Question: Compute the tangent vector and curvature of the plane curve \(r=\left\langle \cos \left( {{e}^{t}} \right),\sin \left( {{e}^{t}} \right) \right\rangle \) for \(t=\ln \left( 2\pi \right)\). Imagine that this curve is graphed for \(0\le t\le 10\) on a calculator where each plotted point is connected to the next by a line segment. What would the picture look like? What would an accurate rendering of the curve be?