The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
Had to make this dumb powerpoint for my algebra II class and I put a lot of work into it for some reason... so yeah it's just been sitting on my laptop doing nothing and I thought why not upload this to help other people? So yeah, hope you guys find it useful...
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
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[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
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[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
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[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxTatianaMajor22
MATH 107 FINAL EXAMINATION
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
A. Domain [–2, 2];
B. Domain [–1, 1];
C. Domain [–1, 3];
D. Domain [–3/2, –1/2];
2. Solve:
A. 3
B. 3,7
C. 9
D. No solution
3. Determine the interval(s) on which the function is increasing.
A. (−1.3, 1.3)
B. (1, 3)
C. (−∞,−1)and (3,∞)
D. (−2.5, 1)and (4.5,∞)
4. Determine whether the graph of y = 2|x| + 1 is symmetric with respect to the origin,
the x-axis, or the y-axis.
A. symmetric with respect to the origin only
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. not symmetric with respect to the origin, not symmetric with respect to the x-axis, and
not symmetric with respect to the y-axis
5. Solve, and express the answer in interval notation: | 9 – 7x | ≤ 12.
A. (–∞, –3/7]
B. (–∞, −3/7] ∪ [3, ∞) C. [–3, 3/7]
D. [–3/7, 3]
6. Which of the following represents the graph of 7x + 2y = 14 ?
A. B.
C. D.
7. Write a slope-intercept equation for a line parallel to the line x – 2y = 6 which passes through the point (10, – 4).
A.
B.
C.
D.
8. Which of the following best describes the graph?
A. It is the graph of a function and it is one-to-one.
B. It is the graph of a function and it is not one-to-one.
C. It is not the graph of a function and it is one-to-one.
D. It is not the graph of a function and it is not one-to-one.
9. Express as a single logarithm: log x + log 1 – 6 log (y + 4)
A.
B.
C.
D.
10. Which of the functions corresponds to the graph?
A.
B.
C.
D.
11. Suppose that a function f has exactly one x-intercept.
Which of the following statements MUST be true?
A. f is a linear function.
B. f (x) ≥ 0 for all x in the domain of f.
C. The equation f(x) = 0 has exactly one real-number solution.
D. f is an invertible function.
12. The graph of y = f(x) is shown at the left and the graph of y = g(x) is shown at the right. (No formulas are given.) What is the relationship between g(x) and f(x)?
y = f (x) y = g(x)
A. g(x) = f (x – 3) + 1
B. g(x) = f (x – 1) + 3
C. g(x) = f (x + 3) – 1
D. g(x) = f (x + 1) .
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.
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.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
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 there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” 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
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.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
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/
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t 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 clients’ 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.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
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.
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.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re 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 “Essentials of Automation” 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.
Here’s what you’ll 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.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
2. Definition of Exponential Functions
The exponential function f with a base
b is defined by f(x) = bx
where b is a
positive constant other than 1 (b > 0,
and b ≠ 1) and x is any real number.
So, f(x) = 2x
, looks like:
3. Graphing Exponential Functions
Four exponential
functions have
been graphed.
Compare the
graphs of functions
where b > 1 to
those where b < 1
2x
y =
7x
y =
1
2
x
y =
1
7
x
y =
4. Graphing Exponential Functions
So, when b > 1, f(x)
has a graph that
goes up to the right
and is an
increasing function.
When 0 < b < 1,
f(x) has a graph
that goes down to
the right and is a
decreasing
function.
5. Characteristics
The domain of f(x) = bx
consists of all real
numbers (-∞, ∞). The range of f(x) = bx
consists of all positive real numbers
(0, ∞).
The graphs of all exponential functions
pass through the point (0,1). This is
because f(o) = b0
= 1 (b≠o).
The graph of f(x) = bx
approaches but
does not cross the x-axis. The x-axis is a
horizontal asymptote.
f(x) = bx
is one-to-one and has an inverse
that is a function.
11. You Do
Graph the function f(x)
= 2(x-3)
+2
Where is the horizontal
asymptote?
y = 2
12. You Do, Part Deux
Graph the function f(x)
= 4(x+5)
- 3
Where is the horizontal
asymptote?
y = - 3
13. The Number e
The number e is known as Euler’s number.
Leonard Euler (1700’s) discovered it’s
importance.
The number e has physical meaning. It
occurs naturally in any situation where a
quantity increases at a rate proportional to
its value, such as a bank account producing
interest, or a population increasing as its
members reproduce.
14. The Number e - Definition
An irrational number, symbolized by the letter
e, appears as the base in many applied
exponential functions. It models a variety of
situations in which a quantity grows or decays
continuously: money, drugs in the body,
probabilities, population studies, atmospheric
pressure, optics, and even spreading rumors!
The number e is defined as the value that
approaches as n gets larger and larger.
1
1
n
n
+ ÷
15. The Number e - Definition
n
1 2
2 2.25
5 2.48832
10 2.59374246
100 2.704813829
1000 2.716923932
10,000 2.718145927
100,000 2.718268237
1,000,000 2.718280469
1,000,000,000 2.718281827
1
1
n
n
+ ÷
0
1
1
n
A
n
+ ÷
1
, 1
n
As n e
n
→∞ + → ÷
The table shows
the values of
as n gets
increasingly large.
n → ∞As , the
approximate
value of e (to 9
decimal places)
is ≈
2.718281827
16. The Number e - Definition
For our purposes,
we will use
e ≈ 2.718.
e is 2nd
function on
the division key on
your calculator.
y = e
1
1
n
y
n
= + ÷
17. The Number e - Definition
Since 2 < e < 3, the
graph of y = ex
is
between the
graphs of y = 2x
and y = 3x
ex
is the 2nd
function
on the ln key on
your calculator
y =e
y = 2x
y = 3x
y = ex
18. Natural Base
The irrational number e, is called the
natural base.
The function f(x) = ex
is called the
natural exponential function.
19. Compound Interest
The formula for compound interest:
( ) 1
= + ÷
nt
r
A t P
n
Where n is the number of times per
year interest is being compounded
and r is the annual rate.
20. Compound Interest - Example
Which plan yields the most interest?
Plan A: A $1.00 investment with a 7.5% annual
rate compounded monthly for 4 years
Plan B: A $1.00 investment with a 7.2% annual
rate compounded daily for 4 years
A:
B:
12(4)
0.075
1 1 1.3486
12
+ ≈ ÷
365(4)
0.072
1 1 1.3337
365
+ ≈ ÷
$1.35
$1.34
21. Interest Compounded Continuously
If interest is compounded “all the
time” (MUST use the word
continuously), we use the formula
where P is the initial principle (initial
amount)
( ) = rt
A t Pe
22. ( ) = rt
A t Pe
If you invest $1.00 at a 7% annual
rate that is compounded
continuously, how much will you have
in 4 years?
You will have a whopping $1.32 in 4
years!
(.07)(4)
1* 1.3231e ≈
23. You Do
You decide to invest $8000 for 6
years and have a choice between 2
accounts. The first pays 7% per year,
compounded monthly. The second
pays 6.85% per year, compounded
continuously. Which is the better
investment?
24. You Do Answer
1st
Plan:
2nd Plan:
0.0685(6)
(6) 8000 $12,066.60P e= ≈
12(6)
0.07
(6) 8000 1 $12,160.84
12
A
= + ≈ ÷