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. $$ x=-\cosh t, \quad y=\sinh t, \quad-\infty < t < \infty $$

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.
$$
x=-\cosh t, \quad y=\sinh t, \quad-\infty < t < \infty
$$

Key Concepts

Graphing and Direction of Motion

Graphing the resulting Cartesian equation helps in visualizing the particle’s path in the xy-plane. Additionally, determining the portion of the graph that is traced by the particle and the direction of motion involves analyzing the parameter's interval and understanding how the coordinates evolve as the parameter changes. This provides a complete picture of the particle's trajectory and the orientation of its movement.

Elimination of the Parameter

Eliminating the parameter involves rewriting a set of parametric equations as a single Cartesian equation in x and y. This process typically uses algebraic manipulation and known identities, such as the hyperbolic identity cosh²t - sinh²t = 1, to relate x and y directly, allowing the curve's shape and properties to be studied in the standard coordinate system.

Hyperbolic Functions

Hyperbolic functions, such as cosh t and sinh t, are analogs of the trigonometric functions but for a hyperbola instead of a circle. They have identities like cosh²t - sinh²t = 1, which are crucial for eliminating the parameter in problems involving hyperbolic functions, and they often appear in problems describing curves with hyperbolic characteristics.

Parametric Equations

Parametric equations represent a set of equations where one or more variables are expressed as functions of an independent parameter. This framework is useful for describing motions and trajectories in the coordinate plane, allowing for the analysis of both the path of a curve and the direction of motion without needing a single equation in x and y.

Transcript

00:01 So in this question, we're giving dramatic equations x equals minus hyperbolic cosine of time and y equals the regular hyperbolic, the abrogelic sign of t, where t is between negative infinity and positive infinity.

00:27 So we've got to find a relation between these two.

00:30 Well, you could try to isolate and solve.

00:36 I don't know any good tricks to find like the hyperbolic cosine of the inverse hyperbolic sign or anything.

00:46 Not like i do with, not like you do with cosine and arc sign.

00:50 However, what we do know about the hyperbolic versions of the cosine and sine is that they satisfy a similar equation.

01:03 Only this time it is a minus sign instead of a plus sign.

01:10 So remember that for the cosine and sine, we have cosine of t squared plus sine ft squared equals one.

01:16 For the hyperbolic cosine and sine, it's highbolic cosine minus hypolytic sine squared equals one.

01:27 Okay, so if we know this, well, then it's easy.

01:32 Or easy, we at least know that x squared minus y squared equals one.

01:43 So let's isolate this a little bit.

01:49 So we could say like x squared equals one plus y squared, x equals plus or minus square root of one plus y squared.

02:06 So how are we going to graph this? so this is a physics trick or a trick i learned in physics.

02:15 If y is very big, 1 plus y squared and y squared are roughly the same thing.

02:22 So basically for y large, x is approximately just plus or minus the absolute value of y.

02:36 So that helps us graph it.

02:39 And for y zero, well, x is plus or minus one, depending on which branch we're in.

02:50 So let's graph this.

02:56 So we have plus one, and then we have like the absolute values.

03:06 So this is, these lines appear.

03:11 Let's draw them to do like i've been doing...

Please give Ace some feedback