Brain Gains (Rapid Recall, Jivin' Generation)

  • For the curve $r(t) = (t,\cos t, \sin t)$, compute $\ds \int_0^a \left|\frac{d\vec r}{dt}\right|dt$.
  • For the same curve $r(t) = (t,\cos t, \sin t)$, compute the arc length parameter $\ds s(t) = \int_0^t \left|\frac{d\vec r}{d\tau}\right|d\tau$.
  • Compute $\ds \frac{d}{dt}\left(\int_0^t x^3 dx\right)$.
  • Compute $\ds \frac{d}{dt}\left(\int_0^t \sqrt{1+\tau^2+\cos(\tau)} d\tau\right)$.
  • Compute $\ds \frac{ds}{dt}$, so compute $\ds \frac{d}{dt}\left(\int_0^t \left|\frac{d\vec r}{d\tau}\right|d\tau\right)$.
  • For the curve $\vec r(t) = (t,t^2)$, compute both $\dfrac{d\vec r}{dt}$ and $\dfrac{d\vec r}{ds}$, giving both in terms of $t$.
  • For the vectors $\vec u = (2,-1,5)$ and $\vec v = (-3, 4,6)$, compute the cross product $\vec u\times \vec v$.
  • Compute $\vec v\times \vec u$, and verify that $\vec v\times \vec u = - \vec u\times \vec v$.

Group Problems

  1. For the curve $r(t) = (2\cos t, 3t, 2\sin t)$, compute the arc length parameter $s(t)$.
    1. Compute $ds/dt$, $d\vec r/dt$, $|d\vec r/dt|$, and $d\vec r/ds$.
    2. State the unit tangent vector $\vec T$ at time $t$, and identify how to obtain it from the parts above.
    3. Compute $d\vec T/dt$, and show that this vector is orthogonal to $\vec T$.
  2. For the curve $r(t) = (t^2, 3t)$, set up an integral that gives the arc length parameter $s(t)$.
    1. Compute $ds/dt$, $d\vec r/dt$, $|d\vec r/dt|$, and $d\vec r/ds$.
    2. State the unit tangent vector $\vec T$ at time $t$, and identify how to obtain it from the parts above.
    3. Compute $d\vec T/dt$, and show that this vector is orthogonal to $\vec T$.
  3. For the vectors $\vec u = (1,2,3)$ and $\vec v = (4, 5,6)$, compute the cross product $\vec u\times \vec v$ and $\vec v\times \vec u$. Then change the vectors to something else, and practice doing this multiple times.