Lesson 19: Related Rates
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The document discusses related rates problems in mathematics. It provides examples of how to solve related rates problems using derivatives and the chain rule. In one example, the radius of an oil slick is increasing and the volume is known to be increasing at a rate of 10,000 liters per second. The problem is to determine the rate of change of the radius. The solution uses derivatives and the geometry of the situation to set up and solve an equation relating the rates of change. A second example involves determining the rate at which two people walking away from each other are increasing their distance apart.
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Level of Detail in
Solution
s: Need to show work leading to final answer
Need
Question 1
(a) Find:
(4 + i2)
(1 + i3)
Use the rules for multiplication and division of complex numbers in rectangular form.
(b) Convert the answer in rectangular form to polar form
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(d) Convert the answer in polar form to rectangular form.
Question 2
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Ph2A Win 2020 Numerical Analysis LabMax YuenMar 2020.docx
Ph2A Win 2020 Numerical Analysis Lab
Max Yuen
Mar 2020
(use g = 9.8m/s2 for all problems.)
Background
Many physics problems cannot be solved directly by hand or analytically. We resort to numerical
methods to give us approximations to the problem. In this lab you will learn the Euler method,
which allows you to solve Newton’s laws of motion. This is done by treating the velocity as a
piecewise linear function with many time intervals and during interval the acceleration is assumed
to be uniform. This allows us to use the kinematic equations we learned about in the first half
of the class to approximate the motion. If we choose to partition the motion into smaller time
intervals, the approximation becomes much better since the differences between adjacent intervals
become smaller. In this lab, this numerical analysis method will be applied to the motion of a
falling object under the influence of gravity and drag force. If you are adventurous, you can even
try to extend this to 2D and compute the realistic trajectory of a baseball. You might even try
some other problems, such as a mass attached to a spring.
Euler’s Method Foundations
This method is well suited for problems where the acceleration is a function of the velocity, as in
the case of a falling object under the influence of gravity and drag:
a = f(v) (1)
Falling object with drag force
The model for drag fits the prescription for using Euler’s method since the net force on a falling
object with drag is given by:
ma = −mg −FD (2)
ma = −mg −
1
2
ρairACDv
2 · sgn(v) (3)
a = −g
(
1 +
ρairACDv
2 · sgn(v)
2mg
)
(4)
a = f(v) ← Equation of Motion (5)
where m is the mass of the falling object, a is the acceleration of the object (which is positive when
pointed up), ρair is the density of air (about 1.29 ·10−3kg/m3), A is the cross-sectional area, CD is
the drag coefficient, v is the object’s velocity, and sgn(v) is the signum function which returns the
sign of the argument. The second signum function is there to guarantee that the direction of the
drag force is always in the opposite direction of the velocity function. Note that we see that the
acceleration is an explicit function of v, which sort of makes this a chicken or egg problem. This is
because we need a to get v, but to get a we need v, so which one do we compute first? Hold that
thought. We’ll talk more on how to program this in EXCEL or Google Sheets later.
1
Figure 1: FBD for an object falling under the pull of gravity and resistance by drag force
Terminal Velocity
In lecture, we talked about how after waiting for some time, if the object started at rest the
speed will increase and the drag force will also become larger and eventually balance out with the
gravitational force. When this happens, we have reached terminal velocity vterm = −v. This can
be solved by setting a = 0:
0 = −mg −
1
2
ρairACDv
2 · sgn(v) (6)
2mg = ρairACDv
2
term (7)
→ vterm =
√
2mg
ρairACD
(8)
Using this definition for the terminal ...
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Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
"Scaling RAG Applications to serve millions of users", Kevin Goedecke
How we managed to grow and scale a RAG application from zero to thousands of users in 7 months. Lessons from technical challenges around managing high load for LLMs, RAGs and Vector databases.
Containers & AI - Beauty and the Beast!?!
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Keywords: AI, Containeres, Kubernetes, Cloud Native
Event Link: https://meine.doag.org/events/cloudland/2024/agenda/#agendaId.4211
GNSS spoofing via SDR (Criptored Talks 2024)
In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
Lee Barnes - Path to Becoming an Effective Test Automation Engineer.pdf
So… you want to become a Test Automation Engineer (or hire and develop one)? While there’s quite a bit of information available about important technical and tool skills to master, there’s not enough discussion around the path to becoming an effective Test Automation Engineer that knows how to add VALUE. In my experience this had led to a proliferation of engineers who are proficient with tools and building frameworks but have skill and knowledge gaps, especially in software testing, that reduce the value they deliver with test automation.
In this talk, Lee will share his lessons learned from over 30 years of working with, and mentoring, hundreds of Test Automation Engineers. Whether you’re looking to get started in test automation or just want to improve your trade, this talk will give you a solid foundation and roadmap for ensuring your test automation efforts continuously add value. This talk is equally valuable for both aspiring Test Automation Engineers and those managing them! All attendees will take away a set of key foundational knowledge and a high-level learning path for leveling up test automation skills and ensuring they add value to their organizations.
Apps Break Data
How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
Demystifying Knowledge Management through Storytelling
The Department of Veteran Affairs (VA) invited Taylor Paschal, Knowledge & Information Management Consultant at Enterprise Knowledge, to speak at a Knowledge Management Lunch and Learn hosted on June 12, 2024. All Office of Administration staff were invited to attend and received professional development credit for participating in the voluntary event.
The objectives of the Lunch and Learn presentation were to:
- Review what KM ‘is’ and ‘isn’t’
- Understand the value of KM and the benefits of engaging
- Define and reflect on your “what’s in it for me?”
- Share actionable ways you can participate in Knowledge - - Capture & Transfer
Principle of conventional tomography-Bibash Shahi ppt..pptx
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Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
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"NATO Hackathon Winner: AI-Powered Drug Search", Taras Kloba
This is a session that details how PostgreSQL's features and Azure AI Services can be effectively used to significantly enhance the search functionality in any application.
In this session, we'll share insights on how we used PostgreSQL to facilitate precise searches across multiple fields in our mobile application. The techniques include using LIKE and ILIKE operators and integrating a trigram-based search to handle potential misspellings, thereby increasing the search accuracy.
We'll also discuss how the azure_ai extension on PostgreSQL databases in Azure and Azure AI Services were utilized to create vectors from user input, a feature beneficial when users wish to find specific items based on text prompts. While our application's case study involves a drug search, the techniques and principles shared in this session can be adapted to improve search functionality in a wide range of applications. Join us to learn how PostgreSQL and Azure AI can be harnessed to enhance your application's search capability.
Getting the Most Out of ScyllaDB Monitoring: ShareChat's Tips
ScyllaDB monitoring provides a lot of useful information. But sometimes it’s not easy to find the root of the problem if something is wrong or even estimate the remaining capacity by the load on the cluster. This talk shares our team's practical tips on: 1) How to find the root of the problem by metrics if ScyllaDB is slow 2) How to interpret the load and plan capacity for the future 3) Compaction strategies and how to choose the right one 4) Important metrics which aren’t available in the default monitoring setup.
Dandelion Hashtable: beyond billion requests per second on a commodity server
This slide deck presents DLHT, a concurrent in-memory hashtable. Despite efforts to optimize hashtables, that go as far as sacrificing core functionality, state-of-the-art designs still incur multiple memory accesses per request and block request processing in three cases. First, most hashtables block while waiting for data to be retrieved from memory. Second, open-addressing designs, which represent the current state-of-the-art, either cannot free index slots on deletes or must block all requests to do so. Third, index resizes block every request until all objects are copied to the new index. Defying folklore wisdom, DLHT forgoes open-addressing and adopts a fully-featured and memory-aware closed-addressing design based on bounded cache-line-chaining. This design offers lock-free index operations and deletes that free slots instantly, (2) completes most requests with a single memory access, (3) utilizes software prefetching to hide memory latencies, and (4) employs a novel non-blocking and parallel resizing. In a commodity server and a memory-resident workload, DLHT surpasses 1.6B requests per second and provides 3.5x (12x) the throughput of the state-of-the-art closed-addressing (open-addressing) resizable hashtable on Gets (Deletes).
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Lesson 19: Related Rates
- 1. Lesson 19 (Section 4.1) Related Rates Math 1a November 7, 2007 Announcements OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
- 3. Today we’ll look at a direct application of the chain rule to real-world problems. Examples of these can be found whenever you have some system or object changing, and you want to measure the rate of change of something related to it.
- 5. Problem Example An oil slick in the shape of a disk is growing. At a certain time, the radius is 1 km and the volume is growing at the rate of 10,000 liters per second. If the slick is always 20 cm deep, how fast is the radius of the disk growing at the same time?
- 8. A solution The volume of the disk is V = 2πr2 h. We are given dV dt , a certain value of r, and the object is to find dr dt at that instant.
- 9. Solution Solution Differentiating with respect to time we have dV dt = 2πrh dr dt =⇒ dr dt = 1 2πrh · dV dt . Now we evaluate: dr dt r=1000 m = 1 2π(1 km)(20 cm) · 10, 000 L s Converting every length to meters we have dr dt r=1 km = 1 2π(1000 km)(0.2 cm) · 10 m3 s = 1 40π m s
- 11. Strategies for Related Rates Problems
- 12. Strategies for Related Rates Problems 1. Read the problem.
- 13. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram.
- 14. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants)
- 15. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Express the given information and the required rate in terms of derivatives
- 16. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Express the given information and the required rate in terms of derivatives 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate all but one of the variables.
- 17. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Express the given information and the required rate in terms of derivatives 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate all but one of the variables. 6. Use the Chain Rule to differentiate both sides with respect to t.
- 18. Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Express the given information and the required rate in terms of derivatives 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate all but one of the variables. 6. Use the Chain Rule to differentiate both sides with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate.
- 20. Another one Example A man starts walking north at 4 ft/sec from a point P. Five minutes later a woman starts walking south at 4 ft/sec from a point 500 ft due east of P. At what rate are the people walking apart 15 min after the woman starts walking?
- 21. Diagram m 500 ww 500 s 4 ft/sec 4 ft/sec s = (m + w)2 + 5002
- 22. 15 minutes after the woman starts walking, the woman has traveled 4 ft sec 60 sec min (15 min) = 3600 ft while the man has traveled 4 ft sec 60 sec min (20 min) = 4800 ft We want to know ds dt when m = 4800, w = 3600, dm dt = 4, and dw dt = 4.
- 23. Differentiation We have ds dt = 1 2 (m + w)2 + 5002 −1/2 (2)(m + w) dm dt + dw dt = m + w s dm dt + dw dt At our particular point in time ds dt = 672 √ 7081 ≈ 7.98587
- 25. Worksheet