Multivariable Calculus 12.1-12.3

Find the domain of the vector-valued function:

$$r\left(t\right)=t^2i\ +\left(\frac{1}{t^2-1}\right)j\ +\ \left(t-3\right)k$$  

Evaluate the vector valued function r(t) = cos t i + sin t j + tan t k at $$t\ =\ \frac{\pi}{3}$$  

Match the vector-valued function to the plane curve that it represents.

Represent the plane curve $$x^2+y^2=16$$  by a vector-valued function.

Find $$\frac{d}{dt}\left[r_1\cdot r_2\right]\$$  if $$r_1\left(t\right)=\cos t\ i\ +\ e^{-t}\ j\ +\ t^3\ k$$  and $$r_2\left(t\right)=t^2\ k$$  

Evaluate the definite integral: $$\int_0^1\left(t^2i\ +e^{-t}j+3tk\right)dt$$  

Given $$r\left(t\right)=\left(3t^2\right)i+\left(e^{2t}\right)j+\left(\ln\ t\right)\ k$$  , find r''(1/3).

Given $$r'\left(t\right)=\left(\sqrt[]{t}\right)i\ +\left(2e^{2t}\right)j\$$  and r(0) = -2i + j, find r(t).

Find the velocity and acceleration vectors at the point (2, 0) given the position vector $$r\left(t\right)=ti+\left(t^2-4\right)j$$  

Sketch the path of the position vector r(t) = t i + (t^2 - 4)j