Estimation quiz!
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07-30-2020, 02:13 PM
Post: #6
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RE: Estimation quiz!
(07-30-2020 10:54 AM)EdS2 Wrote: which is larger, the number of millimetres in a mile or the number of square yards in a square kilometre? We can estimate relative size without multiplying 1 yard = (36 in) (25.4 mm/in) = 914.4 mm → 1 yard ≈ 1 m 1 mile = 1760 yard ≈ 1.760e6 mm // over-estimated 10% 1 km^2 = (1e3 m)^2 ≈ 1e6 yard^2 // under-estimated 20% Since 1.76 > 1.30, we have mile/mm > km^2/yard^2 (07-30-2020 11:27 AM)Paul Dale Wrote: Which is larger \( e^{\pi} \) or \( \pi ^ e \)? e^x = 1 + x + x^2/2! + x^3/3! + ... ≥ 1+x x ≥ ln(1+x) → if x≠0, x > ln(1+x) ln(e^y) = y ln(y^e) = e*(1 + ln(1 + (y/e-1))) ≤ e*(1 + (y/e-1)) = y → if y≠e, ln(y^e) < y ⇒ e^y > y^e → e^pi > pi^e |
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