NMAI054 -- 11. přednáška (22.12.2010) -- Derivace

Funkce spojité na intervalu

Definice   Vnitřními body intervalu $ J $ rozumíme ty body z $ J $, které nejsou krajními. Množinu těchto bodů nazýváme vnitřek~$ J $ ($ \operatorname{int}J $).
Definice   Nechť $ f $ je funkce a $ J $ je interval. Řekneme, že $ f $ je spojitá na $ J $, jestliže je spojitá ve všech vnitřních bodech $ J $. Je-li počáteční bod $ J $ prvkem $ J $, tak požadujeme i spojitost zprava v tomto bodě a je-li koncový bod $ J $ prvkem $ J $, tak požadujeme i spojitost zleva v tomto bodě.
Věta  (Darbouxova vlastnost spojité funkce)   Nechť $ f $ je spojitá na intervalu $ [a,b] $ a platí $ f(a)<f(b) $. Pak pro každé $ y\in(f(a),f(b)) $ existuje $ x\in (a,b) $ tak, že $ f(x)=y $.
Poznámka   Předchozí věta vypadá nenápadně, ale je velmi užitečná. Pomocí ní bychom například ze spojitosti funkce $ x^n $ mohli snadno odvodit existenci odmocniny z každého kladného čísla, což na začátku semestru byla nelehká věta. Obecněji, máme-li spojitou funkci $ f $, pro kterou $ f(a)<0<f(b) $, tak existuje kořen $ x \in (a,b) $ (tj. bod splňující $ f(x)=0 $). Můžeme ho nalézt např. metodou půlení intervalů, pro $ c=(a+b)/2 $ zkontrolujeme, zda $ f(c)=0 $ (pak jsme hotovi), $ f(c)>0 $ (pak položíme $ b := c $), nebo $ f(c)<0 $ (pak $ a := c $). Tento postup je poměrně efektivní, každým krokem získáme novou binární číslici. Hledáním ještě efektivnějších (tj. rychlejších) metod se zabývá tzv. numerická matematika).

Derivace funkce

Dá se říci, že pořádná analýza začíná až tehdy, když se naučíme derivovat. Budeme pak umět mnohem lépe zkoumat chování funkcí (hledat extrémy, zkoumat kde je funkce rostoucí a kde klesající), atd. Následující definice možná vypadá děsivě, ale jedná se vlastně o přirozený pojem. Pokud např. $ f(x) $ označuje polohu bodu na přímce v čase $ x $ (a pokud je $ f $ rostoucí), tak $ (f(a+h)-f(a))/h $ je průměrná rychlost mezi časy $ a $ a $ a+h $. Limitu (pokud existuje) těchto průměrných rychlostí můžeme nazvat okamžitou rychlostí v čase $ a $, obecně derivací v bodě $ a $.

Definice   Nechť $ f $ je reálná funkce a $ a\in\R $. Pak derivací $ f $ v bodě $ a $ budeme rozumět
$$   f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}; $$

derivací $ f $ v bodě $ a $ zprava budeme rozumět

$$   f_+'(a)=\lim_{h\to 0_+}\frac{f(a+h)-f(a)}{h}; $$

derivací $ f $ v bodě $ a $ zleva budeme rozumět

$$   f_-'(a)=\lim_{h\to 0_-}\frac{f(a+h)-f(a)}{h}. $$
Věta  (vztah derivace a spojitosti)   Nechť má funkce $ f $ v bodě $ a\in\R $ derivaci $ f'(a)\in\R $. Pak je $ f $ v bodě $ a $ spojitá.
Věta  (aritmetika derivací)   Nechť $ f'(a) $ a $ g'(a) $ existují.
  1. $ (f+g)'(a)=f'(a)+g'(a) $, MLPSS
  2. Nechť je $ g $ spojitá v $ a $ pak $ (fg)'(a)=f'(a)g(a)+f(a)g'(a) $, MLPSS.
  3. Nechť je $ g $ spojitá v $ a $ a $ g(a)\neq 0 $, pak $ \Bigl(\frac{f}{g}\Bigr)'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{g^2(a)} $, MLPSS.

(MLPSS = Má-li pravá strana smysl.)

(Důkaz jen pro součet a součin.)

Věta  (derivace složené funkce)   Nechť $ f $ má derivaci v bodě $ y_0 $, $ g $ má derivaci v $ x_0 $ a je v $ x_0 $ spojitá a $ y_0=g(x_0) $. Pak
$$   (f\circ g)'(x_0)=f'(y_0) g'(x_0)= f'(g(x_0)) g'(x_0), $$

je-li výraz vpravo definován.

(Důkaz byl až na další přednášce, jen za předpokladu $ g'(x_0) \ne 0 $.)

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