Calculus II - 19
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There are three types of conic sections: parabolas, ellipses, and hyperbolas. A parabola is the set of points equidistant from a fixed point (focus) and line (directrix). An ellipse is the set of points where the sum of distances from two fixed points (foci) is a constant. A hyperbola is the set of points where the difference of distances from two fixed points (foci) is a constant. Equations of parabolas, ellipses, and hyperbolas can be written and their properties like foci, vertices and asymptotes can be determined from the equations.
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This document provides a review of key concepts in Algebra II including:
1) Methods for solving quadratic equations such as factoring and the quadratic formula.
2) Properties and equations of parabolas, circles, ellipses, and hyperbolas including their standard forms and key components.
3) Instructions for using a graphing calculator to graph functions and view graphs.
4) Information about the format of the upcoming test which will include 8-10 questions involving these graphing concepts.
Circles
Conics are curves formed by the intersection of a plane with a double cone. A hyperbola results if the plane cuts both cones, a parabola if the plane is parallel to the edge of the cone, and an ellipse if neither of those cases apply. A circle is a special type of ellipse that occurs when the plane is perpendicular to the altitude of the cone.
Conic Section
This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations and key characteristics for each conic section with varying positions of the vertex. Circles are defined by a center point and radius. Parabolas are defined by a focus, directrix, and vertex. Ellipses are defined by two foci and the sum of distances to these points. Hyperbolas are defined by two foci and the difference of distances to these points. Examples of each conic section in architecture and acoustics are also given.
Conic Section slayerix
This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations for each conic section with variations in positioning, including the location of the center, vertices, foci, and axes. Key characteristics of each type of conic section are described, such as the relationship between the foci and the shape of the curve. Examples of each conic section in architecture and other applications are also mentioned.
Calculus II - 16
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
Calculus II - 15
The document discusses parametric equations and parametric curves. Parametric equations define curves using two equations, where x and y are defined in terms of a parameter t. Common examples of parametric curves are given. Parametric equations can be used to define circles by giving the x and y coordinates in terms of the center point and radius. The parameter t can be eliminated to obtain the Cartesian equation of the curve. Calculus can be applied to parametric curves to find tangents, areas, arc length, and surfaces of revolution. Formulas are given for finding the equations of tangent lines to parametric curves at a given value of t.
Lecture co2 math 21-1
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Conics
The document discusses hyperbolas, which are conic sections that have two disconnected branches. Key elements of a hyperbola include its foci, vertices, transverse/conjugate axes, center, and asymptotes. The eccentricity measures the "bendiness" of the hyperbola. Equations of hyperbolas depend on whether the transverse axis is horizontal or vertical. The document provides examples and exercises on finding equations and properties of hyperbolas given certain information.
Lesson 2: A Catalog of Essential Functions (slides)
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
114333628 irisan-kerucut
The document discusses properties of parabolas, including their definition as the set of points equidistant from a focus point and directrix line. It presents the standard equation for a par
114333628 irisan-kerucut
The document discusses properties and equations of circles, including the standard form of a circle equation with a given center and radius. It also discusses tangent lines to circles, providing the process and equations for finding the equation of a tangent line to a circle at a given
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Math - analytic geometry
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
15.) Line L in the figure below is parallel to the line y=.docx
15.)
Line L in the figure below is parallel to the line y=4x+2. Find the coordinates of the point P
P=
19.)
Match the Lines L1 (blue), L2 (red) and L3 (green) with the slopes by placing the letter of the slopes next to each set listed below
The slope of line L2:
The slope of line L3:
The slope of line L1:
A. m=−1.
B. m=0
C. m=0.2
24.) The height s of a ball (in feet) at time t(in seconds) is given by the formula s(t)=64−16(t−1) s(t)=64−16(t−1)^2 . Consider the following expression.
AV[0.5,1]=s(1)−s(0.5)/1−0.5
Compute the value of AV[0.5,1]=
What does this value measure geometrically?
slope. The slope of the line between the points (0.5,s(0.5))and (1,s(1))(1,s(1)).
curvature. The curvature of the graph of ss on the interval 0.5≤t≤1.
area. The area under the graph of ss on the interval 0.5≤t≤1.
distance. The distance between the points (0.5,s(0.5)) and (1,s(1))(1,s(1)).
What does this value measure physically?
average velocity. The average velocity of the ball during the interval 0.5≤t≤10.5≤t≤1.
distance. The distance that the ball travels during the interval 0.5≤t≤10.5≤t≤1.
direction. The direction that the ball traveling during the interval 0.5≤t≤10.5≤t≤1.
acceleration. The acceleration of the ball during the interval 0.5≤t≤10.5≤t≤1.
What are the units on AV[0.5,1]AV[0.5,1]?
feet per second
feet
seconds
feet per second squared
foot seconds
none of these
15.)
Line L in the figure below is parallel to the line y=4x+2. Find the coordinates of the point P
P=
19.)
Match the Lines L1 (blue), L2 (red) and L3 (green) with the slopes by placing the letter of the slopes next to each set listed below
The slope of line L2:
The slope of line L3:
The slope of line L1:
A. m=−1.
B. m=0
C. m=0.2
24.) The height s of a ball (in feet) at time t(in seconds) is given by the formula s(t)=64−16(t−1) s(t)=64−16(t−1)^2 . Consider the following expression.
AV[0.5,1]=s(1)−s(0.5)/1−0.5
Compute the value of AV[0.5,1]=
What does this value measure geometrically?
slope. The slope of the line between the points (0.5,s(0.5))and (1,s(1))(1,s(1)).
curvature. The curvature of the graph of ss on the interval 0.5≤t≤1.
area. The area under the graph of ss on the interval 0.5≤t≤1.
distance. The distance between the points (0.5,s(0.5)) and (1,s(1))(1,s(1)).
What does this value measure physically?
average velocity. The average velocity of the ball during the interval 0.5≤t≤10.5≤t≤1.
distance. The distance that the ball travels during the interval 0.5≤t≤10.5≤t≤1.
direction. The direction that the ball traveling during the interval 0.5≤t≤10.5≤t≤1.
acceleration. The acceleration of the ball during the interval 0.5≤t≤10.5≤t≤1.
What are the units on AV[0.5,1]AV[0.5,1]?
feet per second
feet
seconds
feet per second squared
foot seconds
none of these
1.) Find the slope and the equation for the line passing through the point (−6,8)(−6,8) with yy-intercept at y=−1y=−1.
a) m=
b) The equation for the line in slope-intercept form:
2..
18Ellipses-x.pptx
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
Hyperbola (Advanced Algebra)
A hyperbola is the set of all points where the absolute difference between the distance to two fixed points (foci) is a constant. It is formed by the intersection of a plane with a double cone. The key parts of a hyperbola include the foci, vertices, center, and asymptotes. Hyperbolas can be written in standard form equations and graphed based on identifying these key parts.
Parabola 091102134314-phpapp01
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
Peperiksaan pertengahan tahun t4 2012 (2)
This document contains 12 mathematics questions testing skills such as solving simultaneous linear equations, quadratic equations, calculating areas and perimeters of shapes, set theory, and logical reasoning. The questions cover topics like functions, sequences, proportions, geometry, and Venn diagrams. Students are required to show their work and provide answers for full marks.
Pre c alc module 1-conic-sections
The document provides an overview of Module 1 of an analytic geometry course, which covers conic sections. Lesson 1 focuses specifically on circles. It defines a circle, discusses the standard form of a circular equation, and how to graph circles. It also provides an example of stating the center and radius of a circle given its equation. The objectives are to illustrate different conic sections including circles, define and work with circular equations, and solve problems involving circles.
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Lesson 2: A Catalog of Essential Functions (slides)
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Lesson 2: A Catalog of Essential Functions (slides)
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15.) Line L in the figure below is parallel to the line y=.docx
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Calculus II - 17
Polar coordinates represent points in a plane using an angle and distance from an origin point, rather than Cartesian x and y distances. In polar coordinates, a point is defined by its radial distance r from the origin and its angle θ measured counterclockwise from the polar axis. Converting between polar and Cartesian coordinates involves simple formulas relating r, θ, x, and y. Polar equations can more simply describe curves like the cardioid than their Cartesian equivalents.
Calculus II - 14
Separable differential equations can be solved explicitly by factoring the expression for the derivative into a function of y and a function of x. Examples of separable equations provided include dy/dx = x, dy/dx = y + x, and dy/dx = y/x. These types of first order differential equations can be solved by separating the variables and integrating both sides.
Calculus II - 13
This document discusses numerical methods for solving differential equations. It introduces direction fields, which provide a graphical approach to studying solutions, and Euler's method, which provides a numerical approach. Euler's method works by approximating the slope of the tangent line at each step using small step sizes to iteratively calculate successive approximations of the solution.
Calculus II - 12
The document discusses differential equations, which are equations containing an unknown function and one or more of its derivatives. A function is a solution to a differential equation if it satisfies the equation. Common types of differential equations include those modeling population growth over time and the motion of springs. These examples show how differential equations can be used to model real-world phenomena and how initial conditions are used to fully define a unique solution to what is called an initial-value problem.
Calculus II - 11
This document discusses finding the center of mass of objects using calculus. It provides examples of finding the center of mass for discrete objects, continuous objects defined by a function, and regions bounded by curves. The key steps are to calculate the moments about the x- and y-axes and then use these to determine the x- and y-coordinates of the center of mass. Examples include finding the center of a semicircular plate and the centroid of a region bounded by a line and parabola.
Calculus II - 10
The document discusses calculating the area of a surface obtained by rotating a curve around an axis. It provides the general formulas for finding the area which involves integrating the function squared plus the integral of its derivative. Several examples are worked out for rotating curves and arcs around the x-axis and y-axis. The document also gives an example problem of finding the force on a dam due to hydrostatic water pressure.
Calculus II - 9
The document discusses the arc length formula and how to calculate arc length. The arc length formula is the integral of the square root of (dx/dt)^2 + (dy/dt)^2 from t1 to t2, where t1 and t2 are the starting and ending points. Examples are provided on calculating the perimeter of a circle using this formula and approximating the arc length of a hyperbola between two points.
Calculus II - 8
This document discusses different types of improper integrals. It defines Type I improper integrals as integrals where the integrand is defined at every point but the limit at one or both endpoints is infinite. Type II integrals have an integrand that is continuous on some interval excluding one or both endpoints. The document provides examples of evaluating various improper integrals and checking whether they converge or diverge. It also introduces a comparison theorem for improper integrals.
Calculus II - 7
This document discusses numerical integration techniques for approximating integrals of functions that have no closed-form antiderivative. It introduces the midpoint rule, trapezoidal rule, and Simpson's rule for approximating integrals using Riemann sums over subintervals. An example applies Simpson's rule to approximate a definite integral and estimates the error bound.
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Calculus II - 19
- 1. 10.5 Conic Sections Three types of conic sections: Parabola Ellipse Hyperbola
- 2. A parabola is the set of points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The point half way between the focus and the directrix is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. axis focus vertex directrix
- 3. An equation of the parabola with focus ( , ) and directrix = is = . An equation of the parabola with focus ( , ) and directrix = is = . y = ( , ) x =
- 4. An ellipse is the set of points in a plane that the sum of whose distances from two fixed points (called the foci) is a constant. The line through the foci is called the major axis of the ellipse. The intersecting points of the major axis and the ellipse are called the vertices. axis vertices foci
- 5. The ellipse + = > has foci (± , ) where = , and vertices (± , ). The ellipse + = > has foci ( , ± ) where = , and vertices ( , ± ). y ( , ) + = ( , ) ( , ) ( , ) ( , ) x ( , )
- 6. An hyperbola is the set of points in a plane that the difference of whose distances from two fixed points (called the foci) is a constant. The line through the foci is called the axis of the ellipse. The intersecting points of the major axis and the ellipse are called the vertices. Both branches of the hyperbola approach the asymptotes. foci axis vertices
- 7. The hyperbola = has foci (± , ) where = + , vertices (± , ) , and asymptotes = ±( / ) . The hyperbola = has foci ( ,± ) where = + , vertices ( , ± ) , and asymptotes = ±( / ) . y ( , ) ( , ) x
- 8. Ex: What can you say about the conic? + + =
- 9. Ex: What can you say about the conic? + + = Complete the square: ( ) ( )+ =
- 10. Ex: What can you say about the conic? + + = Complete the square: ( ) ( )+ = ( + ) ( + )+ =
- 11. Ex: What can you say about the conic? + + = Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + =
- 12. Ex: What can you say about the conic? + + = Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + = ( ) ( ) =
- 13. Ex: What can you say about the conic? + + = Complete the square: ( ) ( )+ = ( + ) ( + )+ = ( ) ( ) + = ( ) ( ) = It is a hyperbola, = , = , = + = . The foci are ( , ± ). The vertices are ( , ) and ( , ). The asymptotes are − = ± ( − ).
Editor's Notes
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