Math Insight

Quiz 4

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Total points: 8

  1. Compute the Jacobian, $D\vc{g}$, of the multivariate function $\vc{g} (x,y)$ below, i.e. compute the matrix of partial derivatives. \[ \vc{g}(x,y) = (- x^{2} + 3 x y^{2} - 0.2, - 2 x^{2} y - y ) \]



    $D\vc{g}=$




  2. Let $f(s,t) = 3 s^{2} t - 9 s - 9 t^{2}$. Calculate the partial derivatives of $f$.

    $\displaystyle \pdiff{ f }{ s } = $

    $\displaystyle \pdiff{ f }{ t } = $

  3. For the two-dimensional linear system \begin{align*} \diff{ a }{t} &= 0.3 a - 1.2 c\\ \diff{ c }{t} &= 1.8 a + 4.3 c, \end{align*} calculate the equilibrium: $(a_e, c_e) = $
    .

    Classify the equilibrium. It is a/an
    .

    After a long time and for most initial conditions, the solution will be moving in the direction:
    . (Enter a vector if the solution will be heading parallel to that vector. Enter none if the solution will not be heading along a single direction.)

    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.)

  4. For the dynamical system \begin{align*} \diff{ x }{t} &= - 4.7 x - 0.5 y\\ \diff{ y }{t} &= 56.8 x + 5.0 y, \end{align*} calculate the equilibrium: $(x_e, y_e) = $
    .

    Classify the equilibrium. It is a/an
    .

    After a long time and for most initial conditions, the solution will be moving in the direction:
    . (Enter a vector if the solution will be heading parallel to that vector. Enter none if the solution will not be heading along a single direction.)

    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.)

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Math 2241, Spring 2016