos x)′ = −sin x, so cos 62° ≈ 0.5 − 0.866 · 0.0349 ≈ 0.4698.
4
Bacteria obey N′ = kN with N(0) = 1000 and reach 4000 after 2 hours. How many are there after 5 hours? (Use N = 1000 · 4^(t/2).)
32000
5
By the model T(t) = 20 + 60e^(−0.1t) (°C, t in minutes) find the tea’s initial temperature and its temperature after a very long time. Which equation does it satisfy?
T(0) = 80 °C, and as t → ∞, T → 20 °C (room temperature). T′ = −6e^(−0.1t) = −0.1(T − 20).