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HW5sol - t minutes as V = 5000 ³ 1-t 40 ´ 2 ≤ t ≤...

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M408C, Homework 5 — Solution Instructor: Guoliang Wu 1. Find the derivative of the function y = ax + p x 2 + b 2 - 2 , where a and b are constants. Solution. Use Chain Rule, y 0 = - 2 ax + p x 2 + b 2 - 3 a + 1 2 x 2 + b 2 2 x = - 2 ax + p x 2 + b 2 - 3 a + x x 2 + b 2 2. Find y 00 by implicit differentiation x 4 + y 4 = 1 . Solution. Taking the first derivative, 4 x 3 + 4 y 3 y 0 = 0 y 0 = - x 3 y 3 Taking one more derivative using the quotient rule, y 00 = - 3 x 2 y 3 - x 3 3 y 2 y 0 y 6 = - 3 x 2 y - 3 + 3 x 3 y - 4 ( - x 3 y - 3 ) = - 3 x 2 y - 3 - 3 x 6 y - 7 3. If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after
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Unformatted text preview: t minutes as V = 5000 ³ 1-t 40 ´ 2 , ≤ t ≤ 40 (1) Find the rate at which water is draining from the tank after 10 minutes. (2) At what time is the water flowing out the fastest? The slowest? 1 Solution. (1) The rate at which water is draining from the tank after t minutes is V = 10000 ± 1-t 40 ² (-1 / 40) = 25 4 ( t-40) . (2) Since V is linear, at t = 0 the water is flowing out the fastest, and at t = 40 the slowest. 2...
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