3 years ago

Rewrite x^2+12x+42 in the form (x+b)^2 +c

Mathematics

1 answer:

Rzqust [24]3 years ago

6 0

This is completing the square:

(x^2+12x+36)+42-36

Answer: (x+6)^2 +6

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Determine whether or not the vector field is conservative. if it is, find a function f such that f = ∇f. if the vector field is

amm1812

$\dfrac{\partial f}{\partial x}=3y^2z^3\implies f(x,y,z)=3xy^2z^3+g(y,z)$

$\dfrac{\partial f}{\partial y}=6xyz^3+\dfrac{\partial g}{\partial y}=6xyz^3$
$\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

$\dfrac{\partial f}{\partial z}=9xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}=9xy^2z^2$
$\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

$\implies f(x,y,z)=3xy^2z^3+C$

The potential function exists, so $\mathbf f$ is indeed conservative.

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4 years ago

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