A space curve is defined by for .

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A space curve is defined by r(t)=⟨4cos⁡t, 4sin⁡t, 3t2⟩\mathbf r(t)=\langle 4\cos t,\,4\sin t,\,3t^{2}\rangle for 0≤t≤2π0\le t\le 2\pi. (a) Compute the speed of the particle at t=0t=0 and at t=π2t=\tfrac{\pi}{2}. (b) Determine the unit tangent vector at t=0t=0 and at t=π2t=\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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