Chapter 8.6, Problem 4E

(a)

To determine

The tangent vector ${\text{x}}^{\prime }\left(t\right)$, how ${\text{x}}^{\prime }\left(0\right)$ and ${\text{x}}^{\prime }\left(1\right)$ related to control points, the geometric descriptions of the directions of these tangent vectors, and what happens when both ${\text{x}}^{\prime }\left(0\right)$ and ${\text{x}}^{\prime }\left(1\right)$ equal 0.

(b)

To determine

The second derivative ${\text{x}}^{″}\left(t\right)$, how ${\text{x}}^{″}\left(0\right)$ and ${\text{x}}^{″}\left(1\right)$ related to control points, and to construct a line segment that points in the direction ${\text{x}}^{″}\left(0\right)$.

To draw: The figure based on B-spline.