Derivatives and Integrals of Vector Functions

1. Consider the function  \(\displaystyle{{\bf r}(t) = (1+t^2) \, {\bf i} + t e^{-t} \,  {\bf j} + \sin (2t) \,  {\bf k}} \).    solution

 (a) Find the derivative.    (b) Find the unit tangent vector at the point where \(t=0 \).

2. Find parametric equations for the line tangent to the graph of the given function at the point where \(t=2\).  solution \[ {\bf r}(t)=\left\langle 2\sin \left(\frac{\pi}{2}t\right),-3e^{-t},3t^2 \right\rangle\]

3. Find parametric equations for the tangent line to the helix with parametric equations  \[ x = 2 \cos t,   \: y = \sin t, \: z =t \] at the point \((0, 1, \pi/2)\).    solution

4. Evaluate the integral:  \( \displaystyle{\int_0^1 \left( {\bf i} + \frac{1}{1+t^2} {\bf j} + \frac{2t}{1+t^2}{\bf k}  \right) \, dt} \)       solution