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

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

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

    2(u − 1) $\displaystyle 2\,\sqrt{5}\,\left(u-1\right)$ $\displaystyle \sqrt{17}\,\left(u-1\right)$ u − 1

  3. Reparametrize [ 0 ,  0 ,  4t ] with respect to arc length from [ 0 ,  0 ,  0 ]

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

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

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

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

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

  6. Find the curvature of $\displaystyle \left[ 4\,t , 3\,\cos t , 3\,\sin t \right] $ at a general point.

    $\displaystyle \frac{1}{3}$ $\displaystyle \frac{1}{2}$ $\displaystyle \frac{3}{25}$ $\displaystyle \frac{1}{10}$

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

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

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

  9. Find the unit tangent vector for $\displaystyle \left[ t^2 , t , t^4 \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] $ $\displaystyle \left[ 0 , 0 , \frac{t}{\left\vert t\right\vert } \right] $ $\displaystyle \left[ \frac{1}{\sqrt{2}} , 0 , \frac{1}{\sqrt{2}} \right] $ $\displaystyle \left[ \frac{3\,t^2}{\sqrt{9\,t^4+1}} , 0 , \frac{1}{\sqrt{9\,t^4+1 }} \right] $

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

    $\displaystyle \frac{2^{\frac{3}{2}}}{\left(8\,t^2+1\right)^{\frac{3}{2}}}$ $\displaystyle \frac{6\,t^2}{\left(9\,t^4+4\,t^2\right)^{\frac{3}{2}}}$ 0 $\displaystyle \frac{6\,t^2}{\left(9\,t^4+4\,t^2\right)^{\frac{3}{2}}}$


Department of Mathematics
Last modified: 2025-10-14