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Pocetni priklady

• Ukazte, ze rovnice {: alt="$e^{xy} = cos(x+y)+y$" type="image/"} urcuje v jistem okoli bodu {: alt="$(0,0)$" type="image/"} implicitne zadanou funkci promenne {: alt="$x$" type="image/"}. Spoctete prvni a druhou derivaci teto funkce v bode 0.

• Spoctete Tayloruv polynom stupne 6 se stredem v bode 0 funkce {: alt="$f(x) = cos(x^2cosx) - cos(x^2)$" type="image/"}. S jeho pomoci spoctete limitu {: alt="$\lim_{x\to 0} \cfrac{cos(x^2cosx) - cos(x^2)}{x^6}$" type="image/"}.
  Teoreticke priklady

• Dokazte z definice Riemannova integralu: je-li {: alt="$f$" type="image/"} riemannovsky integrovatelna na {: alt="$[0,1]$" type="image/"} a {: alt="$g = f$" type="image/"} na ![$(0,1]$](http://latex.codecogs.com/gif.latex?$(0,1]$){: alt="$(0,1]$" type="image/"}, potom {: alt="$g$" type="image/"} je riemannovsky integrovatelna na {: alt="$[0,1]$" type="image/"} a {: alt="$\int_{0}^{1} f= \int_{0}^{1} g$" type="image/"}.

• Necht {: alt="$f,g:\mathbb{R}^3 \rightarrow \mathbb{R}$" type="image/"} maji obe totalni diferencial v bode {: alt="$a$" type="image/"} a necht {: alt="$\Delta f(a) = \Delta g(a) = (0,0,0)$" type="image/"}. Plati, ze i funkce {: alt="$h = f \cdot g$" type="image/"} ma totalni diferencial {: alt="$L$" type="image/"} v bode {: alt="$a$" type="image/"} a plati {: alt="[latex]" type="image/"}$\Delta h(a) = (0,0,0)$?