WS7s

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MATH1821 Mathematical Methods for Actuarial Science I Worksheet 7 1. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3. Solution. The volume of the cone is V = 1 3 πx 2 h , where h = 3 + y and x 2 + y 2 = 9. Hence V ( y ) = π 3 (9 - y 2 )( y + 3) for y [0 , 3]. Differentiating, V ( y ) = π (1 - y )(3 + y ). The only critical point (in the domain of V ) is y = 1. As V (1) = 32 π 3 , V (0) = 9 π and V (3) = 0, maximum volume is 32 π 3 (cubic units). 2. Use the Intermediate Value Theorem to show that f ( x ) = x 3 + 2 x - 4 has a root between x = 1 and x = 2. Then use Newton’s Method with x 0 = 1 to find the root to five decimal places. Solution. Let f ( x ) = x 3 + 2 x - 4. Then f (1) = - 1 < 0 and f (2) = 8 > 0. By the Intermediate Value Theorem, the equation f ( x ) = 0 has a solution between 1 and 2. Since f ( x ) = 3 x 2 + 2, the recursive formula in Newton’s Method is x n +1 = x n - f ( x n ) f ( x n ) = x n - x 3 n + 2 x n - 4 3 x 2 n + 2 = 2 x 3 n + 4 3 x 2 n + 2 . With x 0 = 1, we get successively x 1 = 1 . 2, x 2 = 1 . 179746835, x 3 = 1 . 179509058 and x 4 = 1 . 179509024. Hence the root is 1.17951, correct to 5 decimal places. 3. Use the formula sin h + sin 2 h + sin 3 h + · · · + sin
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