Exercises 1-18 give parametric equations and parameter...

Exercises $1-18$ give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$ x=\sec ^{2} t-1, \quad y=\tan t, \quad-\pi / 2 < t < \pi / 2 $$

Exercises $1-18$ give parametric equations and parameter intervals for
the motion of a particle in the $x y$ -plane. Identify the particle's path by
finding a Cartesian equation for it. Graph the Cartesian equation. (The
graphs will vary with the equation used.) Indicate the portion of the
graph traced by the particle and the direction of motion.
$$
x=\sec ^{2} t-1, \quad y=\tan t, \quad-\pi / 2 < t < \pi / 2
$$

Key Concepts

Parametric Equations

Parametric equations define a set of related quantities as functions of a common independent parameter. In the context of motion in the plane, they describe the coordinates of a particle at any given instance and can represent complex curves that are not easily expressed in conventional Cartesian form.

Elimination of the Parameter

Eliminating the parameter involves expressing the relationship between the coordinates directly by removing the parameter. This process converts parametric equations into a Cartesian equation, allowing for analysis of the geometric properties of the curve such as intercepts, symmetry, and overall shape.

Cartesian Equations

A Cartesian equation represents the same curve as the parametric equations but in terms of x and y only. This form is useful for graphing the curve with standard plotting techniques and for understanding the geometric nature of the path without having to refer back to the parameter.

Graphical Analysis of Parametric Curves

Graphical analysis of parametric curves involves plotting the Cartesian equation, identifying key features such as intercepts, asymptotes, and curves, and understanding how the particle's path is traced over a specific interval. This analysis often requires considering the domain restrictions on the parameter which define which portion of the complete curve is actually represented.

Direction of Motion

The direction of motion is determined by the way the particle moves along the curve as the parameter increases. By analyzing the derivative of the position components or by observing the parameter's influence on x and y, one can determine both the sequence in which points on the curve are visited and the overall orientation of the path.

Transcript

00:01 So we're given a complicated looking barometric equation.

00:08 X as a function of time is secant of time squared minus 1, while the y -coronate function of time is the tangent of t or the tan of t.

00:24 Okay, and we have t going from minus by halves to by halves.

00:36 And this is probably strictly less because the tangent of by half doesn't make sense.

00:44 Okay, so first let's figure out the parametric equation.

00:52 So we're first going to, isolating and substituting in this case is going to be very annoying because the secant of arc tangent, you can definitely do that.

01:08 It's just one over the cosine tangent, which you can do in sort of the regular way.

01:15 But in these cases it's often easier to just manipulate one of them until you see a pattern.

01:21 It's also risky because it might not work out and then you have to do all the isolate and something work anyway.

01:31 But let's do it by just manipulating the equation a bit.

01:36 So x of t equals 1 over cosine of t squared minus 1.

01:44 That says the definition of sequence.

01:48 Now one to fix the fraction can be written as cosine of t squared divided by cosine of t squared.

02:02 So adding these together you get one minus cosine of t squared divided by cosine of t squared.

02:13 Now one minus cosine of t squared equals sine of t.

02:19 Squared divided by cosine of t squared and hey this is exactly tan of t squared which is y squared so x equals y squared you can definitely get this by substitution and isolation or isolation substitution rather i find this easier whenever i have the idea that i can simplify something i try that first um but it's up to you...

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