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Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = t $ , $ y = \frac{1}{(1 + t^2)} $ , $ z = t^2 $

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Graph 5

00:51

Wen Zheng

Calculus 3

Chapter 13

Vector Functions

Section 1

Vector Functions and Space Curves

Missouri State University

Campbell University

Harvey Mudd College

Boston College

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

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So what we're given is that X is equal to t. Why is equal to 1/1 plus t squared and Z is equal to t Square. So with this, we consider the given equations in the xz plane and we see that Z is going to equal X squared because we just plug in this value right here. So since because is X squared, that means that we have a projection that's a parabola. And then we also observe that the winds coordinates are never negative. Um, so we see that Y is going to be greater than or equal to zero, and Z is going to be greater than or equal to zero for all values of X. Um, so both of these properties are going to be only seen in graph five. So based on that, we can assume and predict and know that, given all of our options, these properties right here tell us that graph five is going to be the graph that's best described by these parametric equations

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