In Exercises 19-24, match the parametric equations with the parametric curves labeled A through

step 1 So for this question, we are given parametric equations. So x is n t and y equals 3E to the minu

In Exercises 19-24, match the parametric equations with the...

Key Concepts

Domain Considerations

Domain considerations involve recognizing the allowable values of the parameter for which the functions are defined. For example, the logarithmic function ln(t) necessitates that t be positive, thereby limiting the domain and subsequently affecting the portion of the curve that is drawn in the coordinate plane.

Parametric Equations

Parametric equations represent curves using a parameter, typically denoted by t. They define both x and y (and possibly z) as functions of this single independent variable. This method is useful for representing a wide variety of curves that may not be easily described by a standard Cartesian equation.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to express y directly in terms of x, or vice versa. This process helps in understanding the overall structure and shape of the curve by relating the dependent variables directly without the parameter.

Exponential and Logarithmic Functions

The use of exponential and logarithmic functions in parametric equations introduces unique behaviors such as rapid growth or decay and domain restrictions. Logarithmic functions require their arguments to be positive, which influences the domain of the parameter, and the interaction with exponential functions can result in curves with distinctive characteristics that are useful in many applications.

Transcript

00:01 So for this question, we are given parametric equations.

00:07 So x is n t and y equals 3e to the minus t over 2.

00:20 And we don't know what the domain of t is, but we are asked to identify the graph of this barometric equation along a set of other ones.

00:33 So here we have all the graphs in order and we're asking which one is it.

00:40 So to do that, we are going to look at this and just start to identify some things.

00:46 So x is an ln of t.

00:52 So if t is just allowed to be any positive number, x is going to be any number.

00:59 So we'd expect the graph to extend infinitely in the x directions.

01:08 Looking at the y one, we expect, well, y is a positive function.

01:16 So whatever graph it is, we know y is bigger than a equal to 0.

01:25 In fact, it's actually just straight up bigger than 0.

01:32 So why is bigger than 0? this is something we know.

01:37 And then it might be useful to just find some points.

01:40 So let's say like t equals 1.

01:48 So we'd find x equals 0 and y equals 3 times e to the minus 1⁄22.

01:59 Which is 3 divided by the square root of e.

02:07 Now, you shouldn't try to do this exactly.

02:11 The exact number doesn't even matter that much.

02:15 So e is 2 .7, it's pretty close to 3.

02:18 So this is going to be close to square root of 3, which is going to be something like, i don't know, it's going to be like 1 .6, 1 .7, something like this.

02:31 So let's say it's roughly 1 .6, maybe 1 .7.

02:35 Let's say 1 .7...

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