Sketch the curve represented by the vector-valued function and give the orientation of the curve

Sketch the curve represented by the vector-valued function...

Key Concepts

Vector-valued Function

A vector-valued function assigns a vector to each real number input, commonly representing the position of a point in space over time. This concept is fundamental in describing trajectories and motions in multidimensional space.

Parametric Representation of Curves

This involves expressing the coordinates of points on a curve as functions of a single parameter, typically time. It provides a natural framework for analysis in physics and geometry for curves that cannot be easily described with a single Cartesian equation.

Elimination of the Parameter

Eliminating the parameter converts the parametric equations into a Cartesian equation. This process helps in understanding the overall shape and properties of the curve by relating the coordinates directly without reference to the parameter.

Curve Orientation

The orientation of a curve indicates the direction in which the curve is traced as the parameter increases. This is crucial in applications such as physics and engineering where the direction of movement along the path is important.

Curve Sketching Techniques

Sketching a curve involves analyzing features such as symmetry, intercepts, and behavior at infinity. Understanding these techniques allows for a clear visualization of the curve's geometric and dynamic properties.

Transcript

00:01 Okay, we're going to sketch the curve that is represented by the vector valued function.

00:06 So let's go ahead and make a t chart.

00:09 Our first t listed that is going to be our x component, the t squared, or y component, and the two -thirds, t -to -the -third, our z component.

00:19 So i picked my t values and going from negative 2 to positive 2.

00:24 I did decide to include some negative numbers because of the t -squared.

00:29 The other guys will be affected by negatives and that won't so i think it's important to have some negative numbers and then we're going to go ahead and fill out our chart x is just equal to t so i can rewrite those t values now y is t squared so i can just make sure to square all of my numbers and then the one that's a little messier is my z because i'm doing a two -thirds times a negative two to the third power well negative two the third power is and then when i multiply it by two -thirds, i get that negative 16 over 3.

01:03 I'll be thinking of that as a little below negative 5.

01:08 And then we have our negative 2 -thirds, our 0, and then 2 -thirds and 16 -thirds.

01:15 So that will be a little bit above 5.

01:19 Okay, so i'll jot down my axes.

01:21 Now, my first point, i'm going to be doing x negative 2, and then y -4, and remember i'm going to go down a little bit more than negative five.

01:33 And again, this is a sketch, so everything's just relative to each other.

01:37 Now i'm going to go back negative ones...