Engineering Calculus Notes 245

# Engineering Calculus Notes 245 - 233 3.3. DERIVATIVES →...

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Unformatted text preview: 233 3.3. DERIVATIVES → This formula shows that, if it exists, the aﬃne approximation to f (− ) at x →→ − = − is unique; we call the “linear part” ℓ(△− ) of T f (− ) the → → x x0 x x − → x0 → →→ → derivative or diﬀerential of f (− ) at − = − 0 , and denote it d− 0 f . Note x x x x that this equation can also be interpreted in terms of the derivative at → t = 0 of the composite function f (− (t)): p 1 → → → → → d− 0 f (− ) = lim (f (− 0 + t− ) − f (− 0 )) v x v x x t→0 t d → → [f (− 0 + t− )] . x v = dt t=0 (3.6) For example, if f (x, y ) = x2 − xy and → − = (3, 1) x0 → − = (v , v ) v 12 then, using the limit formula, 1 [f (3 + v1 t, 1 + v2 t) − f (3, 1)] t→0 t 1 = lim (3 + v1 t)2 − (3 + v1 t)(1 + v2 t) − 6 t→0 t 1 2 = lim (9 + 6v1 t + t2 v1 ) − (3 + v1 t + 3v2 t + t2 v1 v2 ) − 6 t→0 t 2 = lim (5v1 − 3v2 ) + t(v1 − v1 v2 ) d(3,1) f ((v1 , v2 )) = lim t→0 = 5v1 − 3v2 or, alternatively, we could use the diﬀerentiation formula: d dt d = dt [f (3 + v1 t, 1 + v2 t)] d(3,1) f ((v1 , v2 )) = t=0 t=0 2 6 + (5v1 − 3v2 )t + (v1 − v1 v2 )t2 2 = (5v1 − 3v2 ) + 2(v1 − v1 v2 ) · 0 = (5v1 − 3v2 ). ...
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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