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Investigate the family of curves defined by the parametric equations $ x = \cos t $, $ y = \sin t - \sin ct $, where $ c > 0 $. Start by letting $ c $ be a positive integer and see what happens to the shape as $ c $ increases. Then explode some of the possibilities that occur when $ c $ is a fraction.

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Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Campbell University

Harvey Mudd College

University of Nottingham

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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Investigate the family of …

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$$ \begin{array}{l}{\text …

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the problem is investigated. The family off curves did find the bus. Parametric equations actually put this society while going to sign the minus sign stayed home steam wire seasick with his and cereal. Start by light and save your positive integer on DH. Say what happens to the ship at sea increases. They explored some of the possibilities that our care What C is a fraction. Look at this graph. Here I see a single one. The curve is a line segment. I say you got to. It's a curve into a sex itself. At one point, my C is equal to three. Curve intersects itself at two points. Must see is it four? The curve intersects itself at three points so we can guess. For safe go to some in and this positive interview the curve intersects the south at a month. One points. Then let's look at some examples. What sees a fraction. If the denominator is equal to three we have. That's a look at this example. If sees it, criticism. One on the denominator is equal to three. Your half the curve dissects itself at fifteen points. It says he got a four one three Andi. If sees you go too far over three, the curve in the sects itself. All right, eighteen points and over half. But if the denominator is not equal to three point damn point if this is too, we have if this is seven over too, the curve in just sects in the south at their team points.

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