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Suppose that the position of one particle at time $ t $ is given by$$ x_1 = 3 \sin t \quad y_1 = 1 + \sin t \quad 0 \leqslant t \leqslant 2 \pi $$and the position of a second particle is given by$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$(a) Graph the paths of both particles. How many points of intersection are there?(b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points.(c) Describe what happens if the path of the second particle given by$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$
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Calculus 2 / BC
Parametric Equations and Polar Coordinates
Curves Defined by Parametric Equations
Missouri State University
Oregon State University
Harvey Mudd College
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.
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).
Suppose that the position …
The paths of two particles…
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