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

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

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

    $\displaystyle \frac{3\,2^{\frac{3}{2}}\,t^2}{\left(9\,t^4+8\,t^2\right)^{\frac{3 }{2}}}$ 0 0 $\displaystyle \frac{2}{\left(4\,t^2+1\right)^{\frac{3}{2}}}$

  3. Reparametrize $\displaystyle \left[ \cos t , 2\,t , \sin t \right] $ with respect to arc length from [ 1 ,  0 ,  0 ]

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

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

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

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

    $\displaystyle \left[ \frac{2}{\sqrt{5}} , -\frac{\sin t}{\sqrt{5}} , \frac{\cos t }{\sqrt{5}} \right] $ $\displaystyle \left[ \frac{4}{5} , -\frac{3\,\sin t}{5} , \frac{3\,\cos t}{5} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{5}} , -\frac{2\,\sin t}{\sqrt{5}} , \frac{2\, \cos t}{\sqrt{5}} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{10}} , -\frac{3\,\sin t}{\sqrt{10}} , \frac{3 \,\cos t}{\sqrt{10}} \right] $

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

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

  7. Find the arc length of $\displaystyle \left[ 3\,\sin t , 3\,\cos t , t \right] $ from $t= 1$ to $t=u$.

    $\displaystyle \sqrt{13}\,\left(u-1\right)$ $\displaystyle \sqrt{10}\,\left(u-1\right)$ 2(u − 1) u − 1

  8. Find the unit tangent vector for $\displaystyle \left[ t^4 , t^3 , 1 \right] $ at a general point.

    $\displaystyle \left[ \frac{1}{\sqrt{9\,t^4+2}} , \frac{3\,t^2}{\sqrt{9\,t^4+2}} , \frac{1}{\sqrt{9\,t^4+2}} \right] $ $\displaystyle \left[ \frac{4\,t^3}{\sqrt{16\,t^6+9\,t^4}} , \frac{3\,t^2}{\sqrt{ 16\,t^6+9\,t^4}} , 0 \right] $ $\displaystyle \left[ \frac{1}{\sqrt{8\,t^2+1}} , \frac{2\,t}{\sqrt{8\,t^2+1}} , \frac{2\,t}{\sqrt{8\,t^2+1}} \right] $ $\displaystyle \left[ \frac{2\,t}{\sqrt{9\,t^4+4\,t^2+1}} , \frac{3\,t^2}{\sqrt{9 \,t^4+4\,t^2+1}} , \frac{1}{\sqrt{9\,t^4+4\,t^2+1}} \right] $

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

    $\displaystyle \left[ -\frac{4\,\sin t}{\sqrt{10}} , -\frac{3}{\sqrt{10}} , \frac{ 4\,\cos t}{\sqrt{10}} \right] $ $\displaystyle \left[ -\frac{2\,\sin t}{\sqrt{5}} , -\frac{1}{\sqrt{5}} , \frac{2 \,\cos t}{\sqrt{5}} \right] $ $\displaystyle \left[ -\frac{4\,\sin t}{5} , -\frac{3}{5} , \frac{4\,\cos t}{5} \right] $ $\displaystyle \left[ -\frac{\sin t}{\sqrt{5}} , -\frac{2}{\sqrt{5}} , \frac{\cos t}{\sqrt{5}} \right] $

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

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


Department of Mathematics
Last modified: 2026-05-07