MA 1024 B '01 Practice Exam 2

  1. Compute the two first-order and three distinct second-order partial derivatives of the function $f(x,y) = \sin(xy)+x^3y^4+\exp(x+y)$.

  2. Compute the directional derivative of the function $f(x,y) = xy^2$ at the point $(1,2)$ in the direction of the vector ${\bf v } = {\bf i} + {\bf j}$.

  3. Find the equation for the tangent plane, or best linear approximation, to the function $f(x,y) = \sin(2x)+\sin(3y)$ at the point $x=0$, $y=0$.

  4. Show that the gradient of a function $f(x,y)$ at a particular point $(x_0,y_0)$ is perpendicular to the level curve through this same point.
  5. Find the stationary points of the function $f(x,y) = 3xy-x^3-y^3$.

  6. Find the absolute minimum and maximum values of the function $f(x,y) = 4xy-x^4-2y^2$ on the square domain $-2 \leq x \leq 2$, $-2
\leq y \leq 2$.

  7. Sketch the contours of the function $z
= x^2-y^2$. Include and label in your sketch contours for $z=0$, $z=1$ and $z=-4$. You may restrict your attention to the rectangle $-4
\leq x \leq 4$ and $-4 \leq y \leq 4$.

  8. Convert the expression

    \begin{displaymath}x^2+y^2+2z^2=2 \end{displaymath}

    to cylindrical coordinates and spherical coordinates.

William W. Farr