Exercise 3.  Find a parametrization of the curve that is a portion of the parabola    that  

3 (a).  joins the origin    to the point  .  

Solution 3 (a).

See text and/or instructor's solution manual.

    If we use the natural parameterization  ,  then

        ,   and

        .

    Therefore,  

           for     

is the portion of the parabola that joins the origin to the point .  

We are done.   

Aside.  We can let Mathematica double check our work.

        The curve     for   ,   where  

        the initial point is    and the terminal point is  .    

We are done.   

Aside.  There are many other parameterizations.  Here is one of them.

    If we use the parameterization  ,  then

        ,   and

        .

Therefore,  

           for     

is the portion of the parabola that joins the origin to the point .  

Aside.  We can let Mathematica double check our work.

        The curve     for   ,   where   

        the initial point is    and the terminal point is  .    

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell