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

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

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

    $\displaystyle \frac{36\,t^3+4\,t}{\sqrt{18\,t^4+4\,t^2}}$ $\displaystyle \frac{18\,t^3+4\,t}{\sqrt{9\,t^4+4\,t^2}}$ $\displaystyle \frac{4\,t}{\sqrt{4\,t^2+1}}$ $\displaystyle \frac{18\,t^3+4\,t}{\sqrt{9\,t^4+4\,t^2}}$

  5. Find the arc length of $\displaystyle \left[ 3\,t , 4\,\sin t , 4\,\cos t \right] $ from $t= 1$ to $t=u$.

    2(u − 1) $\displaystyle \sqrt{13}\,\left(u-1\right)$ 5(u − 1) 4(u − 1)

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

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

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

    $\displaystyle \frac{1}{5}$ $\displaystyle \frac{3}{13}$ 1 $\displaystyle \frac{1}{3}$

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

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

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

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

  10. Find the unit tangent vector for $\displaystyle \left[ t^2 , 1 , t^3 \right] $ at a general point.

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


Department of Mathematics
Last modified: 2026-10-24