Chapter 8 Plane Curves and Parametric Equations. Copyright © Houghton Mifflin Company. All rights reserved.8 | 2 Definition of a Plane Curve.

Slides:

Advertisements

Similar presentations

Functions and Their Graphs

Advertisements

Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables.

Parametric Equations t x y

Chapter 13 Functions of Several Variables. Copyright © Houghton Mifflin Company. All rights reserved.13-2 Definition of a Function of Two Variables.

Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x

10 Conics, Parametric Equations, and Polar Coordinates

© 2010 Pearson Education, Inc. All rights reserved.

Digital Lesson Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and.

10.7 Polar Coordinates Adapted by JMerrill, 2011.

Copyright©2000 by Houghton Mifflin Company. All rights reserved. 1 QUESTION.

Polar Coordinates and Graphs of Polar Equations Digital Lesson.

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

Adapted by JMerrill, Copyright © by Houghton Mifflin Company, Inc. All rights reserved.2 Definition: Conic The locus of a point in the plane which.

Parametric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A pair of parametric equations are equations.

Parametric and Polar Curves; Conic Sections “Parametric Equations; Tangent Lines and Arc Length for Parametric Curves”

Vectors and the Geometry of Space

Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin.

Section 11.4 Areas and Lengths in Polar Coordinates.

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Major theorems, figures,

Polar Coordinates and Graphs of Polar Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The polar coordinate system is formed.

Advance Calculus Diyako Ghaderyan 1 Contents:  Applications of Definite Integrals  Transcendental Functions  Techniques of Integration.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

1.5 Infinite Limits. Copyright © Houghton Mifflin Company. All rights reserved Figure 1.25.

Chapter 1 Functions, Graphs, and Limits. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Figure 1.1: The Cartesian Plane.

Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved.

Chapter 1 Functions, Graphs, and Limits. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Figure 1.5: Pythagorean Theorem.

Chapter 15 Vector Analysis. Copyright © Houghton Mifflin Company. All rights reserved.15-2 Definition of Vector Field.

Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 10 Parametric and Polar Curves.

Conics, Parametric Equations, and Polar Coordinates

Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.

Advance Calculus Diyako Ghaderyan 1 Contents:  Applications of Definite Integrals  Transcendental Functions  Techniques of Integration.

9.6 Polar Coordinates Digital Lesson. HWQ 3/24 Find a set of parametric equations to represent the graph of using the parameter. Sketch a graph on showing.

7.4 Length of a Plane Curve y=f(x) is a smooth curve on [a, b] if f ’ is continuous on [a, b].

JMerrill,  Coordinate Conversion  To convert from polar to rectangular:  x = r cos θ  y = r sin θ  To convert from rectangular to polar: 

Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.

Chapter 14 Multiple Integration. Copyright © Houghton Mifflin Company. All rights reserved.14-2 Figure 14.1.

Chapter 12 Vector-Valued Functions. Copyright © Houghton Mifflin Company. All rights reserved.12-2 Definition of Vector-Valued Function.

Essential Question: How do you use derivatives with natural logs? Students will write a summary describing techniques used for taking the derivative with.

Chapter Seven Applications of Integration. Copyright © Houghton Mifflin Company. All rights reserved. 7 | 2 Figure 7.1.

Chapter 7 Linear Systems and Matrices. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 7–2 Section 7.1, The Method of Substitution,

Polar Coordinates and Graphs of Polar Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The polar coordinate system is formed.

Chapter 1 Functions and Their Graphs. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Section 1.1: Figure 1.1, The Cartesian Plane.

Chapter 2 Differentiation. Copyright © Houghton Mifflin Company. All rights reserved.2 | 2 Tangent Line to a Graph.

Chapter Three Differentiation. Copyright © Houghton Mifflin Company. All rights reserved. 3 | 2 Secant Line.

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions.

Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.

Chapter 4 Trigonometry. Copyright © Houghton Mifflin Company. All rights reserved.4 | 2Copyright © Houghton Mifflin Company. All rights reserved. Section.

Chapter 1 Ingredients of Change: Functions and Models.

Chapter 1 Functions and Their Graphs. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Section 1.1, Slope of a Line.

Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral.

Chapter 1 Functions and Their Graphs. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–2 Section 1.1, Figure 1.1, Illustration.

Trigonometric Functions

Digital Lesson Graphs of Equations.

Polar Coordinates and Graphs of Polar Equations

© 2010 Pearson Education, Inc. All rights reserved

Parametric Equations and Polar Coordinates

10 Conics, Parametric Equations, and Polar Coordinates

The Derivative and the Tangent Line Problems

x = 4y - 4 x = 4y + 4 x = 4y - 1 y = 4x - 4 y = 4x - 1 y = 4x + 4

Polar Coordinates and Graphs of Polar Equations

Parametric and Polar Curves

Digital Lesson Graphs of Equations.

Conic Sections and Polar Coordinates

Digital Lesson Parametric Equations.

Polar Coordinates and Graphs of Polar Equations

Digital Lesson Graphs of Equations.

Applications of Integration

5.3 Solving Trigonometric Equations

Plane Curves and Parametric Equations

Area and Arc Length in Polar Coordinates

Presentation transcript:

Chapter 8 Plane Curves and Parametric Equations

Copyright © Houghton Mifflin Company. All rights reserved.8 | 2 Definition of a Plane Curve

Copyright © Houghton Mifflin Company. All rights reserved.8 | 3 Eliminating the Parameter

Copyright © Houghton Mifflin Company. All rights reserved.8 | 4 Definition of a Smooth Curve

Copyright © Houghton Mifflin Company. All rights reserved.8 | 5 Figure 8.12: Parametric form of the Derivative

Copyright © Houghton Mifflin Company. All rights reserved.8 | 6 Second and Third Derivative

Copyright © Houghton Mifflin Company. All rights reserved.8 | 7 Figure 8.14: A Figure with Two Tangent lines at a Point

Copyright © Houghton Mifflin Company. All rights reserved.8 | 8 Arc Length in Parametric Form

Copyright © Houghton Mifflin Company. All rights reserved.8 | 9 Figure 8.15: An Epicycloid

Copyright © Houghton Mifflin Company. All rights reserved.8 | 10 Area of a Surface of Revolution

Copyright © Houghton Mifflin Company. All rights reserved.8 | 11 Figure 8.18: Polar Coordinates

Copyright © Houghton Mifflin Company. All rights reserved.8 | 12 Figure 8.20: Coordinate Conversion

Copyright © Houghton Mifflin Company. All rights reserved.8 | 13 Figure 8.25: Sketching the Graph of r = 2cos(3  )

Copyright © Houghton Mifflin Company. All rights reserved.8 | 14 Figure 8.27: Slope in Polar Forms

Copyright © Houghton Mifflin Company. All rights reserved.8 | 15 Tangent Lines at the Pole

Copyright © Houghton Mifflin Company. All rights reserved.8 | 16 Figure 8.31: The Area of a Sector of a Circle

Copyright © Houghton Mifflin Company. All rights reserved.8 | 17 Figure 8.32: Area in Polar Coordinates

Copyright © Houghton Mifflin Company. All rights reserved.8 | 18 Arc Length of a Polar Curve

Copyright © Houghton Mifflin Company. All rights reserved.8 | 19 Area of a Surface of Revolution

Copyright © Houghton Mifflin Company. All rights reserved.8 | 20 Review of Conics

Copyright © Houghton Mifflin Company. All rights reserved.8 | 21 Classification of Conics by Eccentricity

Copyright © Houghton Mifflin Company. All rights reserved.8 | 22 Figure 8.40: Graphs of Conics

Copyright © Houghton Mifflin Company. All rights reserved.8 | 23 Figure 8.42: Polar Equation of Conics