Generating...                               quiz03b_n9

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

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

  2. Find the tangential component of acceleration for the position $\displaystyle \left[ t , t^3 , 1 \right] $ at a general point.

    $\displaystyle \frac{18\,t^3}{\sqrt{9\,t^4+1}}$ $\displaystyle \frac{18\,t^3+4\,t}{\sqrt{9\,t^4+4\,t^2}}$ $\displaystyle \frac{18\,t^3+4\,t}{\sqrt{9\,t^4+4\,t^2}}$ $\displaystyle \frac{2^{\frac{3}{2}}\,t}{\left\vert t\right\vert }$

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

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

  4. Find the normal component of acceleration for the position $\displaystyle \left[ t^2 , t^3 , 1 \right] $ at a general point.

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

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

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

  6. Find the arc length of $\displaystyle \left[ 2\,\cos t , 2\,\sin t , t \right] $ from $t= 0$ to $t=u$.

    3u $\displaystyle \sqrt{13}\,u$ $\displaystyle \sqrt{5}\,u$ u

  7. Find the acceleration for the position $\displaystyle \left[ 1 , t , t^3 \right] $ at a general point.

    $\displaystyle \left[ 12\,t^2 , 6\,t , 6\,t \right] $ $\displaystyle \left[ 12\,t^2 , 2 , 6\,t \right] $ [ 0 ,  0 ,  6t ] [ 6t ,  6t ,  0 ]

  8. 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{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] $ $\displaystyle \left[ -\frac{\sin t}{\sqrt{5}} , \frac{\cos t}{\sqrt{5}} , \frac{2 }{\sqrt{5}} \right] $

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

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

  10. Reparametrize $\displaystyle \left[ 2\,t , 4\,\sin t , 4\,\cos t \right] $ with respect to arc length from [ 0 ,  0 ,  4 ]

    $\displaystyle \left[ \frac{s}{\sqrt{5}} , 3\,\sin \left(\frac{s}{2\,\sqrt{5}} \right) , 3\,\cos \left(\frac{s}{2\,\sqrt{5}}\right) \right] $ $\displaystyle \left[ \frac{2\,s}{\sqrt{13}} , 4\,\sin \left(\frac{s}{\sqrt{13}} \right) , 4\,\cos \left(\frac{s}{\sqrt{13}}\right) \right] $ $\displaystyle \left[ \frac{2\,s}{\sqrt{13}} , 3\,\sin \left(\frac{s}{\sqrt{13}} \right) , 3\,\cos \left(\frac{s}{\sqrt{13}}\right) \right] $ $\displaystyle \left[ \frac{s}{\sqrt{5}} , 4\,\sin \left(\frac{s}{2\,\sqrt{5}} \right) , 4\,\cos \left(\frac{s}{2\,\sqrt{5}}\right) \right] $


Department of Mathematics
Last modified: 2025-06-19