Thus there are four variables to consider, the position of the point x,y,z and an independent variable t, which we can think of as time. This means we define both x and y as functions of a parameter. Center the ferris wheel on the vertical axis such that. A point x, y is on the unit circle if and only if there is a value of t such that these two equations generate that point.
This is simply the idea that a point moving in space traces out a. But anyway, i thought a good place to start is the motivation. Writing parametric equations to represent the path of a projectile with sample. Then the parametric equations and define y as a differentiable function of x and. Use them to describe curves in the plane when one function wont do. Repeating what was said earlier, a parametric curve is simply the idea that a point moving in. Apr 03, 2018 parametric equations introduction, eliminating the paremeter t, graphing plane curves, precalculus duration. Because the first time i learned parametric equations i was like, why mess up my nice and simple world of xs and ys by introducing a third parameter, t. Here is a set of practice problems to accompany the parametric equations and curves section of the parametric equations and polar coordinates chapter of the notes for paul dawkins calculus ii course at lamar university. Know how to determine whether two lines in space are parallel, skew, or intersecting. In these examples we shall use the same parametric equations we used above. This is simply the idea that a point moving in space traces out a path over time. Write each pair of parametric equations in rectangular form.
Introduction to parametric equations circles in parametric form eliminating the parameter finding the parametrization of a line introduction to parametric equations graphing parametric equations on the ti84 find parametric equations for ellipse using sine and cosine write parametric equations as a cartesian equation parametric ray intuition. And time tends to be the parameter when people talk about parametric equations. Selection file type icon file name description size revision time user. We can define a plane curve using parametric equations. Repeating what was said earlier, a parametric curve is simply the idea that a point moving in the space traces out a path. Keyconcept parametric equations if f and g are continuous functions of t on the interval then the set of ordered pairs ft, dt represent a parametric curve. Check your work by first graphing the parametric equations on your calculator than graphing the. Be able to nd the parametric equations of a line that satis es certain conditions by nding a point on the line and a vector parallel to the line.
To this point in both calculus i and calculus ii weve looked almost exclusively at functions in. Finding parametric equations from a rectangular equation note that i showed examples of how to do this via vectors in 3d space here in the introduction to vector section. Now we will look at parametric equations of more general trajectories. This zip folder contains noneditable pdf documents. Example 2this is the cartesian equation for the ellipse. They are in the form of pdf documents that can be printed or annotated by students for educational purposes. Parametric curves general parametric equations we have seen parametric equations for lines. When these points are plotted on an xy plane they trace out a curve. For the love of physics walter lewin may 16, 2011 duration. You might need to use any of the pythagorean identities. It is often useful to have the parametric representation of a particular curve.
All sorts of interesting problems come out of using parametric equations, not just in physics. Weve also seen how we can model rectangular equations in parametric form. Parameters are variables used in 2 equations tcompletely describing a function. In this section we will introduce parametric equations and parametric curves i. Parametric equations, differential calculus from alevel. Recognize the parametric equations of basic curves, such as a line and a circle. However, when it comes time to use our mathematical toolbox on real applied problems. Length of a curve and surface area university of utah. If the function f and g are di erentiable and y is also a di erentiable function of x, the three derivatives dy dx, dy dt and dx dt are related by the chain rule.
Even if we examine the parametric equations carefully, we may not be able to tell that the corresponding plane curve is a portion of a parabola. Parametric equations primarily describe motion and direction. Day 1 graphing parametric equations and eliminating the parameter day 2 calculus of parametric equations. Unit 8 conic sections page 2 of 18 precalculus graphical, numerical, algebraic. Sometimes and are given as functions of a parameter.
Math 101 lab 10 notes from the parametric equations x r cos. Write an equation in standard form of a parabola with vertex 0,0 and passes through the point 3,5. Parametric equations definition a plane curve is smooth if it is given by a pair of. We can write rectangular equations that model the height of the ball as a function of. Calculate curvature and torsion directly from arbitrary parametric equations. We will graph several sets of parametric equations and discuss how to eliminate the parameter to get an algebraic equation which will often help with the graphing process. The problems listed in this activity packet build nicely on each other if introduced in the order that they appear. Answers to worksheet on parametrics and calculus 2 2 2 3 3 2 6 3 3. The equations x ft and gt are paramefric equations for this curve, t is the parameter, and i is the parameter interval. By eliminating the parameter, we can write one equation in and that is equivalent to the two parametric equations. Plane curves and parametric equations imagine hitting a golf ball and watching its flight path until it lands. As you probably realize, that this is a video on parametric equations, not physics. Convert the parametric equations of a curve into the form yfx. For each problem, write an integral expression that represents the length of the arc of the curve over the given interval.
On problems 11 12, a curve c is defined by the parametric equations given. Determine the parametric equations which will model the height of a rider starting in the 3 oclock position at t 0. Guidedscaffolded notes and interactivefoldable activities are perfect for precalculus or integrated math students. Unit 8 conic sections page 4 of 18 precalculus graphical, numerical, algebraic. Then, are parametric equations for a curve in the plane. Use point plotting to graph plane curves described by parametric equations.
The unit on parametric equations and vectors takes me six days to cover see the following schedule, not including a test day. Then write a second set of parametric equations that represent the same function, but with a faster speed and an opposite orientation. Just as we describe curves in the plane using equations involving x and y, so can we. Sometimes you may be asked to find a set of parametric equations from a rectangular cartesian formula. Parametric representations of plane curves x t21 y t3t. Vectors and parametric equations guided notes and inb. Curves defined by parametric equations mathematics. Both x and y are given as functions of another variable called a parameter eg t. Calculus with parametric equations let cbe a parametric curve described by the parametric equations x ft. And, if the lines intersect, be able to determine the point of intersection. Each value of the parameter, when evaluated in the parametric equations, corresponds to a point. Calculus ii parametric equations and curves practice problems. A circle centered at h, k h,k h, k with radius r r r can be described by the parametric equation.
Example 1so, to find the cartesian equation use t y2 to get. This is known as a parametric equation for the curve that is traced out by varying the values of the parameter t. When we parameterize a curve, we are translating a single equation in two variables, such as x and y, into an equivalent pair of equations in three variables, x, y, and t. In this section we see how to calculate the derivative dy dx from a knowledge of the socalled parametric derivatives dx dt and dy dt. Parametric equations if there are functions f and g with a common domaint, the equations x ft and y gt, for t in t, areparametric equations of the curve consisting of allpoints ft, gt, for t in t. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors. These math journal activitiesscaffolded notes cover vectors and parametric equations. In this part of the unit we are going to look at parametric curves. The cartesian equation of this curve is obtained by eliminating the parameter t from the parametric equations. Introduction to parametric equations typical, high school precalculus and algebra courses only discuss parametric equations lightly and focus on the fundamental functions polynomials, exponentials, trig, etc. The following links are pdf files of notes we took inclass for each section. If you want to find the cartesian equation for parametric equations involving trigonometric functions, you will probably need to use a trigonometric identity. Thus a pair of equations, called parametric equations, completely describe a single xy function the differentiation of functions given in parametric form is carried out using the chain rule.
Polar coordinates, parametric equations whitman college. Calculus ii parametric equations and curves practice. Parametric equations express both x and y coordinates as a function of a third parameter, often called t. Projectile motion sketch and axes, cannon at origin, trajectory mechanics gives and. Now we can just rearrange to get the equation in terms of y. Parametric equations of lines general parametric equations in this part of the unit we are going to look at parametric curves. We have now seen how both polar equations and parametric equations model complicated curves, especially curves that fail the vertical line test, much more easily. Parametric equations and the parabola extension 1 parametric equations and the parabola extension 1 parametric equations parametric equations are a set of equations in terms of a parameter that represent a relation.
1584 861 1299 764 304 812 1240 1422 81 1520 436 1292 1296 401 316 1260 1593 1004 409 1281 814 516 54 677 1073 1483 36 1259 684 227 1457 1257 27 467 305 804 607