Consider the three–dimensional curve described by for all real .

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

Problem

Consider the three–dimensional curve described by

\mathbf r(t)=\langle e^{t}\cos t,\,e^{t}\sin t,\,e^{t}\rangle

for all real

t

. (a) Find

\mathbf r'(t)

and the magnitude

|\mathbf r'(t)|

. (b) Determine the unit tangent vector

\mathbf T(t)

. (c) Derive an explicit expression for the curvature

\kappa(t)

of the curve, fully simplified. (d) Evaluate

\displaystyle \lim_{t\to\infty} \kappa(t)

and interpret what this limit says about the shape of the curve far from the origin.