An Interesting Introduction to Parametric curves in the plane

Perhaps you are already familiar with the concept of parametric curves in the plane.  You have seen in the last lecture there are many different parameters that can often be used as the basis for writing a set of parametric equations to describe a given curve.  This set of homework exercises are designed to help you review some of the basic ideas from lecture as well as to give you the chance to familiarize yourself with that part of your graphing calculator which allows you to easily plot parametric curves, in some of the problems.  You should make a point of “playing around” with this feature of your calculator until you feel comfortable using it. 

1.         (a)   Show that for a projectile fired from ground level,  Newton’s Law of

                    projectile motion holds true:

 

Assume the only force acting on the projectile after it has been fired is gravity and that x(0)=0, y(0)=0, and that v(0)= v₀, where v is the velocity vector. 

(b)   Show that the relationship between the variables x and y is parabolic – do  

        this by eliminating the parameter in the set of parametric equations in part (a)

(c)    show the maximum value of y is given by :

 

(d)        What initial angle of elevation produces the maximum range of the projectile (horizontal distance traveled before hitting the ground)?

            Hint:  Show that the range as a function of the initial angle is

 

(e)        Show that doubling the initial velocity multiplies the maximum height and range by a factor of 4. 

2.         Eliminate the parameter and thereby write the following curves in their Cartesian form.  Also sketch (using your graphing utility when necessary), indicating the orientation of the parametrically described curves:

           

3.         Parametrize the line y=3x-2, using as parameter each of the following:

            (a)  The x-coordinate

            (b)  The y-coordinate

            (c)  The directed distance from (1, 1)

4.         Parametrize the segment AB where A = (1, 5) and B= (2, 7) so that:

            (a)        The orientation is from A to B for increasing t

            (b)        The orientation is from B to A for increasing t

The Calculus of Parametric Curves

1.         dy/dx and also d²y/dx² for each of the following parametrically described curves at the given value of the parameter.  You may wish to check your results by eliminating the parameter, if possible, and re-working the problem as a non-parametric problem.  You should also use your graphing calculator to examine the graphs and judge whether your calculated information looks graphically plausible.

 

2.         Find the slope of the tangent line to the curves below at the given points:

 

3.         Find all points of horizontal and vertical tangency to the curves below:

4.         Find the arc-length of the following parametric curves over the given intervals:

           

5.         Find the surface area generated by revolving the given curve about the given axis:

           

6.            Find the area of the region in the first quadrant bounded by the given parametric curve , and the line x=2:

           

7.         Find the area of the region bounded by the given parametric curve and the entire x-axis: