Chapter Eight. f : R R

Transcription

1 Chapter Eght f : R R 8. Itroducto We shall ow tur our atteto to the very mportat specal case of fuctos that are real, or scalar, valued. These are sometmes called scalar felds. I the very, but mportat, specal subcase whch the dmeso of the doma space s, we ca actually look at the graph of a fucto. Specfcally, suppose f :R R. The collecto S = {( x, x, x ) R 3 : f ( x, x ) = x } s called the graph of f. If f s a 3 3 reasoably ce fucto, the S s what we call a surface. We shall see more of ths later. Let us ow retur to the geeral case of a fucto f : R R. The dervatve of f s just f a row vector f '( x ) = x ad deoted grad f or f. f x L f x. It s frequetly called the gradet of f 8. The Drectoal Dervatve I the applcatos of scalar felds t s of terest to talk of the rate of chage of the fucto a specfed drecto. Suppose, for stace, the fucto T( x, y, z) gves the temperature at pots ( x, y, z ) space, ad we mght wat to kow the rate at whch the temperature chages as we move a specfed drecto. Let f : R R, let a R, ad let u R be a vector such that u =. The the drectoal dervatve of f at a the drecto of the vector u s defed to be d D f dt f t u ( a) = ( a + u). t =0 8.

2 Now that we are experts o the Cha Rule, we kow at oce how to compute such a thg. It s smply d D f dt f t f u ( a) = ( a + u) = u t =0. Example The surface of a mouta s the graph of f ( x, y) = 700 x 5y. I other words, at the pot (x, y), the heght s f (x, y). The postve y-axs pots North, ad, of course, the the postve x-axs pots East. You are o the mouta sde above the pot (, 4) ad beg to walk Southeast. What s the slope of the path at the startg pot? Are you gog uphll or dowhll? (Whch!?). The aswers to these questos call for the drectoal dervatve. We kow we are at the pot a = (, 4 ), but we eed a ut vector u the drecto we are walkg. Ths s, of course, just u = (, ). Next we compute the gradet f ( x, y) = [ x, 0 y]. At the pot a ths becomes f (, 4) = [, 40 ], ad at last we have = f u + 40 = 38. Ths gves us the slope of the path; t s postve so we are gog uphll. Ca you tell whch drecto the path wll be level? Aother Example The temperature space s gve by T( x, y, z) = x y + yz 3. From the pot (,,), whch drecto does the temperature crease most rapdly? We clearly eed the drecto whch the drectoal dervatve s largest. The drectoal dervatve s smply T u = T cosθ, where θ s the agle betwee T ad u. Ayoe ca see that ths wll be largest whe θ = 0. Thus T creases most rapdly 8.

3 the drecto of the gradet of T. Here that drecto s [ xy, x + z 3, 3yz ]. At (,,), ths becomes [,, 3]. Exercses. Fd the dervatve of f ( x, y, z) = xlog z + xy at (,, ) the drecto of the vector [,, ].. Fd the dervatve of f ( x, y, z) = x cos y + 3z 3 xz at (, π, ) the drecto of the vector [ 3, -, ]. xy 3. Fd the drectos whch g( x, y) = x y + e s y creases ad decreases most rapdly from the pot (, 0) The surface of a hll s the graph of the equato z = x x y. You stad o the hll above the pot (5,3) ad pour out a glass of water. I whch drect wll t beg to ru? Expla. 5. The posto of a partcle at tme t s gve by r( t ) = 3( t s t) + tj costk, ad the posto of aother partcle s R( t) = t + ( t 3 + t ) j + stk. At tme t = π, what s the rate of chage of the dstace betwee the two partcles? Are they gettg closer to oe aother, or are they gettg farther apart? (Whch!) Expla. 8.3 Surface Normals 8.3

4 Let f : R 3 R be a fucto ad let c be some costat. Recall that the set S = {( x, y, z) R 3 : f ( x, y, z) = c} s called a level set, or level surface, of the fucto f. Suppose r( t) = x( t) + y( t) j + z( t) k descrbes a curve R 3 that les o the surface S. Ths meas, of course, that f ( r ( t)) = f ( x( t), y( t), z( t)) = c. Now look at the dervatve wth respect to t of ths equato: d dt f ( r ( t )) = f r '( t ) =0. I other words, the gradet of f ad the taget to the curve are perpedcular. Note there was othg specal about our choce of r(t); t s ay curve o the surface. The gradet f s thus perpedcular, or ormal to the surface f ( x, y, z) = c. Example Suppose we wat to fd a equato of the plae taget to the surface x + 3y + z = at the pot (, -, ). For a equato of a plae, we eed a pot a o the plae ad a vector N ormal to the plae. The the equato we seek s smply N ( x a) = 0, where x = ( x, y, z ). I the case at had, we have a pot o the plae: a = (, -, ). Let s fd a ormal vector N. We have just leared that the gradet of f ( x, y, z) = x + 3y + z does the job. f ( x, y, z) = [ x, 6y, 4 z], 8.4

5 ad so N = f (,, ) = [, 68, ]. The taget plae s thus gve by the equato N ( x a) = 0, whch ths case s ( x ) 6( y + ) + 8( z ) = 0. You should ote that the dscusso here dd t deped o the dmeso of the doma. Thus f f : R R, the the set {( x, y ) R : f ( x, y ) = c } s a level curve of f, ad the gradet of f s ormal to such a curve. Combg these results wth what we kow about the drectoal dervatve, we see that at a pot the value of a fucto creases most rapdly a drecto ormal to the level set passg through that pot. O a cotour map of a porto of the Earth s surface, for example, the steepest path s the drecto ormal to the cotour les. Exercses 6. Fd a equato for the plae taget to the surface z = x + y at the pot (,,3). 7. Fd a equato for the plae taget to the surface z = log( x + y ) at the pot ( 0,, 0 ). xz 8. Fd a equato for the plae taget to the surface cosπ x x y + e + yz = 4 at the pot (0,,). 9. Fd a equato of the straght le taget to the curve of tersecto of the surfaces 3 3 x + 3x y + y + 4xy z = 0 ad x + y + z = at the pot (,, 3). 8.5

6 8.4 Maxma ad Mma Let f : R R. A pot a the doma of f s called a local mmum f there s a ope ball B( a; r) cetered at a such that f ( x) f ( a) 0 for all x B( a; r). If f s a ce fucto, the ths meas the drectoal dervatve D f u ( a) 0 for all ut vectors u. I other words, f ( a) u 0. The t must be true that both f ( a) u 0 ad f ( a) u= f( a) ( u) 0. Ths ca be so for every u oly of f ( a) = 0. Thus f has a local mmum at a pot at whch t has a dervatve oly f the dervatve s zero there. You should guess the defto of a local maxmum ad see why t must be true that the gradet s zero at such a pot. Thus f a s a local mmum or a local maxmum of f, ad f f has a dervatve at a, the the dervatve f ( a) = 0. You should be aware of the fact that here, just as Mrs. Turer s elemetary calculus class, the coverse s ot ecessarly true. We may have f ( a) = 0 wthout a beg ether a local mmum or a local maxmum. Example Let us fd all local maxma ad local mma of the fucto f ( x, y) = x + xy+ y + 3x 3y + 4. Medtate o just how should proceed. Ths fucto clearly has a dervatve everywhere, so at ay local maxmum or mmum, ths dervatve, or gradet, must be zero. So let s beg by fdg all pots at whch f ( a) = 0. I other words, we wat (x, y) at whch f x = 0 ad f y = 0: 8.6

7 f x f y = x + y + 3 = 0 = x + y 3 = 0 We are thus faced wth the border-le trval problem of solvg the system of equatos x + y = 3. x + y = 3 There s just oe soluto: ( x, y ) = ( 3, 3 ). Now let us reflect o what we have here. What we have actually foud s all the pots that caot possbly be local mma or maxma. These are all pots except (-3, 3).. All we kow rght ow s that ths pot s the oly possble caddate. Let s fd out what we have by the hammer ad togs method of examg the quatty f ( 3 + x, 3 + y) f ( 33, ) : f ( 3 + x, 3 + y) f ( 33, ) = f ( 3+ x, 3+ y) ( 5) = ( 3+ x) + ( 3+ x)( 3+ y) + ( 3+ y) + 3( 3+ x) 3( 3+ y) + 9 y = x + xy + y = x + + 3y 4 It s therefore clear that f ( 3 + x, 3 + y) f ( 33, ) 0, whch meas that (-3, 3) s a local mmum. Exercses I each of the followg, fd all local maxma ad mma: 8.7

8 0. f ( x, y) = x + 3xy + 3y 6x + 3y 6. f ( x, y) = x + xy + 3x + y + 5. f ( x, y) = xy 5x y + 4x 4 3. f ( x, y) = x + xy 4. f ( x, y) = y x 8.5 Least Squares We shall ext look at some very smple, yet mportat, applcatos whch the locato of a mmum value of a fucto s sought. Suppose we have a set of pots the plae, say ( x, y),( x, y ), K,( x, y), ad we seek the straght le that "best" fts ths collecto of pots. We frst decde what we mea by "best". Let's say we mea the le that mmzes the sum of the squares of the vertcal dstaces from the pots to the le. We ca descrbe all overtcal les the world by meas of two varables, tradtoally called m ad b. Thus every such le has the form y = mx + b. Our quest s thus for the values of m ad b at whch the fucto f ( m, b) = ( mx + b y ) has ts mmum value. Kowg these values wll gve us our le. = We smply apply our vast ad growg kowledge of calculus ad fd where the gradet of f s 0: 8.8

9 f = ( f f = m, b ) 0. Now, f m = = = = = x ( mx + b y ) = [ m x + b x x y ], ad f b = ( mx + b y ) = [ m x + b y ]. = = = We are thus faced wth solvg the x lear system m x + b x = x y = = = m x + b = y = = Medtate suffcetly to covce yourself that there s always exactly oe soluto to ths system, ad cotue medtatg suffcetly to covce yourself that there must be a hoest-to-goodess mmum of the orgal fucto at ths soluto. Let's have a go at a example. Suppose we have the followg table of values: x y

10 The lear system for m ad b s 78m + 76b = m +b = 5.5 Solvg ths system gves us m = ad b =. I other words, the le that best fts the data the sese of least squares s y = 55 4 x Here s a pcture of ths le together wth the data pots: 8.0

11 Looks pretty good! Exercses 5. Here s a table of Köchel umbers versus year of composto for the compostos of W. A. Mozart. Fd the "least squares" straght le approxmato to ths table ad use t to estmate the year whch Mozart's Sfoa Cocertate E-flat major was composed. Köchel Number Year composed [Ths problem s take from Calculus ad Aalytc Geometry (8th Edto), by Thomas & Fey.] 8.

12 6. Fd some data somewhere (The Statstcal Abstract of the Uted States s a good source of terestg data.), fd the least squares lear approxmato to the data, ad say somethg tellget about your results. 8.6 More Maxma ad Mma I real lfe, oe s most lkely terested fdg the places at whch the largest ad smallest values of a fucto f :D R occur, rather tha smply fdg local maxma ad mma. (Here D s a subset of R.). To beg, let's thk a momet about how we ca tell f there s a maxmum or mmum value of f o D. Frst, we suppose that f s cotuous otherwse, aythg ca happe! Next, what propertes of D wll sure the exstece of a bggest ad smallest value of f? The aswer s farly smple. Certaly D must be a closed subset of R ; cosder, for example the fucto f :( 0, ) R gve smply by f ( x) = x, whch has ether a maxmum or a mmum o D = ( 0, ). Havg the doma be closed, however, s ot suffcet to guaratee the exstece of a maxmum ad mmum. Cosder, for example f : R R aga wth f :( 0, ) R gve by f ( x) = x. The doma R s certaly closed, but f has ether a maxmum or a mmum. We eed also to have the doma be bouded. It turs out that for cotuous f, f the doma D s both closed ad bouded, the there must ecessarly be a maxmum ad a mmum value for f o D. Let's thk a momet about what the caddates for such pots are. If the bggest or smallest value of f occurs the teror of D, the surely the pot at whch t occurs s a local maxmum (or mmum). If f has a gradet there, the the gradet must be 0. The pots at whch the largest or smallest values occur must therefore be ether )pots the teror of D at whch the gradet of f vashes, )pots the teror at whch the gradet of f does ot exst, or )pots D but ot the teror of D (that s, pots o the boudary of D). Hark back to Mrs. Turer's thrd grade calculus class. How dd you fd the maxmum value of a fucto f whose doma D s a closed terval [ a, b] R? Recall 8.

13 foud all pots the teror (that s, the ope terval (a,b)) at whch the dervatve vashes. You the smply evaluated f at these pots, evaluated f at ay pots (a,b) at whch there s o dervatve, evaluated f at the two ed pots of the terval ( ths oe dmesoal case, the boudary of D s partcularly smple.), ad the pcked out the bggest ad smallest umbers you computed. The stuato hgher dmesos s a bt more complcated, mostly because the boudary of eve a ce doma D s ot a ce fte set as the case of a terval, but s a fte set. Let's look at a example. Example A flat crcular plate has the shape of the rego {( x, y) R : x + y }. The temperature at the pot ( x, y) o the plate s gve by T( x, y) = x + y x. Our assgmet s to fd the hottest ad coldest pots o the plate. Accordg to our prevous dscusso, caddates for the hottest ad coldest pots are all pots sde the crcular boudary at whch the gradet of T s 0 ad all pots o the boudary. (Note that T has a gradet at all pots sde the crcle.) Frst, let's fd where amog all pots ( x, y ) such that x + y <, the oes at whch T = ( x 4, y) = 0. Ths s easy; t should be clear there s just oe such pot: (, 0 ). Now for the more dffcult part, fdg the caddates o the boudary. Note that the boudary may be descrbed by the vector equato r( t ) = cost + s t j, where 0 t π. The temperature o ths set s the gve by T( t) = T( r ( t)), 0 t π [Here we are abusg the otato, as we have doe before, by usg the same ame for the fucto T( x, y) ad the composto T( r ( t )).] We are ow faced wth the oe dmesoal problem of fdg the maxmum ad mmum values of a ce dfferetable fucto of oe varable o a closed terval. Frst, we kow the edpots of the terval are caddates: t = 0, ad t = π. We have at ths pot added oe more pot to our lst 8.3

14 of caddates: r( 0) = r( π ) = ( 0, ). Now for caddates sde the terval, we seek places at whch the dervatve dt dt = 0. From the Cha Rule, we kow dt dt = T( r( t )) r'( t ) = ( cos t 4, s t ) ( s t,cos t) = cost s t + st. The equato dt dt = 0 ow becomes cost st + s t = 0, s t( cos t + ) = 0 or Thus st = 0, or cos t + = 0. We have, other words, y = 0, or x =. Whe y = 0, the x = or x = ; ad whe x =, the y = 3 or y = 3. Thus our ew caddates are ( 0, ), ( 3 0, ), (-, ), ad (, 3 ). These together wth the oe we have already foud, (, 0 ), make up our etre lst of possbltes for the hottest ad coldest pots o the plate. All we eed do ow s to compute the temperature at each of these pots: T(, ). 0 = = 4 4 T( 0, ) = = 0 T( 0, ) = + = T(, ) = T(, ) = + + = Fally, we have our aswer. The coldest pot s (, 0 ), ad the hottest pots are ( 3, ) 3 ad (, ). 8.4

15 Exercses 7. Fd the maxmum ad mmum value of f ( x, y) = x xy + y + 4 o the closed area the frst quadrat bouded by the tragle formed by the les x = 0, y = 4, ad y = x. 8. Fd the maxmum ad mmum values of f ( x, y) = ( 4 y y )cos x o the closed π π area bouded by the rectagle y 3, x Eve More Maxma ad Mma It should be clear ow that the really troublesome part of fdg maxma ad mma s dealg wth the costraed problem; that s, the problem of fdg the maxma ad mma of a gve fucto o a set of lower dmeso tha the doma of the fucto. I the problems of the prevous secto, we were fortuate that t was easy to fd parametrc represetatos of the these sets; geeral, ths, of course, could be qute dffcult. Let's see what we mght do about ths dffculty. Suppose we are faced wth the problem of fdg the maxmum or mmum value of the fucto f : D R, where D = {( x, y) R : g( x, y) = 0 }, where g s a ce fucto. (I other words, D s a level curve of g.) Suppose r( t ) s a vector descrpto of the curve D. Now the, we are seekg a maxmum or mmum of the fucto F ( t) = f ( r ( t)). At a maxmum or mmum, we must have df dt = 0. (Here g s suffcetly ce to sure that g( x, y) = 0 s a closed curve, ad so there are o edpots to worry about.) The Cha Rule tells us that df dt = f r' = 0. Thus at a maxmum or mmum, the gradet of f must be perpedcular to the taget to g( x, y) = 0. But f f s perpedcular to the taget to the level curve g( x, y) = 0, the t must have the 8.5

16 same drecto as the ormal to ths curve. Ths s just what we eed to kow, for the gradet of g s ormal to ths curve. Thus at a maxmum or mmum, f ad g must "le up". Thus f = λ g, ad there s o eed actually to kow a vector represetato r for g( x, y) = 0. Let's see ths dea acto. Suppose we wsh to fd the largest ad smallest values of f ( x, y) = x + y o the curve x x + y 4y = 0. Here, we may take g( x, y) = x x + y 4 y. The f = x + yj, ad g = ( x ) + ( y 4) j, ad our equato f = λ g becomes x = λ ( x ) y = λ ( y 4) We obta a thrd equato from the requremet that the pot ( x, y) be o the curve g( x, y) = 0. I other words, we eed to fd all solutos to the system of equatos The frst two equatos become x = λ ( x ) y = λ ( y 4) x x + y 4y = 0 x( λ -) = λ y( λ ) = λ Thus x = λ λ ad y = λ λ. (What about the possblty that λ = 0?). The last equato the becomes λ λ 4λ ( λ ) λ + ( λ ) 8λ 0 λ = ; or, λ λ λ( λ ) = 0, λ = 0 We have two solutos: λ = 0 ad λ =. What do you make of the soluto λ = 0? These values of λ gve us two caddates for places at whch extrema occur: x = 0 ad y = 0 ; ad x = ad y = 4. Now the f ( 00, ) = 0, ad f (, 4) = = 0. There 8.6

17 we have them the mmum value s 0 ad t occurs at (0,0); ad the maxmum value s 0, ad t occurs at (,4). Ths method for fdg "costraed" extrema s geerally called the method of Lagrage Multplers. (The varable λ s called a Lagrage multpler.) Exercses 9. Use the method of Lagrage multplers to fd the largest ad smallest values of f ( x, y) = 4x + 3 y o the crcle x + y =. 0. Fd the pots o the ellpse x + y = at whch f ( x, y) = xy has ts extreme values.. Fd the pots o the curve x + xy + y = that are earest to ad farthest from the org. 8.7