Mathematics

Problem

A space curve is defined by

\mathbf r(t)=\langle 4\cos t,\,4\sin t,\,3t^{2}\rangle

for

0\le t\le 2\pi

. (a) Compute the speed of the particle at

t=0

and at

t=\tfrac{\pi}{2}

. (b) Determine the unit tangent vector at

t=0

and at

t=\tfrac{\pi}{2}

. (c) Calculate the curvature at each of the two times listed above. (d) Compare the two curvature values and interpret which point on the curve is bending more sharply. Provide a brief justification based on your calculations.

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