MUSIC THEORY Music 16
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This 2 page Class Notes was uploaded by Shad Aufderhar on Saturday September 12, 2015. The Class Notes belongs to Music 16 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/201837/music-16-university-of-california-irvine in Music at University of California - Irvine.
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Date Created: 09/12/15
LECTURE 16 21 Critical Points Properties and Applications 211 Absolute Maxima and Minima De nition De nition Suppose the domain of a function f includes a closed interval lab A point c is said to be 0 an absolute maximum of f on ab if fc Z for all z in lab 0 an absolute minimum of f on ab if fc g for all z in lab Note Absolute maxima and minima of a function on an open interval and on an arbitrary set of points are de ned analogously eg see absolute mazima in SampM Examples 0 on the interval 047r the function sinz has two absolute maxima and two absolute minima nd them 0 on the interval 12 the function fltzgti 171 for 0 zlt1 7 1 for 1 g I g 2 has absolute maxima at the point z 0 and at all points in the interval 12 on the zaxis but no absolute minima 212 Finite Differences De nition Suppose a function f is de ned in a neighborhood of a point 6 Exercise This exercise is an ef cient way to clarify the limit formulas for the rightsided and leftsided deriva tives Let h be a small positive number such that the closed interval 6 7 hc h is contained in the domain of f Write down the slope of the line passing through 0 the points 6 and C h fc o the points 6 and c 7 h fc 7 De nition For a small It gt 0 the expressions fC h HQ 11 and f C 7 f C i h h are called respectively the right7sia ea and left7sia ea nite ali e39rences of f at c with step Another term for these is a 7ight7sia ea left7sia ea fc h7a i e39rence Note Finite differences have the following geometric interpretation A rightsided nite f c hdifference is the slope of the line passing through the points CfC 6hfch Similarly a leftsided fc hdifference is the slope of the line passing through the points CfC 6 hfC 0 Note that for both differences It is positive 213 The Derivative at an Extremum Analysis by the lst Derivative Theorem Maxima of Di 39erentiable Functions Suppose a function is differentiable on the open interval lab and suppose c is a point in lab at which f attains a a maximum on lab ie fc 2 for all z in lab 24 THEN f c 0 Veri cation To see this consider the rightsided nite difference f6 h 0 f lt25 where h is positive and small enough that C h lies in la b By 24 we have fc 1117 m S 0 26 Letting h tend to 0 from the right and taking the limit we obtain by Nonpositivity Lecture 5 section 462 that NC S 0 27 Carrying out a similar procedure for the leftsided nite difference fc ee my b0 11 we obtain NC 2 0 28 The inequalities 27 and 28 must hold simultaneously and this is possible only if f C 0 Corollary Minima of Di 39erentiable Functions Suppose a function is differentiable on the open interval lab and suppose c is a point in lab at which f attains a minimum on lab ie fc g for all z in lab 29 THEN f c 0 Veri cation To see this introduce the function Then f has a maximum at 6 hence satis es f c 0 as we veri ed above It follows that f c 0 Terminology Extrema Absolute maxima and minima are collectively called absolute extrema How to Picture the Results in this Section Drop a horizontal bar onto a 2D hill77 The bar will hit the highest point of the hill How to Remember the Results in this Section At an absolute extremum the tangent is horizontal
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