21-356 Lecture 3

Since x0 is an interior point there exists b x0 r e

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Unformatted text preview: directional derivatives of f at x0 exist and !f (x0 ) = dfx0 (v) , !v (ii) for every direction v, N # !f !f (x0 ) = (x0 ) vi . !v ! xi i=1 (1) Proof. Since x0 is an interior point, there exists B (x0 , r) $ E . Let v # RN be a direction and x = x0 + tv. Note that for |t| < r, we have that &x % x0 & = &x0 + tv % x0 & = &tv& = |t| &v& = |t| 1 < r and so x0 + tv # B (x0 , r) $ E . Moreover, x " x0 as t " 0 and so, since f is di!erentiable at x0 , f (x0 + tv) % f (x0 ) % T (x0 + tv % x0 ) f (x) % f (x0 ) % T (x % x0 ) = lim t!0 &x % x0 & &x0 + tv % x0 & f (x0 + tv) % f (x0 ) % tT (v) = lim . t!0 |t| 0 = lim x!x0 Since lim t!0 |t| t is bounded by one, it follows that f (x0 + tv) % f (x0 ) % tT (v) |t| f (x0 + tv) % f (x0 ) % tT (v) = lim = 0. t!0 t t |t| But then 0 = lim t!0 f (x0 + tv) % f (x0 ) % tT (v) f (x0 + tv) % f (x0 ) = lim % T (v) , t!0 t t 7 !f which shows that there exists ! v (x0 ) = T (v). !N Part (ii) follows from the linearity of T . Indeed, writing v = i=1 vi ei , by the linearity of T , "N $ N N # # # !f !f (x0 ) = T (v) = T vi ei = vi T (ei ) = (x0 ) vi . !v ! xi i=1 i=1 i=1 Remark 11 If in the previous theorem x0 is not an interior point but for some direction v # RN , the point x0 is an accumulation point of the set E ) L, where L is the line through x0 in the direction v, then as in the first part of the proof !f we can show that there exists the directional derivative ! v (x0 ) and !f (x0 ) = dfx0 (v) . !v 8...
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This document was uploaded on 03/31/2014.

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