This document discusses functions of several variables. It introduces notation for functions with multiple independent variables, defines the domain of such functions, and explains how to graph functions of two and three variables by sketching traces in coordinate planes and parallel planes. Level curves and contour maps are presented as ways to visualize functions of two variables in the xy-plane. Examples demonstrate finding domains, sketching graphs using traces and level curves, and interpreting contour maps.
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
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College Entrance Test Review
Math Session 6 - part 2 of 2
FUNCTIONS
How to evaluate
Operations on functions
Composite functions
Trigonometric Functions
Pythagorean Theorem
30 60 90 triangle
45 45 90 triangle
Exponential Functions
Logarithmic Functions
Discuss and apply comprehensively the concepts, properties and theorems of functions, limits, continuity and the derivatives in determining the derivatives of algebraic functions
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
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The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. Whatâs changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
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Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overviewâ
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During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
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Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
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Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But thereâs more:
In a second workflow supporting the same use case, youâll see:
Your campaign sent to target colleagues for approval
If the âApproveâ button is clicked, a Jira/Zendesk ticket is created for the marketing design team
Butâif the âRejectâ button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
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Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
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The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties â USA
Expansion of bot farms â how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks â Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
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Clients donât know what they donât know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clientsâ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
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In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
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Are you looking to streamline your workflows and boost your projectsâ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, youâre in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part âEssentials of Automationâ series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Hereâs what youâll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
Weâll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Donât miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Essentials of Automations: Optimizing FME Workflows with Parameters
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1. Chapter 8: Functions of Several Variables Section 8.1 Introduction to Functions of Several Variables Written by Karen Overman Instructor of Mathematics Tidewater Community College, Virginia Beach Campus Virginia Beach, VA With Assistance from a VCCS LearningWare Grant
2.
3. Notation for Functions of Several Variables Previously we have studied functions of one variable, y = f(x) in which x was the independent variable and y was the dependent variable. We are going to expand the idea of functions to include functions with more than one independent variable. For example, consider the functions below:
4. The function z = f(x, y) is a function of two variables. It has independent variables x and y, and the dependent variable z. Likewise, the function w = g(x, y, z) is a function of three variables. The variables x, y and z are independent variables and w is the dependent variable. The function h is similar except there are four independent variables. Hopefully you can see the notation for functions of several variables is similar to the notation youâve used with single variable functions.
5. When finding values of the several variable functions instead of just substituting in an x-value, we will substitute in values for each of the independent variables: For example, using the function f on the previous slide, we will evaluate the function f(x, y) for (2, 3) , (4, -3) and (5, y).
6. A Function of Two Variables : A function f of two variables x and y is a rule that assigns to each ordered pair (x, y) in a given set D, called the domain, a unique value of f. Functions of more variables can be defined similarly. The operations we performed with one-variable functions can also be performed with functions of several variables. For example, for the two-variable functions f and g: In general we will not consider the composition of two multi-variable functions.
7. Domains of Functions of Several Variables : Unless the domain is given, assume the domain is the set of all points for which the equation is defined. For example, consider the functions The domain of f(x,y) is the entire xy-plane. Every ordered pair in the xy-plane will produce a real value for f. The domain of g(x, y) is the set of all points (x, y) in the xy-plane such that the product xy is greater than 0. This would be all the points in the first quadrant and the third quadrant.
8. Example 1 : Find the domain of the function: Solution : The domain of f(x, y) is the set of all points that satisfy the inequality: or You may recognize that this is similar to the equation of a circle and the inequality implies that any ordered pair on the circle or inside the circle is in the domain. x y The highlighted area is the domain to f.
9. Example 2 : Find the domain of the function: Solution : Note that g is a function of three variables, so the domain is NOT an area in the xy-plane. The domain of g is a solid in the 3-dimensional coordinate system. The expression under the radical must be nonnegative, resulting in the inequality: This implies that any ordered triple outside of the sphere centered at the origin with radius 4 is in the domain.
10. Example 3 : Find the domain of the function: Solution : We know the argument of the natural log must be greater than zero. So, This occurs in quadrant I and quadrant III. The domain is highlighted below. Note the x-axis and the y-axis are NOT in the domain. x y
11. Graphs of Functions of Several Variables As you learned in 2-dimensional space the graph of a function can be helpful to your understanding of the function. The graph gives an illustration or visual representation of all the solutions to the equation. We also want to use this tool with functions of two variables. The graph of a function of two variables, z = f(x, y), is the set of ordered triples, (x, y, z) for which the ordered pair, (x, y) is in the domain. * The graph of z = f(x, y) is a surface in 3-dimensional space. The graph of a function of three variables, w = f(x, y, z) is the set of all points (x, y, z, w) for which the ordered triple, (x, y, z) is in the domain. * The graph of w = f(x, y, z) is in 4 dimensions. We canât draw this graph or the graphs of any functions with 3 or more independent variables.
12. Example 4 : Find the domain and range of the function and then sketch the graph. Solution : From Example 1 we know the domain is all ordered pairs (x, y) on or inside the circle centered at the origin with radius 5. All ordered pairs satisfying the inequality: The range is going to consist of all possible outcomes for z. The range must be nonnegative since z equals a principle square root and furthermore, with the domain restriction: , the value of the radicand will only vary between 0 and 25. Thus, the range is .
13. Solution to Example 4 Continued : Now letâs consider the sketch of the function: Squaring both sides and simplifying: You may recognize this equation from Chapter 7 - A sphere with radius 5. This is helpful to sketching the function, but we must be careful !! The function and the equation are not exactly the same. The equation does NOT represent z as a function of x and y â meaning there is not a unique value for z for each (x, y). Keep in mind that the function had a range of , which means the function is only the top half of the sphere.
14. As you have done before when sketching a surface in 3-dimensions it may be helpful for you to use the traces in each coordinate plane. 1. The trace in the xy-plane, z = 0, is the equation: The circle centered at the origin with radius 5 in the xy-plane. 2. The trace in the yz-plane, x = 0, is the equation: The circle centered at the origin with radius 5 in the yz-plane. 3. The trace in the xz-plane, y = 0, is the equation: The circle centered at the origin with radius 5 in the xz-plane.
15. Along with sketching the traces in each coordinate plane, it may be helpful to sketch traces in planes parallel to the coordinate planes. 4. Let z = 3: So on the plane z = 3, parallel to the xy-plane, the trace is a circle centered at (0,0,3) with radius 4. 5. Let z = 4: So on the plane z = 4, parallel to the xy-plane, the trace is a circle centered at (0,0,4) with radius 3.
16. Here is a SKETCH with the three traces in the coordinate planes and the additional two traces in planes parallel to the xy-plane. Keep in mind that this is just a sketch. It is giving you a rough idea of what the function looks like. It may also be helpful to use a 3-dimensional graphing utility to get a better picture. z = 3 z = 4
17. Here is a graph of the function using the 3-dimensional graphing utility DPGraph. x y z
18. Example 5 : Sketch the surface: Solution : The domain is the entire xy-plane and the range is . 1. The trace in the xy-plane, z = 0, is the equation: 2. The trace in the yz-plane, x = 0, is the equation: 3. The trace in the xz-plane, y = 0, is the equation: Circle Parabola Parabola
19. Solution to Example 5 Continued : Traces parallel to the xy-plane include the following two. 4. The trace in the plane, z = 5, is the equation: 5. The trace in the plane, z = -7, is the equation: Circle centered at (0, 0, 5) with radius 2. Circle centered at (0, 0, -7) with radius 4.
20. x y z Here is a sketch of the traces in each coordinate plane. This paraboloid extends below the xy-plane.
22. In the previous two examples, traces in the coordinate planes and traces parallel to the xy-plane were used to sketch the function of two variables as a surface in the 3-dimensional coordinate system. When the traces parallel to the xy-plane or in other words, the traces found when z or f(x,y) is set equal to a constant, are drawn in the xy-plane, the traces are called level curves . When several level curves, also called contour lines, are drawn together in the xy-plane the image is called a contour map . Level Curves
23. Example 6 : Sketch a contour map of the function in Example 4, using the level curves at c = 5,4,3,2,1 and 0. Solution : c = a means the curve when z has a value of a.
24. Solution to Example 6 Continued : Contour map with point (0, 0) for c = 5 and then circles expanding out from the center for the remaining values of c. Note : The values of c were uniformly spaced, but the level curves are not. When the level curves are spaced far apart (in the center), there is a gradual change in the function values. When the level curves are close together (near c = 5), there is a steep change in the function values.
25. Example 7 : Sketch a contour map of the function, using the level curves at c = 0, 2, 4, 6 and 8. Solution : Set the function equal to each constant.
26. Solution to Example 7 Continued : Contour map with point (0, 0) for c = 0 and then ellipses expanding out from the center for the remaining values of c.