Question

Considere uma funçao y=f(x) tal que f(0) = -1 e dy/dx =-2x³+9x ² -1 , encontre a funçao y=f(x) e calcule o valor de falor de f ( -2)

Answer 1

Resposta:

$\boxed{\mathtt{- 31}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{\frac{dy}{dx} = - 2x^3 + 9x^2 - 1} \\\\ \mathsf{dy = (- 2x^3 + 9x^2 - 1) \, dx} \\\\ \mathsf{\int dy = \int - 2x^3 + 9x^2 - 1 \, dx} \\\\ \mathsf{y = \frac{- x^4}{2} + 3x^3 - x + C} \\\\ \boxed{\mathsf{f(x) = - \frac{x^4}{2} + 3x^3 - x + C}}$

Determinemos o valor da constante C.

$\\ \displaystyle \mathsf{f(0) = 0 + 0 + 0 + C} \\\\ \boxed{\mathsf{C = - 1}}$

Portanto,

$\\ \displaystyle \mathsf{f(x) = - \frac{x^4}{2} + 3x^3 - x - 1} \\\\ \mathsf{f(- 2) = - \frac{(- 2)^4}{2} + 3 \cdot (- 2)^3 - (- 2) - 1} \\\\ \mathsf{f(- 2) = - \frac{16}{2} + 3 \cdot (- 8) + 2 - 1} \\\\ \mathsf{f(- 2) = - 8 - 24 + 1} \\\\ \boxed{\boxed{\mathsf{f(- 2) = - 31}}}$