First major revision
Please provide suggestions for improvement! I think I may have made the first major revision to this article. I reorganized the old material into the intro and added a definition and properties as well as reference. I may add more later. Gray's definition of a tangent vector is actually prior to his definition of a directional derivative. I may add this material which talks about a vector part and a point of application as comprising a tangent vector which is just these two pair of points in Euclidean space. I don't like this definition however. I need a little more mathematical sophistication, but I intend to cover tangent vectors as elements of a tangent space [in] a differentiable manifold. I do not think that this material is too technical. A tangent is part of elementary calculus, but a tangent vector is part of vector calculus. Therefore, our readers will have some very small bit of mathematical sophistication (like me).Stewart.M.Nash (talk) 02:34, 15 May 2015 (UTC)
Motivation sample with error
Here the bottom part is not equal to the vector length. AFAIK it must be
Expression in denominator of example
The example only makes sense if the denominator evaluates to the value 2 at t=0; e.g., (4t2 + 4e4t + sin2t )0.5 = 2. Arkadi is correct.