Brain Gains (Rapid Recall, Jivin' Generation)

  • Let $z = f(x,y) = x^2-y$. Construct a contour plot (using the outputs $0,-1,4$), and then construct a surface plot.
  • Let $z = f(x,y) = |x|-2$.
    • Construct a contour plot that includes the level curves corresponding to $z = -1$, $z = 0$, and $z=4$.
    • Construct a surface plot.
    • Compute the gradient.
    • In your contour plot, add the gradient $\vec \nabla f(P)$ at the points $P = (2,0)$, $P = (-2,0)$, $P = (2,4)$, $P = (6,1)$, and $P = (-6,-4)$.
  • Let $f(x,y) = e^{2x}\cos(3y)+3x-7y^2$. Compute $f_x$ and $\frac{\partial f}{\partial y}$.

Group Problems

  1. Consider the function $z = f(x,y)=4-y^2$.
    • Compute $\vec \nabla f$.
    • Construct a 2D contour plot by hand. So pick several values for $z$ and plot the resulting curves. If you end up with lots of horizontal lines in the $xy$-plane, you're doing this correctly. Write the height on each horizontal line you draw.
    • Construct a 3D surface plot by hand.
    • Remember that the gradient is a vector field. At a few points in your contour plot, add the gradient vector.
    • Check your work with software.
  2. Repeat the previous exercise using $f(x,y) = x-y^2$.
  3. Pick your own function $f(x,y)$, and use the Mathematica notebook above to compare the level curve plot with the gradient field plot. Repeat with several functions, looking for relationships between the contour plot and gradient field plot.