Generating...                               quiz03b_n18

  1. Find the normal component of acceleration for the position $\displaystyle \left[ t , t^2 , 1 \right] $ at a general point.

    0 0 $\displaystyle \frac{2}{\sqrt{4\,t^2+1}}$ 0

  2. Find the curvature of $\displaystyle \left[ \sin t , 4\,t , \cos t \right] $ at a general point.

    $\displaystyle \frac{1}{10}$ $\displaystyle \frac{1}{17}$ 1 $\displaystyle \frac{1}{2}$

  3. Find the curvature of $\displaystyle \left[ t , t^3 , t^3 \right] $ at a general point.

    $\displaystyle \frac{3\,2^{\frac{3}{2}}\,\left\vert t\right\vert }{\left(18\,t^4+1\right) ^{\frac{3}{2}}}$ $\displaystyle \frac{3\,2^{\frac{3}{2}}\,t^2}{\left(18\,t^4+4\,t^2\right)^{\frac{3 }{2}}}$ 0 $\displaystyle \frac{6\,t^2}{\left(9\,t^4+4\,t^2\right)^{\frac{3}{2}}}$

  4. Find the unit tangent vector for $\displaystyle \left[ t^2 , t , t^3 \right] $ at a general point.

    $\displaystyle \left[ \frac{4\,t^3}{\sqrt{16\,t^6+2}} , \frac{1}{\sqrt{16\,t^6+2}} , \frac{1}{\sqrt{16\,t^6+2}} \right] $ $\displaystyle \left[ \frac{4\,t^3}{\sqrt{16\,t^6+18\,t^4}} , \frac{3\,t^2}{\sqrt{ 16\,t^6+18\,t^4}} , \frac{3\,t^2}{\sqrt{16\,t^6+18\,t^4}} \right] $ [ 1 ,  0 ,  0 ] $\displaystyle \left[ \frac{2\,t}{\sqrt{9\,t^4+4\,t^2+1}} , \frac{1}{\sqrt{9\,t^4+ 4\,t^2+1}} , \frac{3\,t^2}{\sqrt{9\,t^4+4\,t^2+1}} \right] $

  5. Find the tangential component of acceleration for the position $\displaystyle \left[ \sin t , 2\,t^2 , \cos t \right] $ at a general point.

    $\displaystyle \frac{16\,t}{\sqrt{16\,t^2+4}}$ $\displaystyle \frac{4\,t}{\sqrt{4\,t^2+1}}$ $\displaystyle \frac{16\,t}{\sqrt{16\,t^2+1}}$ $\displaystyle \frac{4\,t}{\sqrt{4\,t^2+4}}$

  6. Find the unit tangent vector for $\displaystyle \left[ \cos t , \sin t , 3\,t \right] $ at a general point.

    $\displaystyle \left[ -\frac{\sin t}{\sqrt{5}} , \frac{\cos t}{\sqrt{5}} , \frac{2 }{\sqrt{5}} \right] $ $\displaystyle \left[ -\frac{\sin t}{\sqrt{17}} , \frac{\cos t}{\sqrt{17}} , \frac{4}{\sqrt{17}} \right] $ $\displaystyle \left[ -\frac{\sin t}{\sqrt{10}} , \frac{\cos t}{\sqrt{10}} , \frac{3}{\sqrt{10}} \right] $ $\displaystyle \left[ -\frac{2\,\sin t}{\sqrt{13}} , \frac{2\,\cos t}{\sqrt{13}} , \frac{3}{\sqrt{13}} \right] $

  7. Find the normal vector for $\displaystyle \left[ 4\,t , -2\,\sin t , -2\,\cos t \right] $ at a general point.

    $\displaystyle \left[ 0 , \sin t , -\cos t \right] $ $\displaystyle \left[ 0 , -\sin t , \cos t \right] $ $\displaystyle \left[ 0 , \sin t , \cos t \right] $ $\displaystyle \left[ 0 , -\sin t , -\cos t \right] $

  8. Find the binormal vector for $\displaystyle \left[ 2\,t , 3\,\sin t , 3\,\cos t \right] $ at a general point.

    $\displaystyle \left[ -\frac{1}{\sqrt{17}} , \frac{2\,\cos t}{\sqrt{17}} , -\frac{ 2\,\sin t}{\sqrt{17}} \right] $ $\displaystyle \left[ -\frac{1}{\sqrt{5}} , \frac{2\,\cos t}{\sqrt{5}} , -\frac{2 \,\sin t}{\sqrt{5}} \right] $ $\displaystyle \left[ -\frac{3}{\sqrt{13}} , \frac{2\,\cos t}{\sqrt{13}} , -\frac{ 2\,\sin t}{\sqrt{13}} \right] $ $\displaystyle \left[ -\frac{3}{5} , \frac{4\,\cos t}{5} , -\frac{4\,\sin t}{5} \right] $

  9. Reparametrize $\displaystyle \left[ \cos t , \sin t , 0 \right] $ with respect to arc length from [ 1 ,  0 ,  0 ]

    $\displaystyle \left[ \cos \left(\frac{s}{4}\right) , \sin \left(\frac{s}{4} \right) , 0 \right] $ $\displaystyle \left[ 4\,\cos s , 4\,\sin s , 0 \right] $ $\displaystyle \left[ 4\,\cos \left(\frac{s}{4}\right) , 4\,\sin \left(\frac{s}{4} \right) , 0 \right] $ $\displaystyle \left[ \cos s , \sin s , 0 \right] $

  10. Find the normal component of acceleration for the position $\displaystyle \left[ \sin t , t^2 , \cos t \right] $ at a general point.

    $\displaystyle \frac{\sqrt{16\,t^2+32}}{\sqrt{4\,t^2+4}}$ $\displaystyle \frac{\sqrt{64\,t^2+80}}{\sqrt{16\,t^2+4}}$ $\displaystyle \frac{\sqrt{4\,t^2+5}}{\sqrt{4\,t^2+1}}$ $\displaystyle \frac{\sqrt{16\,t^2+17}}{\sqrt{16\,t^2+1}}$


Department of Mathematics
Last modified: 2025-12-21