1.3 SYMMETRY
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The document discusses three types of symmetry for graphs: x-axis symmetry, where (x,y) corresponds to (x,-y); y-axis symmetry, where (x,y) corresponds to (-x,y); and origin symmetry, where (x,y) corresponds to (-x,-y). It provides tests for each type of symmetry by replacing variables in the graphing equation. An example at the end finds no x-axis or origin symmetry but y-axis symmetry.
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The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
Parallel and perpendicular Lines
This document discusses parallel and perpendicular lines. Parallel lines have equal slopes and will never intersect. Perpendicular lines intersect at a right angle, and their slopes are negative reciprocals of each other. The document provides examples of determining if lines are parallel or perpendicular based on their slopes, graphing parallel and perpendicular lines, and finding equations of lines parallel or perpendicular to a given line through a specified point.
Parallel and perpendicular lines
This document discusses parallel and perpendicular lines. Parallel lines have equal slopes and will never intersect. Perpendicular lines intersect at a right angle, and their slopes are negative reciprocals of each other. The document provides examples of determining if lines are parallel or perpendicular based on their slopes, graphing parallel and perpendicular lines, and finding equations of lines parallel or perpendicular to a given line through a specified point.
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Chapter 5 Slopes of Parallel and Perpendicular Lines
Chapter 5 Slopes of Parallel and Perpendicular Lines
More from Sharon Henry
6.1 Law of Sines Ambiguous Case
This document discusses the ambiguous case of the Law of Sines when given an acute angle (A) and sides a and c of a triangle, where a < c. It explains that:
1) When a < h, where h is the altitude of the triangle, no triangle can be formed.
2) When a = h, one right triangle is formed.
3) When a > h, two triangles are formed - one acute triangle and one obtuse triangle.
So in summary, given the measurements A, a and c where a < c, there will be 0 triangles formed if a < h, 1 triangle if a = h, and 2 triangles if a > h.
6.2 law of cosines
This document provides instruction on using the Law of Cosines to solve triangles. It begins by explaining that the Law of Sines cannot be used to solve triangles when the side-side-side (SSS) or side-angle-side (SAS) configurations are given. The Law of Cosines formulas are then introduced. Examples are worked through demonstrating how to use the Law of Cosines to solve triangles in SSS and SAS formations. Key steps like verifying a triangle can be formed from given SSS measurements and properly applying order of operations are emphasized. Students are assigned homework problems applying these concepts.
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Definite Integral Review
This document provides an overview of key concepts in calculus including the definite integral, properties of definite integrals, the Mean Value Theorem for Integrals, the Average Value Theorem, the Fundamental Theorem of Calculus parts 1 and 2, and the Trapezoidal Rule for approximating definite integrals. It defines integrals, discusses how to evaluate them on a TI-84 calculator, and lists properties such as additivity, constant multiples, and order of integration. It also introduces concepts like finding the average value of a function over an interval using the integral, and relating the derivative of an integral to the original function.
4.4 Fundamental Theorem of Calculus
This document discusses the mean value theorem and how it can be used to find the average value of a function over an interval. It states that for a continuous function, there exists some value c within the interval where the area under the curve between the bounds is equal to the value of the function at c multiplied by the width of the interval. It then works through an example of finding the average value of the function f(x)=3x^2-2x over the interval [1,4].
4.3 The Definite Integral
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
Lesson 3.3 First Derivative Information
The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
Lesson 3.2 Rolle and Mean Value Theorems
Rolle's Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in the open interval where the derivative f'(c) is equal to 0. The Mean Value Theorem similarly guarantees the existence of at least one value c where the derivative is equal to the average rate of change of the function over the interval, (f(b)-f(a))/(b-a).
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval.
2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point.
3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
3.5 3.6 exp-log models 13-14
1. This document discusses several common mathematical models involving exponential and logarithmic functions, including exponential growth/decay models, logistic growth models, Gaussian models, logarithmic models, and Newton's Law of Cooling.
2. Examples are provided to illustrate how to set up and solve problems using each type of model. For instance, an example uses an exponential growth model to predict when the world's population will reach 7.1 billion people.
3. Guidance is also given on using graphing utilities and calculators to analyze exponential and logarithmic relationships. Instructions for performing exponential regression on a TI-84 calculator are provided.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
2.6 Related Rates
This document defines and provides examples of related rate problems. Related rate problems involve calculating the rates of change of two or more related variables with respect to time. The document outlines the steps to solve related rate problems: 1) identify changing quantities, 2) write an equation relating the quantities, 3) differentiate both sides with respect to time, 4) solve for the required quantity and substitute known values. Examples include calculating the rate of change of a dog's circular puddle area, a melting snowball's radius, the top of a slipping ladder, and the height of a conical sand pile.
3.5 EXP-LOG MODELS
The document discusses various mathematical models involving exponential and logarithmic functions, including exponential growth and decay models, logistic growth models, Gaussian models, and logarithmic models. It provides examples of how each type of model can be used, such as modeling world population growth with an exponential function, modeling the spread of a virus on a college campus with a logistic function, and modeling cooling of a heated object using Newton's Law of Cooling. The document also discusses how to graph and solve problems involving these different mathematical models.
Larson 4.4
This document discusses evaluating trigonometric functions of any angle by using reference angles. It defines trigonometric functions for any real value of the angle θ. For angles greater than 90° or less than 0°, the values can be determined from the acute reference angle. Examples show how to find the reference angle and then use it to evaluate trigonometric functions in different quadrants, including for nonacute angles.
Larson 4.3
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
Larson 4.2
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
Larson 4.1
The document describes angles and their measurement in radians and degrees. It defines an angle, measures angles in radians using central angles of a circle, and defines the relationship between radians and degrees. Examples show converting between radians and degrees, finding coterminal angles, complementary and supplementary angles, and calculating arc length and linear speed using angular measurements.
Derivative rules ch2
This document provides the basic derivative rules from Chapter 2. It lists various expressions and their derivatives on subsequent slides, including the derivatives of sums, products, quotients, composite functions, powers, trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, and constants. The derivatives covered are: du/dv, df(g(x))/dx, dn/dx, d/dx cot(u), d/dx cu, d/dx tan(u), d/dx (u*v), d/dx c, d/dx cos(u), d/dx sec(u), dx/dx, and d/dx csc(
1.4 Continuity
The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
Chapter 5 Definite Integral Topics Review
This document discusses several topics in calculus including RAM theory for approximating functions, computing definite integrals using formulas and a TI-89 calculator, properties of definite integrals like additivity and the mean value theorem, the fundamental theorem of calculus in two parts, and the trapezoidal rule for approximating definite integrals which overestimates for increasing functions and underestimates for decreasing functions.
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Attributes & automation parameters enable the automation author to pass data values from one automation component to the next. During this webinar, our FME Flow Specialists will cover leveraging the three types of these output attributes & parameters in FME Flow: Event, Custom, and Automation. As a bonus, they’ll also be making use of the Split-Merge Block functionality.
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JavaLand 2024: Application Development Green Masterplan
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Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
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2. Understanding Edge (IoT)
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3. What is ArgoCD?
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4. Deployment Using ArgoCD for Edge Devices
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9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.
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“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
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1.3 SYMMETRY
- 1. 1.3Symmetry
- 2. A graph is said to be symmetric with respect to the x-axis if for every point (x,y) on the graph, the point (x,-y) is on the graph. x-Axis Symmetry
- 3. A graph is said to be symmetric with respect to the y-axis if for every point (x,y) on the graph, the point (-x,y) is on the graph. y-Axis Symmetry
- 4. A graph is said to be symmetric with respect to the origin if for every point (x,y) on the graph, the point (-x,-y) is on the graph. origin Symmetry
- 5. x-axis Replace y by -y in the equation. If an equivalent equation results, the graph is symmetric with respect to the x-axis. Tests for Symmetry y-axis Replace x by -x in the equation. If an equivalent equation results, the graph is symmetric with respect to the y-axis. origin Replace x by -xandy by -y in the equation. If an equivalent equation results, the graph is symmetric with respect to the origin.
- 6. Not symmetric with respect to the x-axis.
- 7. Symmetric with respect to the y-axis.
- 8. Not symmetric with respect to the origin.