Continuity, Derivatives, and All That

Transcription

1 Chater Seve Cotiuity, Derivatives, ad All That 7 imits ad Cotiuity et x R ad r > The set B( a; r) { x R : x a < r} is called the oe ball of radius r cetered at x The closed ball of radius r cetered at x is the set B( a; r) { x R : x a r} Now suose D R A oit a D is called a iterior oit of D if there is a oe ball B( a; r) D The collectio of all iterior oits of D is called the iterior of D, ad is usually deoted it D A set U is said to be oe if U it U Suose f :D R, where D R ad suose a R is such that every oe ball cetered at a meets the domai D If y R is such that for every ε >, there is a δ > so that f ( x) y < ε wheever < x a < δ, the we say that y is the limit of f at a This is writte lim f ( x) y, x a ad y is called the limit of f at a Notice that this agrees with our revious defiitios i case ad,, or 3 The usual roerties of limits are relatively easy to establish: lim( f ( x) + g( x)) lim f ( x) + lim g ( x), ad x a x a x a lim af ( x) a lim f ( x) x a x a where D Now we are ready to say what we mea by a cotiuous fuctio f :D R, R Agai this defiitio will ot cotradict our revious lower dimesioal 7

2 defiitios Secifically, we say that f is cotiuous at a D if lim f ( x) f ( a) If f is cotiuous at each oit of its domai D, we say simly that f is cotiuous x a Examle Every liear fuctio is cotiuous To see this, suose f :R R is liear ad a R et ε > Now let M max{ f ( e ), f ( e ), K, f ( e ) } ad let δ The for x such that < x a < δ, we have ε M f ( x) f ( a) f ( x e + x e + K+ x e ) f ( a e + a e + K+ a e ) ( x a ) f ( e ) + ( x a ) f ( e ) + K+ ( x a ) f ( e ) x a f ( e ) + x a f ( e ) + K+ x a f ( e ) ( x a + x a + K+ x a ) M x a M < ε Thus lim f ( x) f ( a) ad so f is cotiuous x a Aother Examle x x et f : R R be defied by f ( x ) f ( x, x ) x + x, et s see about lim f ( x) et x α(, ) The for all α, we have x (, ), for x + x otherwise f ( x ) f ( α, α) α α + α 7

3 Now let x α(, ) ( α, ) It follows that all α, f ( x ) What does this tell us? It tells us that for ay δ >, there are vectors x with < x (, ) < δ such that f ( x ) ad such that f ( x ) This, of course, meas that lim f ( x) does ot x (, ) exist 7 Derivatives et f : D R, where D R, ad let x it D The f is differetiable at x if there is a liear fuctio such that lim [ f ( x ) ( ) ( )] h + h f x h h The liear fuctio is called the derivative of f at x It is usual to idetify the liear fuctio with its matrix reresetatio ad thik of the derivative at a matrix Note that i case, the matrix is simly the matrix whose sole etry is the every day grammar school derivative of f Now, how do fid the derivative of f? Suose f has a derivative at x First, let h te (,, K,, t,, K, ) The f( x, x, K, x + t, K, x ) f ( x, x,, x + t,, x) f ( x + h) f ( x, x,, x + t,, x) K K K K, M f ( x, x, K, x + t, K, x ) ad 73

4 m m m h m m m M m m m m t M m t, t M M m t where x ( x, x, K, x ), etc Now the, h [ f (x + h) f (x ) (h)] f, x,k, x + t,k,x ) f,x,k, x ) m t f, x,k, x + t,k, x ) f, x,k, x ) m t t M f, x,k, x + t,k,x ) f, x,k, x ) m t f,x,k, x + t,k, x ) f, x,k,x ) m t f, x,k, x + t,k, x ) f,x,k,, x ) m t M f,x,k, x + t,k, x ) f,x,k,, x ) m t Meditate o this vector For each comoet, f lim i, x,k,x + t,k, x ) f i,x,k,, x ) t t d ds f i, x,k, s,k, x ) s x This derivative has a ame It is called the artial derivative of f i with resect to the th variable There are may differet otatios for the artial derivatives of a fuctio g( x, x, K, x ) The two most commo are: 74

5 g, ( x, x, K, x) x g ( x, x,, x K ) The requiremet that lim [ f ( x ) ( ) ( )] h + h f x h ow traslates ito h m i f x i, ad, mirabile dictu, we have foud the matrix! Examle et f : R 3x si x R be give by f ( x x ) 3 x + x x Assume f is differetiable ad let s fid the derivative (more recisely, the matrix of the derivative This matrix will, of course, be : m m m m Now f ( x, x ) 3x si x, 3 f ( x, x ) x + x x ad Comute the artial derivatives: f x f x 3si x, 3x + x 75

6 ad f x f x 3x x x cosx The derivative is thus 3si x 3x cosx 3x + x xx We ow kow how to fid the derivative of f at x if we kow the derivative exists; but how do we kow whe there is a derivative? The fuctio f is differetiable at x if the artial derivatives exist ad are cotiuous It should be oted that it is ot sufficiet ust for the artial derivatives to exist Exercises Fid all artial derivatives of the give fuctios: a) f ( x, y) x y 3 b) f ( x, y, z) x yz + zcos( xy) c) g( x, x, x ) x x x + x d) h( x, x, x, x ) x x3 si( e ) x + x 4 Fid the derivative of the liear fuctio whose matrix is What is the derivative a liear fuctio whose matrix is A? 76

7 4 Fid the derivative of R( t) costi + si t + tk 5 Fid the derivative of f ( x, y) x y 3 6 Fid the derivative of x x 3 + e x x f, x, x 3 ) 3 log + x ) x x x The Chai Rule Recall from elemetary oe dimesioal calculus that if a fuctio is differetiable at a oit, it is also cotiuous there The same is true here i the more geeral settig of fuctios f : R R et s see why this is so Suose f is differetiable at a with derivative et h x a The lim f ( x) lim f ( a + h) Now, x a h f ( a + h) f ( a) ( h) f ( a + h) f ( a) ( h) Now look at the limit of this as h : f lim ( a + h ) f ( a ) ( h ) h 77

8 because f is differetiable at a, ad lim ( h ) ( ) because the liear fuctio is h cotiuous Thus lim( f ( a + h) f ( a)), or lim f ( a + h) f ( a), which meas f is h cotiuous at a Next, let s see what the celebrated chai rule looks like i higher dimesios et h f : R R ad g: R R q Suose the derivative of f at a is ad the derivative of g at f ( a ) is M We go o a quest for the derivative of the comositio g o f : R R q at a et r g o f, ad look at r( a + h) r( a) g( f ( a + h)) g( f ( a)) Next, let k f ( a + h) f ( a) The we may write r( a + h) r( a) M( h) g( f ( a + h)) g( f ( a)) M( h) g( f ( a) + k) g( f ( a)) M( k) + M( k) M( h) g( f ( a) + k) g( f ( a)) M( k) + M( k ( h)) Thus, r( a + h) r( a) M( h) g( f ( a) + k) g( f ( a)) M( k) ( ) M( k + h ) Now we are ready to see what haes as h look at the secod term first: ( ) ( ) ( ) ( ) lim M( k h f ) lim M a + h f a h f (lim ( a + h ) f ( a ) M ( h ) ) h h h M( ) sice is the derivative of f at a ad M is liear, ad hece cotiuous Now we eed to see what haes to the term g( f ( a) + k) g( f ( a)) M( k) lim h 78

9 This is a bit tricky Note first that because f is differetiable at a, we kow that k f ( a + h) f ( a) behaves icely as h Next, g( f ( a) + k) g( f ( a)) M( k) k lim h k lim h g( f ( a) + k) g( f ( a)) M( k) k k sice the derivative of g at f ( a ) is M, ad k is well-behaved Fially at last, we have show that r lim ( a + h ) r ( a ) M ( h ) h, which meas the derivative of the comositio r g o f is simly the comositio, or matrix roduct, of the derivatives What could be more leasig from a esthetic oit of view! Examle et f ( t) ( t, + t 3 3 ) ad g( x, x ) ( x x ), ad let r go f First, we shall fid the derivative of r at t usig the Chai Rule The derivative of f is 79

10 t 3, t ad the derivative of g is [ 6( ) 3( ) ] M x x x x At t, 4 ; ad at g f g ( ( )) ( 4, 9) comositio is [ ], M [ ] M [ ] Now for fu, let s fid a exlicit recie for r ad differetiate: 6 3 Thus the derivative of the r( t ) g( f ( t)) g( t, + t ) ( t t ) Thus r'( t) 3( t t ) ( 4t 3t ), ad so r'( ) 3( )(8 ) It is, of course, very comfortig to get the same aswer as before There are several differet otatios for the matrix of the derivative of f : R R at x R The most usual is simly f '( x ) Exercises 7 et g( x, x, x ) ( x x, x x + ) ad f ( x, x ) ( x x si x, x + 3x, x x ) Fid the derivative of g o f at (,-4) 8 et u( x, y, z) ( x + y, xy, x si y, x 3 y ) ad v( r, s, t, q) ( r + s q 3,( r t) e s ) a)which, if either, of the comositio fuctios u o v or vo u is defied? Exlai b)fid the derivative of your aswer to art a) ( x + y ) ( x y ) 9 et f ( x, y) ( e, e ) ad g( x, y) ( x y 3, x + y) 7

11 a)fid the derivative of f b)fid the derivative of g o g at the oit (,-) o f at the oit (,-) c) Fid the derivative of f o f at the oit (,-) d) Fid the derivative of g o g at the oit (,-) Suose r t cos t ad t x 3 y Fid the artial derivatives r x ad r y 74 More Chai Rule Stuff I the everyday cruel world, we seldom comute the derivative of the comositio of two fuctios by exlicitly multilyig the two derivative matrices Suose, as usual, we have r g o f : R R q The the derivative is, as we ow kow, r r r r r r r'( x ) r'( x, x, K, x) x M r r r x We ca thus fid the derivative usig the Chai Rule oly i the very secial case i which the comsite fuctio is real valued Secifically, suose g:r R ad f : R R et r g o f The r is simly a real-valued fuctio of x ( x, x, K, x ) et s use the Chai Rule to fid the artial derivatives 7

12 r r'( x ) r r g y g y f f f g f f f x y M f f f x Thus makes it clear that r g f g f g f y y y Frequetly, egieers ad other malefactors do ot use a differet ame for the comositio g o f, ad simly use the ame g to deote both the comositio g o f ( x, x, K, x ) g( f ( x, x, K, x ), f ( x, x, K, x ), K, f ( x, x, K, x )) ad the fuctio g give by g( y ) g( y, y, K, y ) Sice y f x x x (,, K, ), these same folks also frequetly ust use y to deote the fuctio f The Chai Rule give above the looks eve icer: g g y g y g y y y y Examle Suose g( x, y, z) x y + ye z ad x s + t, y st 3, ad z s + 3 t et us fid the artial derivatives g r ad g We kow that t 7

13 g s g x s + g y y s + g z z s xy() +(x + e z )t 3 + ye z (s) xy +(x + e z )t 3 + sye z Similarly, g t g x t + g y y t + g z z t xy() + (x + e z )3st + ye z (6t) xy + 3(x + e z )st + 6tye z These otatioal shortcuts are fie ad everyoe uses them; you should, however, be aware that it is a ractice sometimes fraught with eril Suose, for istace, you have g( x, y, z) x + y + z, ad x t + z, y t + z, ad z t 3 Now it is ot at all clear what is meat by the symbol g Meditate o this z Exercises Suose g( x, y) f ( x y, y s) Fid g x + g y Suose the temerature T at the oit ( x, y, z ) i sace is give by the fuctio T( x, y, z) x + xyz zy Fid the derivative with resect to t of a article movig alog the curve described by r( t ) costi + si t + 3 tk 73

14 3 Suose the temerature T at the oit ( x, y, z ) i sace is give by the fuctio T( x, y, z) x + y + z A article moves alog the curve described by r( t ) siπti + cos πt + ( t t + ) k Fid the coldest oit o the traectory 4 et r( x, y) f ( x) g( y), ad suose x t ad y t Use the Chai Rule to fid dr dt 74