7-14 Sketch the curve with the given vector equation. Indicate with an arrow the direction in wh

7-14 Sketch the curve with the given vector equation....

Key Concepts

Vector-Valued Functions

Vector-valued functions express curves or surfaces in the plane or space by assigning each parameter value a vector that represents a point. They are often written as r(t) = ?x(t), y(t)? (or with more components in higher dimensions), where each component is a function of the parameter. This representation is used to trace the path of a particle or describe geometric shapes.

Parametric Equations

Parametric equations provide a way to represent curves by defining each coordinate as a separate function of a common parameter. This method is especially useful for describing curves that cannot be represented as functions of x or y alone, allowing one to analyze the behavior and properties of the curve by varying the parameter.

Trigonometric Functions in Parametrizations

Trigonometric functions, such as sine and cosine, are often used in parametric equations to introduce periodic or oscillatory behavior. When incorporated into the coordinate functions, they can create patterns like cycles, loops, or waves, enriching the geometric and analytic properties of the curve.

Curve Orientation and Direction of Motion

Indicating the direction in which the parameter increases is crucial for understanding the orientation and behavior of a curve. An arrow drawn on the curve shows the path that a particle would follow as the parameter increases, thereby highlighting the dynamic aspects of the curve and clarifying which end corresponds to earlier or later values in the parameter interval.

Transcript

00:01 So they want us to sketch this vector -valued function that they give us right here.

00:07 So let's go ahead and get some intuition as to what we're actually going to be drawn.

00:12 So first, we have that x is really equal to sine of t, and then y is equal to t.

00:22 So i guess we could technically could say like x of t and y of t, but this is essentially what we're going to be working with.

00:29 Now, notice over here, x is going to be bounded between 1 and negative 1, just because sine is bounded between 1 and negative 1.

00:40 And then t, well, that's unbounded, so it's just going to be increasing forever in the y and decreasing forever in the negative y direction.

00:49 So if we were to kind of sketch this before we actually do anything, it looks like x is going to be, so it's just bounded in between.

01:00 These two functions like this and then the y is just going to keep on increasing or decreasing depending on which way we go so this is kind of the general shape of what our curve should look like so let's actually make it look a little bit prettier because they also want us to plot and say what the direction is so we already know x is going to be bounded between negative one and one so let's just go ahead and put those down negative one and one and now let's just plug in some points in c so it's going to be r of 0 is equal to.

01:35 So sine of 0 is 0.

01:38 T, oh, that's just going to be 0.

01:41 So we're going to start at this point.

01:44 And then we can plug in some other point, something that probably doesn't give us the 0.

01:50 So let's do like r of pi half.

01:58 So doing this will give one pi half.

02:02 So that means, well, where's pi off? so it would be like 1, 2, 3.

02:10 And so pi half is going to be a little above 1 or so.

02:19 So probably like around here is going to be where r of pi half is.

02:28 So remember earlier we said this is going to be like this kind of squiggly line.

02:33 So we're going to start here...