Para resolver a questão, vamos analisar as partes (a) e (b) separadamente. ### Parte (a) Temos a função \( g(u, v) = f(u - 2v, v + 2u) \). Para calcular \( \frac{\partial g}{\partial u} \), usamos a regra da cadeia: \[ \frac{\partial g}{\partial u} = \frac{\partial f}{\partial x} \cdot \frac{\partial (u - 2v)}{\partial u} + \frac{\partial f}{\partial y} \cdot \frac{\partial (v + 2u)}{\partial u} \] Calculando as derivadas parciais: - \( \frac{\partial (u - 2v)}{\partial u} = 1 \) - \( \frac{\partial (v + 2u)}{\partial u} = 2 \) Substituindo na fórmula: \[ \frac{\partial g}{\partial u} = \frac{\partial f}{\partial x} \cdot 1 + \frac{\partial f}{\partial y} \cdot 2 \] Agora, substituímos os valores dados: \[ \frac{\partial g}{\partial u}(1, 0) = -2 \cdot 1 + (-1) \cdot 2 = -2 - 2 = -4 \] ### Parte (b) Para calcular \( \frac{\partial^2 g}{\partial u^2} \), novamente usamos a regra da cadeia, mas agora precisamos calcular a derivada de \( \frac{\partial g}{\partial u} \): \[ \frac{\partial^2 g}{\partial u^2} = \frac{\partial}{\partial u} \left( \frac{\partial g}{\partial u} \right) \] Usando a regra da cadeia novamente: \[ \frac{\partial^2 g}{\partial u^2} = \frac{\partial^2 f}{\partial x^2} \cdot \left( \frac{\partial (u - 2v)}{\partial u} \right)^2 + \frac{\partial f}{\partial x} \cdot \frac{\partial^2 (u - 2v)}{\partial u^2} + \frac{\partial^2 f}{\partial x \partial y} \cdot \frac{\partial (u - 2v)}{\partial u} \cdot \frac{\partial (v + 2u)}{\partial u} + \frac{\partial^2 f}{\partial y^2} \cdot \left( \frac{\partial (v + 2u)}{\partial u} \right)^2 \] Substituindo as derivadas: - \( \frac{\partial^2 (u - 2v)}{\partial u^2} = 0 \) - \( \frac{\partial^2 (v + 2u)}{\partial u^2} = 0 \) Assim, a expressão se simplifica para: \[ \frac{\partial^2 g}{\partial u^2} = \frac{\partial^2 f}{\partial x^2} \cdot 1^2 + 0 + \frac{\partial^2 f}{\partial x \partial y} \cdot 1 \cdot 2 + \frac{\partial^2 f}{\partial y^2} \cdot 2^2 \] Substituindo os valores dados: \[ \frac{\partial^2 g}{\partial u^2}(1, 0) = 0 + (-1) \cdot 2 + 1 \cdot 4 = -2 + 4 = 2 \] ### Resumo das Respostas (a) \( \frac{\partial g}{\partial u}(1, 0) = -4 \) (b) \( \frac{\partial^2 g}{\partial u^2}(1, 0) = 2 \)