To finish the sketch of the parametric curve we also need the direction of motion for the curve. This is not the only range that will trace out the curve however.
Any of them would be acceptable answers for this problem. All travel must be done on the path sketched out. This means that we had to go back up the path. The only way for that to happen on this particular this curve will be for the curve to be traced out in both directions.
Contrast this with the ellipse in Example 4. However, the curve only traced out in one direction, not in both directions. So, to finish this problem out, below is a sketch of the parametric curve. Note that we put direction arrows in both directions to clearly indicate that it would be traced out in both directions. It is more than possible to have a set of parametric equations which will continuously trace out just a portion of the curve.
We will often use parametric equations to describe the path of an object or particle. Completely describe the path of this particle. Eliminating the parameter this time will be a little different. This gives,. This time the algebraic equation is a parabola that opens upward. So, again we only trace out a portion of the curve. Here is a quick sketch of the portion of the parabola that the parametric curve will cover.
Here is that work. Here are a few of them. We should give a small warning at this point. Because of the ideas involved in them we concentrated on parametric curves that retraced portions of the curve more than once. Do not, however, get too locked into the idea that this will always happen. Many, if not most parametric curves will only trace out once. The first one we looked at is a good example of this. That parametric curve will never repeat any portion of itself.
There is one final topic to be discussed in this section before moving on. However, there are times in which we want to go the other way. Given a function or equation we might want to write down a set of parametric equations for it. In these cases we say that we parameterize the function. If we take Examples 4 and 5 as examples we can do this for ellipses and hence circles.
Given the ellipse. Every curve can be parameterized in more than one way. Any of the following will also parameterize the same ellipse. Note as well that the last two will trace out ellipses with a clockwise direction of motion you might want to verify this.
There are many more parameterizations of an ellipse of course, but you get the idea. Each parameterization may rotate with different directions of motion and may start at different points. One possible way to parameterize a circle is,. In these cases we parameterize them in the following way,.
At this point it may not seem all that useful to do a parameterization of a function like this, but there are many instances where it will actually be easier, or it may even be required, to work with the parameterization instead of the function itself. Unfortunately, almost all of these instances occur in a Calculus III course.
Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 1 Sketch the parametric curve for the following set of parametric equations.
Here is the sketch of this parametric curve. Example 2 Sketch the parametric curve for the following set of parametric equations. Example 3 Eliminate the parameter from the following set of parametric equations. Example 4 Sketch the parametric curve for the following set of parametric equations.
Clearly indicate direction of motion. The direction the point moves is again called the orientation and is indicated on the graph. Then we can apply any previous knowledge of equations of curves in the plane to identify the curve.
These steps give an example of eliminating the parameter. Eliminate the parameter for each of the plane curves described by the following parametric equations and describe the resulting graph. This is the equation of a parabola opening upward.
The graph of this plane curve follows. Sometimes it is necessary to be a bit creative in eliminating the parameter. The parametric equations for this example are. This gives. Recall from the section opener that the orbit of Earth around the Sun is also elliptical. This is a perfect example of using parameterized curves to model a real-world phenomenon. Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.
So far we have seen the method of eliminating the parameter, assuming we know a set of parametric equations that describe a plane curve. What if we would like to start with the equation of a curve and determine a pair of parametric equations for that curve? This is certainly possible, and in fact it is possible to do so in many different ways for a given curve.
The process is known as parameterization of a curve. This gives the parameterization. We have complete freedom in the choice for the second parameterization. Imagine going on a bicycle ride through the country.
The tires stay in contact with the road and rotate in a predictable pattern. Now suppose a very determined ant is tired after a long day and wants to get home. So he hangs onto the side of the tire and gets a free ride. A cycloid generated by a circle or bicycle wheel of radius a is given by the parametric equations.
To see why this is true, consider the path that the center of the wheel takes. Next, consider the ant, which rotates around the center along a circular path. If the bicycle is moving from left to right then the wheels are rotating in a clockwise direction.
A possible parameterization of the circular motion of the ant relative to the center of the wheel is given by. The negative sign is needed to reverse the orientation of the curve. If the negative sign were not there, we would have to imagine the wheel rotating counterclockwise. Adding these equations together gives the equations for the cycloid.
In this graph, the green circle is traveling around the blue circle in a counterclockwise direction. A point on the edge of the green circle traces out the red graph, which is called a hypocycloid. These equations are a bit more complicated, but the derivation is somewhat similar to the equations for the cycloid. Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus.
For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve? If the position of the baseball is represented by the plane curve x t , y t , x t , y t , then we should be able to use calculus to find the speed of the ball at any given time.
Furthermore, we should be able to calculate just how far that ball has traveled as a function of time. We start by asking how to calculate the slope of a line tangent to a parametric curve at a point. Consider the plane curve defined by the parametric equations. The graph of this curve appears in Figure 1. Substituting this into y t , y t , we obtain. This is no coincidence, as outlined in the following theorem. Then the derivative d y d x d y d x is given by.
This theorem can be proven using the Chain Rule. Differentiating both sides of this equation using the Chain Rule yields. Equation 1. Calculate the derivative d y d x d y d x for each of the following parametrically defined plane curves, and locate any critical points on their respective graphs. Find the equation of the tangent line to the curve defined by the equations.
First find the slope of the tangent line using Equation 1. Calculating x 2 x 2 and y 2 y 2 gives. Now use the point-slope form of the equation of a line to find the equation of the tangent line:. Our next goal is to see how to take the second derivative of a function defined parametrically. This gives us. From Example 1. Using Equation 1. Now that we have seen how to calculate the derivative of a plane curve, the next question is this: How do we find the area under a curve defined parametrically?
Suppose we want to find the area of the shaded region in the following graph. To derive a formula for the area under the curve defined by the functions. We use rectangles to approximate the area under the curve. This follows from results obtained in Calculus 1 for the function y t i - 1 x t i - x t i - 1 t i - t i - 1.
Taking the limit as n n approaches infinity gives. Consider the non-self-intersecting plane curve defined by the parametric equations.
0コメント