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(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.

$ x = \sin\dfrac{1}{2}\theta $, $ \quad y = \cos\dfrac{1}{2}\theta $, $ \quad -\pi \leqslant \theta \leqslant \pi $

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(a) (a) $x=\sin \frac{1}{2} \theta, y=\cos \frac{1}{2} \theta,-\pi \leq \theta \leq \pi$$x^{2}+y^{2}=\sin ^{2} \frac{1}{2} \theta+\cos ^{2} \frac{1}{2} \theta=1 .$ For $-\pi \leq \theta \leq 0,$ we have$-1 \leq x \leq 0$ and $0 \leq y \leq 1 .$ For $0<\theta \leq \pi,$ we have $0<x \leq 1$and $1>y \geq 0 .$ The graph is a semicircle.(b) Graph

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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(a) Eliminate the paramete…

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a) Eliminate the parameter…

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the problem is part of a eliminated the perimeter. Tto find a Kardashian question of occurred. I say we have exploited a sly squire. It's the control floor. What did I say between the two pie and pie we have? Why is between zero one? This is our petition question off the roof had to be sketch the curve on indicated ism. Arrow's direction. Inv Aged. The curve is traced as parameter increases, so we can't scratch the crew as follows. This is half a circle once here and once. So here On's Carol is the direction based the curve. It's traced as permitted increases.

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