Question

Find the parametric equations for the tangent line to the curve
x=t^4−1 y=t^4+1 z=t^4
at the point (808281). Use the variable t for your parameter.

Answer 1

\[(t^4-1,t^4+1,t^4)=(80,82,81)\Rightarrow t=\pm 3\\\vec R(t)=(t^4-1)\mathrm i+(t^4+1)\mathrm j + t^4\mathrm k\\\vec {R}{'}(t)=4t^3\mathrm i + 4t^3\mathrm j+4t^3\mathrm k\\\vec{R}{'}(3)=108\mathrm i + 108 \mathrm j +108\mathrm k \]

equation of line:
\[\large \frac{x-80}{108} = \frac{y-82}{108} = \frac{z-81}{108}\]