The document discusses key concepts in set theory and functions, including:
- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
Double integrals over Rectangle, Fubini’s Theorem,Properties of double integrals, Double integrals over a general region, Double integrals in polar region
Double integrals over Rectangle, Fubini’s Theorem,Properties of double integrals, Double integrals over a general region, Double integrals in polar region
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
JMeter webinar - integration with InfluxDB and GrafanaRTTS
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 Overview
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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
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
2. The symbol means ‘is an element of’. Introduction to Set Theory NOTE In Mathematics, the word set refers to a group of numbers or other types of elements. Sets are written as follows: Examples { 1, 2, 3, 4, 5, 6 } { -0.7, -0.2, 0.1 } { red, green, blue } 4 { 1, 2, 3, 4, 5 } Î Ï Î 7 { 1, 2, 3 } { 6, 7, 8 } { 6, 7, 8, 9 } Î Ï If A = { 0, 2, 4, 6, 8, … 20 } and B = { 1, 2, 3, 4, 5 } then B A Sets can also be named using letters: P = { 2, 3, 5, 7, 11, 13, 17, 19, 23, … } Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART
3. N W Z { 1, 2, 3, 4, 5, ... } { 0, 1, 2, 3, 4, 5, ... } { ... -3, -2, -1, 0, 1, 2, 3, ... } The Basic Number Sets NOTE Q Rational numbers Includes all integers, plus any number which can be written as a fraction. R √ 7 π Includes all rational numbers, plus irrational numbers such as or . Real numbers C Complex numbers Includes all numbers, even imaginary ones which do not exist. Whole numbers Integers Natural numbers N Î W Z Î Q Î Î R Î C 2 3 Î Q Ï √ - 1 R Examples Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART
4. Set Theory and Venn Diagrams NOTE Venn Diagrams are illustrations which use overlapping circles to display logical connections between sets. Blue Animal Food Pig Blueberry Pie Blue Whale ? Rain Red Yellow Orange Juice Sun Strawberries Aardvark N W Z Q R C √ 82 5 7 Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART
5. Function Domain and Range NOTE Any function can be thought of as having an input and an output. The ‘input’ is sometimes also known as the domain of the function, with the output referred to as the range . f ( x ) domain range Each number in the domain has a unique output number in the range. The function has the domain { -2, -1, 0, 1, 2, 3 } Find the range. Imporant Example f ( x ) = x 2 + 3 x f ( - 2 ) = 4 – 6 = - 2 f ( - 1 ) = 1 – 3 = - 2 f ( 0 ) = 0 + 0 = 0 f ( 1 ) = 1 + 3 = 4 f ( 2 ) = 4 + 6 = 10 Range = { -2, 0, 4, 10 } Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART
6. Composite Functions NOTE It is possible to combine functions by substituting one function into another. f ( x ) g ( x ) g ( ) f ( x ) is a composite function and is read ‘ ’. g ( ) f ( x ) g of f of x Important ≠ In general Given the functions Example g ( x ) = x + 3 f ( x ) = 2 x and find and . = 2 ( ) x + 3 = 2 x + 6 = ( ) + 3 2 x = 2 x + 3 f ( g ( x )) g ( f ( x )) g ( f ( x )) f ( g ( x )) Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART g ( ) f ( x ) f ( ) g ( x )
7. = x Inverse of a Function NOTE If a function also works backwards for each output number, it is possible to write the inverse of the function. f ( x ) f ( x ) = x 2 f ( 4 ) = 16 f ( -4 ) = 16 Not all functions have an inverse, e.g. Every output in the range must have only one input in the domain. does not have an inverse function. f ( x ) = x 2 domain range Note that = x x and f ( ) f ( x ) Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART f ( 16 ) = ? - 1 f ( ) - 1 f ( x ) f ( x ) - 1 - 1
8. Find the inverse function for . Finding Inverse Functions NOTE g ( x ) = 5 x 3 – 2 Example g ( x ) 3 × 5 – 2 g x x + 2 √ 3 ÷ 5 + 2 g x x x + 2 5 x + 2 5 3 g ( x ) = + 2 ÷ 5 3 Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART - 1 - 1 - 1
9. Graphs of Inverse Functions NOTE To sketch the graph of an inverse function , reflect the graph of the function across the line . f ( x ) y = x x y y = x f ( x ) x y y = x g ( x ) - 1 g ( x ) Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART f ( x ) - 1 f ( x ) - 1
10. Basic Functions and Graphs NOTE x y y x y x x x x y y y f ( x ) = ax f ( x ) = a sin bx f ( x ) = a tan bx f ( x ) = ax ² f ( x ) = ax ³ Linear Functions Quadratic Functions Trigonometric Functions Cubic Functions Inverse Functions Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART f ( x ) = a x
11. Exponential and Logartithmic Functions NOTE 1 f ( x ) = a x 1 ( 1 , a ) x y ( 1 , a ) x y is called an exponential function with base . Exponential Functions f ( x ) = a x a The inverse function of an exponential function is called a logarithmic function and is written as . Logarithmic Functions Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART f ( x ) = log x a f ( x ) = log x a
12. Finding Equations of Exponential Functions NOTE Higher Maths 1 2 1 Sets and Functions UNIT OUTCOME SLIDE PART It is possible to find the equation of any exponential function by substituting values of and for any point on the line. y = a + b x 2 ( 3 , 9 ) x y Example The diagram shows the graph of y = a + b Find the values of a and b . Substitute (0,2): x 2 = a + b 0 = 1 + b b = 1 x y Substitute (3,9): 9 = a + 1 3 a = 2 a = 8 3 y = 2 + 1 x