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Problem 47 Easy Difficulty

Investigate the family of curves defined by the parametric equations $ x = t^2 $, $ y = t^3 - ct $. How does the shape change as $ c $ increases? Illustrate by graphing several members of the family.


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Related Topics

Parametric Equations

Polar Coordinates

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Top Calculus 2 / BC Educators
Grace He
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

16:57

Graphing

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.

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01:59

Polar Coordinates - Intro

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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Watch More Solved Questions in Chapter 10

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Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
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Problem 13
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Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
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Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52

Video Transcript

the problem is investigate the family of curves you find about the parametric equations actually boarded his square. Why go to this Cuban minus city? How does the shape change? As said increases illustrated by graphing several members of the family? Little Cat's a graph here. This curve is sick or too negative one. And this one this they could. Cyril, this is safe. What? You three. This is sick with selling so we can see that one sees last and cereal. The function is an increasing function I see is the greatest and Cyril, the functions of first is secreting on, Then increasing and I see increases. We can see that the local minimum while you is getting smaller and smaller.

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Calculus: Early Transcendentals

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Related Topics

Parametric Equations

Polar Coordinates

Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

16:57

Graphing

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.

Video Thumbnail

01:59

Polar Coordinates - Intro

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).

Join Course
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