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Unformatted text preview: Math 31A 2010.02.02 MATH 31A DISCUSSION JED YANG Applications of the Derivative 1. Related Rates 1.1. Exercise 3.9.44. A wheel of radius r is centred at the origin. As it rotates, the rod of length L attached at the point P drives a piston back and forth in a straight line. Let x be the distance from the origin to the point Q at the end of the rod. (a) Use the Pythagorean Theorem to show that L 2 = ( x- r cos θ ) 2 + r 2 sin 2 θ. (b) Differentiate part (a) with resepct to t to prove that 2( x- r cos θ ) parenleftbigg dx dt + r sin θ dθ dt parenrightbigg + 2 r 2 sin θ cos θ dθ dt = 0 . (c) Calculate the speed of the piston when θ = π 2 , assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 re volutions per minute. Solution. Parts (a) and (b) are straightforward. 4 revolutions per minute means dθ dt = 4 · 2 π per minute. From part (a), we get 30 2 = x 2 + 10 2 , so x = 20 √ 2....
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This note was uploaded on 09/19/2011 for the course MATH 31A taught by Professor Jonathanrogawski during the Winter '07 term at UCLA.
- Winter '07