Generating...                               quiz03b_n6

  1. Find the unit tangent vector for $\displaystyle \left[ t^3 , 1 , 1 \right] $ at a general point.

    $\displaystyle \left[ \frac{2\,t}{\sqrt{16\,t^6+4\,t^2+1}} , \frac{1}{\sqrt{16\,t^ 6+4\,t^2+1}} , \frac{4\,t^3}{\sqrt{16\,t^6+4\,t^2+1}} \right] $ [ 1 ,  0 ,  0 ] $\displaystyle \left[ \frac{4\,t^3}{\sqrt{32\,t^6+4\,t^2}} , \frac{4\,t^3}{\sqrt{ 32\,t^6+4\,t^2}} , \frac{2\,t}{\sqrt{32\,t^6+4\,t^2}} \right] $ $\displaystyle \left[ \frac{t}{\sqrt{2}\,\left\vert t\right\vert } , \frac{t}{\sqrt{2}\, \left\vert t\right\vert } , 0 \right] $

  2. Find the normal vector for $\displaystyle \left[ -3\,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] $

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

    $\displaystyle \frac{3\,2^{\frac{3}{2}}\,t^2}{\sqrt{9\,t^4+8\,t^2}}$ 0 $\displaystyle \frac{\sqrt{36\,t^4+36\,t^2+4}}{\sqrt{9\,t^4+4\,t^2+1}}$ $\displaystyle \frac{\sqrt{36\,t^4+36\,t^2+4}}{\sqrt{9\,t^4+4\,t^2+1}}$

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

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

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

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

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

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

  7. Find the curvature of $\displaystyle \left[ t^3 , t^2 , 1 \right] $ at a general point.

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

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

    $\displaystyle \frac{2}{5}$ $\displaystyle \frac{1}{3}$ $\displaystyle \frac{1}{2}$ $\displaystyle \frac{3}{10}$

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

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

  10. Reparametrize $\displaystyle \left[ 3\,\cos t , 3\,\sin t , 4\,t \right] $ with respect to arc length from [ 3 ,  0 ,  0 ]

    $\displaystyle \left[ 0 , 0 , \frac{4\,s}{5} \right] $ $\displaystyle \left[ 3\,\cos \left(\frac{s}{4}\right) , 3\,\sin \left(\frac{s}{4} \right) , s \right] $ $\displaystyle \left[ 3\,\cos \left(\frac{s}{5}\right) , 3\,\sin \left(\frac{s}{5} \right) , \frac{4\,s}{5} \right] $ [ 0 ,  0 ,  s ]


Department of Mathematics
Last modified: 2025-10-30