Quiz 4

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Total points: 1

For the dynamical system \begin{align*} \diff{ v }{t} &= - 0.2 v - 3.4 w\\ \diff{ w }{t} &= 1.4 v + 5.0 w, \end{align*} calculate the equilibrium: $(v_e, w_e) = $
.
Classify the equilibrium. It is a/an
.

The solution moves toward the equilibrium for initial conditions along which direction(s)?
(If a single direction, enter a vector. If all directions, enter all. If no directions, enter none.)
Compute the Jacobian, $D\vc{g}$, of the multivariate function $\vc{g} (x,y)$ below, i.e. compute the matrix of partial derivatives. \[ \vc{g}(x,y) = (x^{2} y - 0.6 x - 2, - x^{2} y - 0.4 x ) \]

$D\vc{g}=$
For the two-dimensional linear system \begin{align*} \diff{ p }{t} &= 1.1 p + 1.5 r\\ \diff{ r }{t} &= - 4.8 p - 2.6 r, \end{align*} calculate the equilibrium: $(p_e, r_e) = $
.
Classify the equilibrium. It is a/an
.

The solution moves toward the equilibrium for initial conditions along which direction(s)?
(If a single direction, enter a vector. If all directions, enter all. If no directions, enter none.)
Let $g(z,y) = 10 y z^{2} + 10 y z - 8 z^{2}$. Calculate the partial derivatives of $g$.

$\displaystyle \pdiff{ g }{ z } = $

$\displaystyle \pdiff{ g }{ y } = $

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