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mthsc206-fall-2010-notes-15.5

mthsc206-fall-2010-notes-15.5 - 15.5 The Chain Rule Warm-Up...

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15.5 The Chain Rule Warm-Up Problem: The temperature (in C) at a point ( x, y ) on a metal plate in the xy -plane is T ( x, y ) = x 3 + 2 y 2 + x. Assume that distance is measured in centimeters. Find the rate at which temperature changes with respect to distance if we start at the point (1 , 2) and move 1. to the right and parallel to the x -axis 2. upward and parallel to the y -axis. Chain Rule (Version 1): Let z = f ( x, y ) be a di ff erentiable function of x and y , with x = x ( t ) and y = y ( t ), where x and y are di ff erentiable functions of t . Then z is a di ff erentiable function of t and dz dt = f x dx dt + f y dy dt . Proof: Since f is a di ff erentiable function of x and y , we know that there exist functions 1 and 2 such that Δ z = f x Δ x + f y Δ y + 1 Δ x + 2 Δ y, with lim ( Δ x, Δ y ) (0 , 0) 1 = 0 = lim ( Δ x, Δ y ) (0 , 0) 2 . Thus we have dz dt = lim Δ t 0 f x · Δ x + f y · Δ y + 1 Δ x + 2 Δ y Δ t = lim Δ t 0 f x Δ x Δ t + f y Δ y Δ t + 1 Δ x Δ t + 2 Δ y Δ t = f x dx dt + f y dy
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