Cole Vaughn
05/24/2023 · Elementary School
incontre grav dos \( 8 x^{2}-3 x y+9 y^{2}-4 \) in
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Step-by-step Solution
Para encontrar o gradiente da função \( f(x, y) = 8x^2 - 3xy + 9y^2 - 4 \), precisamos calcular as derivadas parciais em relação a \( x \) e \( y \).
1. **Derivada parcial em relação a \( x \)**:
\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(8x^2) - \frac{\partial}{\partial x}(3xy) + \frac{\partial}{\partial x}(9y^2) - \frac{\partial}{\partial x}(4)
\]
\[
= 16x - 3y + 0 - 0 = 16x - 3y
\]
2. **Derivada parcial em relação a \( y \)**:
\[
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(8x^2) - \frac{\partial}{\partial y}(3xy) + \frac{\partial}{\partial y}(9y^2) - \frac{\partial}{\partial y}(4)
\]
\[
= 0 - 3x + 18y - 0 = -3x + 18y
\]
Agora, podemos escrever o gradiente \( \nabla f \):
\[
\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (16x - 3y, -3x + 18y)
\]
Portanto, o gradiente da função \( 8x^2 - 3xy + 9y^2 - 4 \) é:
\[
\nabla f = (16x - 3y, -3x + 18y)
\]
Quick Answer
O gradiente da função \( 8x^2 - 3xy + 9y^2 - 4 \) é \( \nabla f = (16x - 3y, -3x + 18y) \).
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