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Use the graphs of $ x = f(t) $ and $ y = g(t) $ to sketch the parametric curve $ x = f(t) $, $ y = g(t) $. Indicate with arrows the direction in which the cuve is traced as $ t $ increases.

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01:28

Wen Zheng

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Missouri State University

Campbell University

Harvey Mudd College

Boston College

Lectures

16:57

In mathematics, a graph is a representation of a set of objects where some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called "vertices" or "nodes", and the relations between them are represented by mathematical abstractions called "edges" or "arcs". The basic notion of a graph was developed by the 17th-century French mathematician Pierre de Fermat, and the term "graph" was coined by the 19th-century mathematician James Joseph Sylvester. The more general mathematical concept of a graph "in which any kind of relation between elements of the set is expressed as an edge, is called a network" (Kolmogorov, "1956, p. 111"). In other words, an undirected graph is a graph in which the edges have no direction associated with them. The most familiar examples of graphs are the graphs of equations. In general, the vertices of a graph can represent concepts and the edges can represent real-valued functions on the concepts, so one can speak of the graph as a function's graph or of the edge as a function's edge.

01:59

Polar coordinates are a two-dimensional coordinate system that specifies a point in terms of distance from a reference direction (the pole) and angle from a reference direction (the polar axis).

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So if we want to sketch this graph in the Cartesian plane, let's first go ahead and maybe just graph a couple of these points. That might be easy for us to find Well, over here. Um, so they give us 11 so we can estimate negative one is probably around here. Negative one is here, so let's just make a little chart. So when t is equal to negative one, it looks like First X is equal to one. And why is going to be equal to one also? Okay, um, when tea is able to zero, it looks like ax zero. It also looks like Why is zero And then when t is equal to one X is one again, and then it looks like why is equal to, uh, negative one? So let's come over here now and just graph those points that we have X Y, uh, I'll just put all those are just gonna be one for the tick marks. So at negative one, we have X is equal to one. Why is it good to one? So that would be right here, and this is T is equal to negative one. Now, when t 00 both X and wire zero. Alright. T is zero. And then when t is he going to one Access one again and then y is negative one. So this is t is he took one. Now we can kind of see So X is going to grow a lot quicker than why it's so when we're going through this or actually I should say access first, going to be decreasing to the left and why is always going to be decreasing So we should know that why should always be going down and then X was going to decrease up until we hit zero and they will start increasing again. So that means we can just go ahead and connect these points and it will kind of look like this quadratic here now for us to figure out what the direction is, we're just going to follow the points. So it goes from negative 1 to 0 and then 0 to 1. So this here is going to be a sketch of what this would look like in the XY plane

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