A spline is a long flexible strip of plastic or the like that is used in drawing smooth curves.  Derivatives can be used for the same purpose.
Suppose you have two functions and , and you have a number .  Create a new function, , by setting
      for
and
      for .
If , then most folks say that the new function is a spline of and knotted at
       .

B.2.a)

Here's a spline of
      and
knotted at , plotted on an interval including .

Discuss reasons for the smoothness of the spline curve as it passes through the knot at .

Answer:

To see why is so smooth at the knot at , look at:

The smoothness of the spline at the knot at comes from the fact that and have order of contact at the knot at . Just to confirm, look at a plot of and together.

Yep.
The fact that and have order of contact makes for a smooth transition from to at .  This smooth transition is what's responsible for the smoothness in the spline at .  
Now look at a plot of , , and the spline :

To get the full effect, grab the last two plots and animate.

Nice transition.

B.2.b)

Here's a spline of
      and  
knotted at , plotted on an interval including .

Discuss reasons for the smoothness of the spline curve as it passes through the knot at .

Answer:

Look at:

The smoothness of the spline at the knot at comes from the fact that and have order of contact at . Just to confirm, here's a plot of and :

Yep.  A beautiful transition at .
Now see a plot of , , and the the spline :

To get the full effect, grab both plots and animate.

Smooth transition.

B.2.c)

Here's a spline of
      and
knotted at , plotted on an interval including .

Discuss reasons for the lack of smoothness of the spline curve as it passes through the knot at .

Answer:

Look at:

The order of contact at is .  This explains the lack of smoothness of the spline at the knot at .  Just to confirm, look at a plot of and .

Terrible transition at .
Now see a plot of , , and :

To get the full effect, grab both plots and animate.

This type of transition would make a very dangerous interstate highway connection.

B.2.d)

What practical good are smooth splines?

Answer:

Good question.
First of all, the idea of a spline smooth at its knots makes for interesting plots and nice art.  But there's a serious aspect as well:  
Laying out the curves on expressways and railroads amounts to joining curved plots to relatively straight plots at knots of a spline.  It is unacceptable to have a hitch (corner) in the middle of the roadway.
The higher the order of contact at the knots, the safer the highway or roadbed.
Any order of contact less than is considered unsafe, and this may be pushing it.