Brain Gains (Rapid Recall, Jivin' Generation)

  1. A curve passes through the point $P = (x,y,z)$ and has normal vector $(a,b,c)$ with curvature $\kappa$ at that point. State the center of curvatuve.
  2. Give an equation of a plane that passes through the points $P = (1,2,3)$, $(-1,3,0)$, and $(0,4,2)$.
  3. For the function $V = \pi r^2 h$, assume that both $r$ and $h$ depend on a parameter $t$. Then compute $\frac{dV}{dt}$ using implicit differentiation.

Group Problems

  1. For the function $z = x^2y+y^2$, compute $\frac{dz}{dt}$ (assuming that both $x$ and $y$ depend on the parameter $t$).
  2. The computations for the TNB frame for the curve $r(t) = (t,0,t^2)$ get rather ugly really quickly, if we leave everything in terms of $t$. At a single point, we can simplify things, provided we evaluate AFTER all differentiation has occurred. Compute $\vec T(0)$, $\vec N(0)$, $\vec B(0)$, and $\kappa(0)$, as well as state the center of curvature at $t=0$.
  3. Pick an exercise from section 3.3, exercises 113-151. Tackle it together.