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7.1: Introduction to Conics


In this chapter, we study the Conic Sections - literally `sections of a cone'. Imagine a double-napped cone as seen below being `sliced' by a plane.

If we slice the cone with a horizontal plane the resulting curve is a circle.

Tilting the plane ever so slightly produces an ellipse.

If the plane cuts parallel to the cone, we get a parabola.

If we slice the cone with a vertical plane, we get a hyperbola.

If the slicing plane contains the vertex of the cone, we get the so-called `degenerate' conics: a point, a line, or two intersecting lines.

We will focus the discussion on the non-degenerate cases: circles, parabolas, ellipses, and hyperbolas, in that order. To determine equations which describe these curves, we will make use of their definitions in terms of distances.


Thinkwell's Trigonometry with Professor Edward Burger

Thinkwell's Trigonometry homeschool course has engaging online video lessons and step-by-step exercises that teach you what you'll need to be successful in Calculus. The Trigonometry curriculum covers trig functions, identities, ratios and more. Instead of trying to learn what you need from an old-fashioned textbook, you can watch easy-to-understand trigonometry videos.

Thinkwell's award-winning math teacher, Edward Burger, can explain and demonstrate trig clearly to anyone, so trigonometry basics are easy to understand and remember. Professor Burger shares the tricks and tips so your students will remember them when they begin calculus.

The workbook (optional) comes with lecture notes, sample problems, and exercises so that you can study even when away from the computer.

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Trigonometry

Thinkwell's Trigonometry has high-quality online video lessons and step-by-step exercises that teach you what you'll need to be successful in Calculus. Thinkwell's Trigonometry covers the same topics that the most popular textbooks cover, so it's a perfect study aid. Instead of trying to learn what you need from an old-fashioned textbook, you can watch easy-to-understand trigonometry videos by one of our nation's best math teachers.

Thinkwell's award-winning math teacher, Edward Burger, can explain and demonstrate trig clearly to anyone, so trigonometry basics are easy to understand and remember. Professor Burger shares the tricks and tips so your students will remember them when they begin calculus. And because it's available 24/7 for one fixed price, instead of by the hour, it's better than a tutor.

  • 12-month Online Subscription to our complete Trigonometry course with video lessons, automatically graded trigonometry problems, and much more.
  • Workbook (optional) with lecture notes, sample problems, and exercises so that you can study even when away from the computer.

Trigonometry Materials

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  • High-quality video lessons explain all of the Trigonometry math topics and concepts
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Workbook, Notes, sample problems, exercises, and practice problems

Study without a computer. Our workbook companion contains the same lecture notes and sample problems that are delivered online, as well as some additional exercises, all in a convenient print format. Answers to the odd-numbered exercises are in the back of the book. Online Subscription is required workbook not sold separately.

Trigonometry Details

Thinkwell's Trigonometry has all the features your home school needs:

  • Equivalent to 11th- or 12th-grade trigonometry
  • More than 180 video lessons
  • 1000+ interactive trigonometry problems with immediate feedback allow you to track your progress
  • Trigonometry practice tests and final tests for all 8 chapters, as well as a midterm and a final
  • Printable illustrated notes for each topic
  • Real-world application examples in both lectures and exercises
  • Closed captioning for all videos
  • Glossary of more than 200 mathematical terms
  • Engaging content to help students advance their mathematical knowledge:
    • Review of graphs and functions
    • Trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant
    • Inverse trigonometric functions
    • Trigonometric identities
    • Law of Sines and Law of Cosines
    • Vectors and unit vectors
    • Complex numbers and polar coordinates
    • Exponential and logarithmic functions
    • Conic sections: parabolas, ellipses, and hyperbolas

    Table of Contents

    • 1.1 Graphing Basics
      • 1.1.1 Using the Cartesian System
      • 1.1.2 Thinking Visually
      • 1.2 Relationships between Two Points
        • 1.2.1 Finding the Distance between Two Points
        • 1.2.2 Finding the Second Endpoint of a Segment
        • 1.3 Relationships among Three Points
          • 1.3.1 Collinearity and Distance
          • 1.3.2 Triangles
          • 1.4 Circles
            • 1.4.1 Finding the Center-Radius Form of the Equation of a Circle
            • 1.4.2 Decoding the Circle Formula
            • 1.4.3 Finding the Center and Radius of a Circle
            • 1.4.4 Solving Word Problems Involving Circles
            • 1.5 Graphing Equations
              • 1.5.1 Graphing Equations by Locating Points
              • 1.5.2 Finding the x- and y-Intercepts of an Equation
              • 1.6 Function Basics
                • 1.6.1 Functions and the Vertical Line Test
                • 1.6.2 Identifying Functions
                • 1.6.3 Function Notation and Finding Function Values
                • 1.7 Working with Functions
                  • 1.7.1 Determining Intervals Over Which a Function Is Increasing
                  • 1.7.2 Evaluating Piecewise-Defined Functions for Given Values
                  • 1.7.3 Solving Word Problems Involving Functions
                  • 1.8 Function Domain and Range
                    • 1.8.1 Finding the Domain and Range of a Function
                    • 1.8.2 Domain and Range: One Explicit Example
                    • 1.8.3 Satisfying the Domain of a Function
                    • 1.9 Linear Functions: Slope
                      • 1.9.1 An Introduction to Slope
                      • 1.9.2 Finding the Slope of a Line Given Two Points
                      • 1.9.3 Interpreting Slope from a Graph
                      • 1.9.4 Graphing a Line Using Point and Slope
                      • 1.10 Equations of a Line
                        • 1.10.1 Writing an Equation in Slope-Intercept Form
                        • 1.10.2 Writing an Equation Given Two Points
                        • 1.10.3 Writing an Equation in Point-Slope Form
                        • 1.10.4 Matching a Slope-Intercept Equation with Its Graph
                        • 1.10.5 Slope for Parallel and Perpendicular Lines
                        • 1.11 Graphing Functions
                          • 1.11.1 Graphing Some Important Functions
                          • 1.11.2 Graphing Piecewise-Defined Functions
                          • 1.11.3 Matching Equations with Their Graphs
                          • 1.12 Manipulating Graphs: Shifts and Stretches
                            • 1.12.1 Shifting Curves along Axes
                            • 1.12.2 Shifting or Translating Curves along Axes
                            • 1.12.3 Stretching a Graph
                            • 1.12.4 Graphing Quadratics Using Patterns
                            • 1.13 Manipulating Graphs: Symmetry and Reflections
                              • 1.13.1 Determining Symmetry
                              • 1.13.2 Reflections
                              • 1.13.3 Reflecting Specific Functions
                              • 1.14 Quadratic Functions: Basics
                                • 1.14.1 Deconstructing the Graph of a Quadratic Function
                                • 1.14.2 Nice-Looking Parabolas
                                • 1.14.3 Using Discriminants to Graph Parabolas
                                • 1.14.4 Maximum Height in the Real World
                                • 1.15 Quadratic Functions: The Vertex
                                  • 1.15.1 Finding the Vertex by Completing the Square
                                  • 1.15.2 Using the Vertex to Write the Quadratic Equation
                                  • 1.15.3 Finding the Maximum or Minimum of a Quadratic
                                  • 1.15.4 Graphing Parabolas
                                  • 1.16 Composite Functions
                                    • 1.16.1 Using Operations on Functions
                                    • 1.16.2 Composite Functions
                                    • 1.16.3 Components of Composite Functions
                                    • 1.16.4 Finding Functions That Form a Given Composite
                                    • 1.16.5 Finding the Difference Quotient of a Function
                                    • 1.16.6 Calculating the Average Rate of Change
                                    • 1.17 Rational Functions
                                      • 1.17.1 Understanding Rational Functions
                                      • 1.17.2 Basic Rational Functions
                                      • 1.18 Graphing Rational Functions
                                        • 1.18.1 Vertical Asymptotes
                                        • 1.18.2 Horizontal Asymptotes
                                        • 1.18.3 Graphing Rational Functions
                                        • 1.18.4 Graphing Rational Functions: More Examples
                                        • 1.18.5 Oblique Asymptotes
                                        • 1.18.6 Oblique Asymptotes: Another Example
                                        • 1.19 Function Inverses
                                          • 1.19.1 Understanding Inverse Functions
                                          • 1.19.2 The Horizontal Line Test
                                          • 1.19.3 Are Two Functions Inverses of Each Other?
                                          • 1.19.4 Graphing the Inverse
                                          • 1.20 Finding Function Inverses
                                            • 1.20.1 Finding the Inverse of a Function
                                            • 1.20.2 Finding the Inverse of a Function with Higher Powers

                                            2. The Trigonometric Functions

                                            • 2.1 Angles and Radian Measure
                                              • 2.1.1 Finding the Quadrant in Which an Angle Lies
                                              • 2.1.2 Finding Coterminal Angles
                                              • 2.1.3 Finding the Complement and Supplement of an Angle
                                              • 2.1.4 Converting between Degrees and Radians
                                              • 2.1.5 Using the Arc Length Formula
                                              • 2.2 Right Angle Trigonometry
                                                • 2.2.1 An Introduction to the Trigonometric Functions
                                                • 2.2.2 Evaluating Trigonometric Functions for an Angle in a Right Triangle
                                                • 2.2.3 Finding an Angle Given the Value of a Trigonometric Function
                                                • 2.2.4 Using Trigonometric Functions to Find Unknown Sides of Right Triangles
                                                • 2.2.5 Finding the Height of a Building
                                                • 2.3 The Trigonometric Functions
                                                  • 2.3.1 Evaluating Trigonometric Functions for an Angle in the Coordinate Plane
                                                  • 2.3.2 Evaluating Trigonometric Functions Using the Reference Angle
                                                  • 2.3.3 Finding the Value of Trigonometric Functions Given Information about the Values of Other Trigonometric Functions
                                                  • 2.3.4 Trigonometric Functions of Important Angles
                                                  • 2.4 Graphing Sine and Cosine Functions
                                                    • 2.4.1 An Introduction to the Graphs of Sine and Cosine Functions
                                                    • 2.4.2 Graphing Sine or Cosine Functions with Different Coefficients
                                                    • 2.4.3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine
                                                    • 2.4.4 Solving Word Problems Involving Sine or Cosine Functions
                                                    • 2.5 Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
                                                      • 2.5.1 Graphing Sine and Cosine Functions with Phase Shifts
                                                      • 2.5.2 Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift
                                                      • 2.6 Graphing Other Trigonometric Functions
                                                        • 2.6.1 Graphing the Tangent, Secant, Cosecant, and Cotangent Functions
                                                        • 2.6.2 Fancy Graphing: Tangent, Secant, Cosecant, and Cotangent
                                                        • 2.6.3 Identifying a Trigonometric Function from its Graph
                                                        • 2.7 Inverse Trigonometric Functions
                                                          • 2.7.1 An Introduction to Inverse Trigonometric Functions
                                                          • 2.7.2 Evaluating Inverse Trigonometric Functions
                                                          • 2.7.3 Solving an Equation Involving an Inverse Trigonometric Function
                                                          • 2.7.4 Evaluating the Composition of a Trigonometric Function and Its Inverse
                                                          • 2.7.5 Applying Trigonometric Functions: Is He Speeding?

                                                          3. Trigonometric Identities

                                                          • 3.1 Basic Trigonometric Identities
                                                            • 3.1.1 Fundamental Trigonometric Identities
                                                            • 3.1.2 Finding All Function Values
                                                            • 3.2 Simplifying Trigonometric Expressions
                                                              • 3.2.1 Simplifying a Trigonometric Expression Using Trigonometric Identities
                                                              • 3.2.2 Simplifying Trigonometric Expressions Involving Fractions
                                                              • 3.2.3 Simplifying Products of Binomials Involving Trigonometric Functions
                                                              • 3.2.4 Factoring Trigonometric Expressions
                                                              • 3.2.5 Determining Whether a Trigonometric Function Is Odd, Even, or Neither
                                                              • 3.3 Proving Trigonometric Identities
                                                                • 3.3.1 Proving an Identity
                                                                • 3.3.2 Proving an Identity: Other Examples
                                                                • 3.4 Solving Trigonometric Equations
                                                                  • 3.4.1 Solving Trigonometric Equations
                                                                  • 3.4.2 Solving Trigonometric Equations by Factoring
                                                                  • 3.4.3 Solving Trigonometric Equations with Coefficients in the Argument
                                                                  • 3.4.4 Solving Trigonometric Equations Using the Quadratic Formula
                                                                  • 3.4.5 Solving Word Problems Involving Trigonometric Equations
                                                                  • 3.5 The Sum and Difference Identities
                                                                    • 3.5.1 Identities for Sums and Differences of Angles
                                                                    • 3.5.2 Using Sum and Difference Identities
                                                                    • 3.5.3 Using Sum and Difference Identities to Simplify an Expression
                                                                    • 3.6 Double-Angle Identities
                                                                      • 3.6.1 Confirming a Double-Angle Identity
                                                                      • 3.6.2 Using Double-Angle Identities
                                                                      • 3.6.3 Solving Word Problems Involving Multiple-Angle Identities
                                                                      • 3.7 Other Advanced Identities
                                                                        • 3.7.1 Using a Cofunction Identity
                                                                        • 3.7.2 Using a Power-Reducing Identity
                                                                        • 3.7.3 Using Half-Angle Identities to Solve a Trigonometric Equation

                                                                        4. Applications of Trigonometry

                                                                        • 4.1 The Law of Sines
                                                                          • 4.1.1 The Law of Sines
                                                                          • 4.1.2 Solving a Triangle Given Two Sides and One Angle
                                                                          • 4.1.3 Solving a Triangle (SAS): Another Example
                                                                          • 4.1.4 The Law of Sines: An Application
                                                                          • 4.2 The Law of Cosines
                                                                            • 4.2.1 The Law of Cosines
                                                                            • 4.2.2 The Law of Cosines (SSS)
                                                                            • 4.2.3 The Law of Cosines (SAS): An Application
                                                                            • 4.2.4 Heron's Formula
                                                                            • 4.3 Vector Basics
                                                                              • 4.3.1 An Introduction to Vectors
                                                                              • 4.3.2 Finding the Magnitude and Direction of a Vector
                                                                              • 4.3.3 Vector Addition and Scalar Multiplication
                                                                              • 4.4 Components of Vectors and Unit Vectors
                                                                                • 4.4.1 Finding the Components of a Vector
                                                                                • 4.4.2 Finding a Unit Vector
                                                                                • 4.4.3 Solving Word Problems Involving Velocity or Forces

                                                                                5. Complex Numbers and Polar Coordinates

                                                                                • 5.1 Complex Numbers
                                                                                  • 5.1.1 Introducing and Writing Complex Numbers
                                                                                  • 5.1.2 Rewriting Powers of i
                                                                                  • 5.1.3 Adding and Subtracting Complex Numbers
                                                                                  • 5.1.4 Multiplying Complex Numbers
                                                                                  • 5.1.5 Dividing Complex Numbers
                                                                                  • 5.2 Complex Numbers in Trigonometric Form
                                                                                    • 5.2.1 Graphing a Complex Number and Finding Its Absolute Value
                                                                                    • 5.2.2 Expressing a Complex Number in Trigonometric or Polar Form
                                                                                    • 5.2.3 Multiplying and Dividing Complex Numbers in Trigonometric or Polar Form
                                                                                    • 5.3 Powers and Roots of Complex Numbers
                                                                                      • 5.3.1 Using DeMoivre's Theorem to Raise a Complex Number to a Power
                                                                                      • 5.3.2 Roots of Complex Numbers
                                                                                      • 5.3.3 More Roots of Complex Numbers
                                                                                      • 5.3.4 Roots of Unity
                                                                                      • 5.4 Polar Coordinates
                                                                                        • 5.4.1 An Introduction to Polar Coordinates
                                                                                        • 5.4.2 Converting between Polar and Rectangular Coordinates
                                                                                        • 5.4.3 Converting between Polar and Rectangular Equations
                                                                                        • 5.4.4 Graphing Simple Polar Equations
                                                                                        • 5.4.5 Graphing Special Polar Equations

                                                                                        6. Exponential and Logarithmic Functions

                                                                                        • 6.1 Exponential Functions
                                                                                          • 6.1.1 An Introduction to Exponential Functions
                                                                                          • 6.1.2 Graphing Exponential Functions: Useful Patterns
                                                                                          • 6.1.3 Graphing Exponential Functions: More Examples
                                                                                          • 6.2 Applying Exponential Functions
                                                                                            • 6.2.1 Using Properties of Exponents to Solve Exponential Equations
                                                                                            • 6.2.2 Finding Present Value and Future Value
                                                                                            • 6.2.3 Finding an Interest Rate to Match Given Goals
                                                                                            • 6.3 The Number e
                                                                                              • 6.3.1 e
                                                                                              • 6.3.2 Applying Exponential Functions
                                                                                              • 6.4 Logarithmic Functions
                                                                                                • 6.4.1 An Introduction to Logarithmic Functions
                                                                                                • 6.4.2 Converting between Exponential and Logarithmic Functions
                                                                                                • 6.5 Solving Logarithmic Functions
                                                                                                  • 6.5.1 Finding the Value of a Logarithmic Function
                                                                                                  • 6.5.2 Solving for x in Logarithmic Equations
                                                                                                  • 6.5.3 Graphing Logarithmic Functions
                                                                                                  • 6.5.4 Matching Logarithmic Functions with Their Graphs
                                                                                                  • 6.6 Properties of Logarithms
                                                                                                    • 6.6.1 Properties of Logarithms
                                                                                                    • 6.6.2 Expanding a Logarithmic Expression Using Properties
                                                                                                    • 6.6.3 Combining Logarithmic Expressions
                                                                                                    • 6.7 Evaluating Logarithms
                                                                                                      • 6.7.1 Evaluating Logarithmic Functions Using a Calculator
                                                                                                      • 6.7.2 Using the Change of Base Formula
                                                                                                      • 6.8 Applying Logarithmic Functions
                                                                                                        • 6.8.1 The Richter Scale
                                                                                                        • 6.8.2 The Distance Modulus Formula
                                                                                                        • 6.9 Solving Exponential and Logarithmic Equations
                                                                                                          • 6.9.1 Solving Exponential Equations
                                                                                                          • 6.9.2 Solving Logarithmic Equations
                                                                                                          • 6.9.3 Solving Equations with Logarithmic Exponents
                                                                                                          • 6.10 Applying Exponents and Logarithms
                                                                                                            • 6.10.1 Compound Interest
                                                                                                            • 6.10.2 Predicting Change
                                                                                                            • 6.11 Word Problems Involving Exponential Growth and Decay
                                                                                                              • 6.11.1 An Introduction to Exponential Growth and Decay
                                                                                                              • 6.11.2 Half-Life
                                                                                                              • 6.11.3 Newton's Law of Cooling
                                                                                                              • 6.11.4 Continuously Compounded Interest

                                                                                                              7. Conic Sections

                                                                                                              • 7.1 Conic Sections: Parabolas
                                                                                                                • 7.1.1 An Introduction to Conic Sections
                                                                                                                • 7.1.2 An Introduction to Parabolas
                                                                                                                • 7.1.3 Determining Information about a Parabola from Its Equation
                                                                                                                • 7.1.4 Writing an Equation for a Parabola
                                                                                                                • 7.2 Conic Sections: Ellipses
                                                                                                                  • 7.2.1 An Introduction to Ellipses
                                                                                                                  • 7.2.2 Finding the Equation for an Ellipse
                                                                                                                  • 7.2.3 Applying Ellipses: Satellites
                                                                                                                  • 7.2.4 The Eccentricity of an Ellipse
                                                                                                                  • 7.3 Conic Sections: Hyperbolas
                                                                                                                    • 7.3.1 An Introduction to Hyperbolas
                                                                                                                    • 7.3.2 Finding the Equation for a Hyperbola
                                                                                                                    • 7.3.3 Applying Hyperbolas: Navigation
                                                                                                                    • 7.4 Conic Sections
                                                                                                                      • 7.4.1 Identifying a Conic
                                                                                                                      • 7.4.2 Name That Conic
                                                                                                                      • 7.4.3 Rotation of Axes
                                                                                                                      • 7.4.4 Rotating Conics

                                                                                                                      About the Author

                                                                                                                      Edward Burger is an award-winning professor with a passion for teaching mathematics.

                                                                                                                      Since 2013, Edward Burger has been President of Southwestern University, a top-ranked liberal arts college in Georgetown, Texas. Previously, he was Professor of Mathematics at Williams College. Dr. Burger earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

                                                                                                                      Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.


                                                                                                                      An Introduction to NURBS

                                                                                                                      The latest from a computer graphics pioneer, An Introduction to NURBS is the ideal resource for anyone seeking a theoretical and practical understanding of these very important curves and surfaces. Beginning with Bézier curves, the book develops a lucid explanation of NURBS curves, then does the same for surfaces, consistently stressing important shape design properties and the capabilities of each curve and surface type. Throughout, it relies heavily on illustrations and fully worked examples that will help you grasp key NURBS concepts and deftly apply them in your work. Supplementing the lucid, point-by-point instructions are illuminating accounts of the history of NURBS, written by some of its most prominent figures.

                                                                                                                      Whether you write your own code or simply want deeper insight into how your computer graphics application works, An Introduction to NURBS will enhance and extend your knowledge to a degree unmatched by any other resource.

                                                                                                                      The latest from a computer graphics pioneer, An Introduction to NURBS is the ideal resource for anyone seeking a theoretical and practical understanding of these very important curves and surfaces. Beginning with Bézier curves, the book develops a lucid explanation of NURBS curves, then does the same for surfaces, consistently stressing important shape design properties and the capabilities of each curve and surface type. Throughout, it relies heavily on illustrations and fully worked examples that will help you grasp key NURBS concepts and deftly apply them in your work. Supplementing the lucid, point-by-point instructions are illuminating accounts of the history of NURBS, written by some of its most prominent figures.

                                                                                                                      Whether you write your own code or simply want deeper insight into how your computer graphics application works, An Introduction to NURBS will enhance and extend your knowledge to a degree unmatched by any other resource.


                                                                                                                      CONIC SECTIONS - PowerPoint PPT Presentation

                                                                                                                      PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

                                                                                                                      You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

                                                                                                                      For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

                                                                                                                      presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!


                                                                                                                      Conic Sections: The Ellipse - PowerPoint PPT Presentation

                                                                                                                      PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

                                                                                                                      You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

                                                                                                                      For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

                                                                                                                      presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!


                                                                                                                      7.1 Integration by Parts

                                                                                                                      Introduction: In this lesson we will learn to integrate functions using the integration by parts technique. Thus far, the only integration technique that you have seen is a u-substitution. The u-substitution works by “undoing” the Chain Rule. In this lesson, we will learn the integration by parts technique is “undoing” the Product Rule.

                                                                                                                      Objectives: After this lesson you should be able to:

                                                                                                                      Video & Notes: Fill out the note sheet for this lesson (7-1-Integration-by-Parts) as you watch the video. If you prefer, you could read Section 7.1 of your textbook and work out the problems on the notes on your own as practice. Remember, notes must be uploaded to Blackboard weekly for a grade! If for some reason the video below does not load you can access it on YouTube here.

                                                                                                                      Homework: Go to WebAssign and complete the 𔄟.1 Integration by Parts” assignment.

                                                                                                                      Practice Problems: # 5, 9, 17, 19, 23, 27, 31, 35, 37, 39

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                                                                                                                      Conic Sections - Circles

                                                                                                                      A series of free, online video lessons with examples and solutions to help Algebra students learn about circle conic sections.

                                                                                                                      The following diagram shows how to derive the equation of circle (x - h) 2 + (y - k) 2 = r 2 using Pythagorean Theorem and distance formula. Scroll down the page for examples and solutions.

                                                                                                                      Circle Conic Section

                                                                                                                      When working with circle conic sections, we can derive the equation of a circle by using coordinates and the distance formula.
                                                                                                                      The equation of a circle is (x - h) 2 + (y - k) 2 = r 2 where r is equal to the radius, and the coordinates (x,y) are equal to the circle center.
                                                                                                                      The variables h and k represent horizontal or vertical shifts in the circle graph.
                                                                                                                      Examples:
                                                                                                                      1. Find the center and the radius
                                                                                                                      a) x 2 + (y + 2) 2 = 121
                                                                                                                      b) (x + 5) 2 + (y - 10) 2 = 9

                                                                                                                      2. Find the equation the circle with
                                                                                                                      a) center(-11, -8) and radius 4
                                                                                                                      b) center (2, -5) and point on circle(-7, -1)

                                                                                                                      How To Graph A Circle In Standard Form And General Form?

                                                                                                                      Identify the equation of a circle.
                                                                                                                      Write the standard form of a circle from general form.
                                                                                                                      Graph a circle.
                                                                                                                      A circle is the set of points (x,y) which are a fixed distance r, the radius, away from a fixed point (h,k), the center.
                                                                                                                      (x - h) 2 + (y - k) 2 = r 2
                                                                                                                      Examples:
                                                                                                                      1. Graph the circle
                                                                                                                      a) (x - 3) 2 + (y + 2) 2 = 16
                                                                                                                      b) x 2 + (y - 1) 2 = 4

                                                                                                                      2. Write in standard form and then graph
                                                                                                                      2x 2 + 2y 2 - 12x + 8y - 24 = 0

                                                                                                                      Conic Sections

                                                                                                                      Introduction to Circles
                                                                                                                      Understand the equation of a circle

                                                                                                                      Graph And Write Equations Of Circles

                                                                                                                      Example:
                                                                                                                      Graph the equation
                                                                                                                      (x - 1) 2 + (y + 2) 2 = 9

                                                                                                                      Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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                                                                                                                      ArcGIS ( Desktop, Server) 10.7.1 Belge Lambert 1972 Equivalency Patch

                                                                                                                      This patch adds the ability to handle both versions of the Belge Lambert 1972 projected coordinate systems as if they are equivalent.

                                                                                                                      Esri® announces the ArcGIS ( Desktop, Server) 10.7.1 Belge Lambert 1972 Equivalency Patch. Esri software has been using a very early definition of the Belge Lambert 1972 projected coordinate system, EPSG:31370. The false easting and false northing values are more precise than the current official NGI/IGN definition. The values differ by 0.44 mm and 0.2 mm. For ArcMap 10.7.0, Esri fixed the definition and created a second definition for the previous definition. Existing data, particularly web services, lost their WKID information, causing data handling problems. This patch ensures both definitions are treated as if they are equal either one will be identified as EPSG:31370. This patch deals specifically with the issue listed below under Issues Addressed with this patch.

                                                                                                                      Issues Addressed with this patch

                                                                                                                      • BUG-000130263 - Handle the old and new versions of the Belge Lambert 1972 projected coordinate systems as if they are equivalent.

                                                                                                                      Installing this patch on Windows

                                                                                                                      Installation Steps:

                                                                                                                      The ArcGIS product listed in the table must be installed on your system before you can install a patch. Each patch setup is specific to the ArcGIS product in the list. To determine which products are installed on your system, please see the How to identify which ArcGIS products are installed section. Esri recommends that you install the patch for each product that is on your system.

                                                                                                                        Download the appropriate file to a location other than your ArcGIS installation location.

                                                                                                                      NOTE: If double clicking on the MSP file does not start the setup installation, you can start the setup installation manually by using the following command:

                                                                                                                      Installing this patch on Linux

                                                                                                                      Installation Steps:

                                                                                                                      Complete the following install steps as the ArcGIS Install owner. The Install owner is the owner of the arcgis folder.

                                                                                                                      The ArcGIS product listed in the table must be installed on your system before you can install a patch. Each patch setup is specific to the ArcGIS product in the list. To determine which products are installed on your system, please see the How to identify which ArcGIS products are installed section. Esri recommends that you install the patch for each product that is on your system.

                                                                                                                        Download the appropriate file to a location other than your ArcGIS installation location.

                                                                                                                      ArcGIS 10.7.1 Checksum (Md5)
                                                                                                                      ArcGIS Server ArcGIS-1071-S-BLE-Patch-linux.tar BB9642CB29AD2F683157DDBC24C287F4

                                                                                                                      Uninstalling this patch on Windows

                                                                                                                        To uninstall this patch on Windows, open the Windows Control Panel and navigate to installed programs. Make sure that "View installed updates" (upper left side of the Programs and Features dialog) is active. Select the patch name from the programs list and click Uninstall to remove the patch.

                                                                                                                      Uninstalling this patch on Linux

                                                                                                                        To remove this patch on versions 10.7 and higher, navigate to the /tmp directory and run the following script as the ArcGIS Install owner:

                                                                                                                      The removepatch.sh script allows you to uninstall previously installed patches or hot fixes. Use the -s status flag to get the list of installed patches or hot fixes ordered by date. Use the -q flag to remove patches or hot fixes in reverse chronological order by date they were installed. Type removepatch -h for usage help.

                                                                                                                      Patch Updates

                                                                                                                      Check the Esri Support Downloads page periodically for the availability of additional patches. New information about this patch will be posted here.

                                                                                                                      How to identify which ArcGIS products are installed

                                                                                                                      To determine which ArcGIS products are installed, choose the appropriate version of the PatchFinder utility for your environment and run it from your local machine. PatchFinder will list all products, hot fixes, and patches installed on your local machine.

                                                                                                                      Getting Help

                                                                                                                      Domestic sites, please contact Esri Technical Support at 1-888-377-4575, if you have any difficulty installing this patch. International sites, please contact your local Esri software distributor.


                                                                                                                      Watch the video: Conic Section 3D Animation (November 2021).