{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mthsc206-fall-2010-notes-15.5

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}