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Home » The area $ A $ of the circular dome is: - 500apps

The area $ A $ of the circular dome is: - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Question: Define $ L(u) = u - rac{u^3}{6} $ for all real $ u $. If $ n $ is a positive integer, define $ a_n $ by $ a_1 = rac{\pi}{2} $, $ a_{n+1} = L(a_n) $. Find $ \lim_{n o \infty} a_n $.

📌 Solution: The recurrence $ a_{n+1} = a_n - rac{a_n^3}{6} $ resembles the Taylor series for $ rctan(u) $, where $ rac{d}{du} rctan(u) = rac{1}{1 + u^2} $. However, the recurrence is not exact. Assume the limit $ L $ exists. Then $ L = L - rac{L^3}{6} \Rightarrow rac{L^3}{6} = 0 \Rightarrow L = 0 $. To confirm convergence, note $ a_1 = \pi/2 pprox 1.57 > 1 $, and $ a_{n+1} = a_n(1 - rac{a_n^2}{6}) $. Since $ a_1 < \sqrt{6} $, $ a_n $ is decreasing and bounded below by 0. By monotone convergence, $ a_n o 0 $.

📌 Question: Find the center of the hyperbola $ 4x^2 - 12x - 9y^2 + 18y = 27 $.

📌 Solution: Complete the square for $ x $ and $ y $. For $ x $: $ 4x^2 - 12x = 4(x^2 - 3x) = 4\left[(x - rac{3}{2})^2 - rac{9}{4}

📌 ight] = 4(x - rac{3}{2})^2 - 9 $. For $ y $: $ -

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