Brain Gains (Rapid Recall, Jivin' Generation)

  • For the function $f(x,y) = \sin(3xy^2)+5x^3$, compute the gradient $\vec \nabla f(x,y)$ and the total derivative $Df(x,y)$.
  • For the vector field $\vec F(x,y) = (2x+3y, 4xy^3)$, compute the differential (as a linear combination of partial derivative), state $\frac{\partial \vec F}{\partial y}$, and then state the total derivative $D\vec F(x,y)$.
  • For the function $f(x,y) = x^2+4y^2$, graph the level curves that passes through the points $(0,1)$ and $(0,2)$.
  • To your plot above, add the gradient at the points $(0,1)$ and $(0,2)$, as well as a few other points along the curves you drew.

Group Problems

  1. Let $f(x,y,z) = 9x-4yz^3+3xz - y^2$.
    • Compute $f_x$ and $\frac{\partial f}{\partial z}$.
    • State $\vec \nabla f(x,y,z)$ and $Df(x,y,z)$.
  2. Let $f(x,y) = x^2-9$.
    • Construct a contour plot of $f$. (The contour corresponding to $f=0$ is a great start. Would $f=5$ or $f=-5$ be simpler to add to your plot? You get to pick values for the output that help. )
    • At various points $P$ in your plot, add the gradient $\vec \nabla f(P)$.
    • Construct a surface plot of $f$.
  3. Let $f(x,y) = 4-4x^2-y^2$.
    • Construct a contour plot of $f$. (Contours corresponding to $f=0$ and $f=4$ are a good start. )
    • At various points $P$ in your plot, add the gradient $\vec f(P)$.
    • Construct a surface plot of $f$.