Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
JEE Physics/ Lakshmikanta Satapathy/ Wave Motion QA part 3/ JEE question on fundamental frequencies of Open and Closed pipes solved with the related concepts
Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 12: Linear Approximations and Differentials (slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Exponential functions change addition into multiplication. Different bases for exponentials produce different functions but they share similar characteristics. One base--a number we call e--is an especially good one.
Exponential functions change addition into multiplication. Different bases for exponentials produce different functions but they share similar characteristics. One base--a number we call e--is an especially good one.
Exponential functions change addition into multiplication. Different bases for exponentials produce different functions but they share similar characteristics. One base--a number we call e--is an especially good one.
Exponential functions change addition into multiplication. Different bases for exponentials produce different functions but they share similar characteristics. One base--a number we call e--is an especially good one.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
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.
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.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s 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.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
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.
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
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.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
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.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
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
-------------------------------------------
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.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
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/
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 4
Lesson 13: Exponential and Logarithmic Functions (slides)
1. Sec on 3.1–3.2
Exponen al and Logarithmic
Func ons
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
March 9, 2011
.
2. Announcements
Midterm is graded.
average = 44, median=46,
SD =10
There is WebAssign due
a er Spring Break.
Quiz 3 on 2.6, 2.8, 3.1, 3.2
on March 30
3. Midterm Statistics
Average: 43.86/60 = 73.1%
Median: 46/60 = 76.67%
Standard Devia on: 10.64%
“good” is anything above average and “great” is anything more
than one standard devia on above average.
More than one SD below the mean is cause for concern.
4. Objectives for Sections 3.1 and 3.2
Know the defini on of an
exponen al func on
Know the proper es of
exponen al func ons
Understand and apply
the laws of logarithms,
including the change of
base formula.
5. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
6. Derivation of exponentials
Defini on
If a is a real number and n is a posi ve whole number, then
an = a · a · · · · · a
n factors
7. Derivation of exponentials
Defini on
If a is a real number and n is a posi ve whole number, then
an = a · a · · · · · a
n factors
Examples
23 = 2 · 2 · 2 = 8
34 = 3 · 3 · 3 · 3 = 81
(−1)5 = (−1)(−1)(−1)(−1)(−1) = −1
8. Anatomy of a power
Defini on
A power is an expression of the form ab .
The number a is called the base.
The number b is called the exponent.
9. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
10. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
11. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
12. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
13. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
14. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
Proof.
Check for yourself:
ax+y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = ax ay
x + y factors x factors y factors
15. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
16. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
!
an = an+0 = an · a0
17. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
18. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
Defini on
If a ̸= 0, we define a0 = 1.
19. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
Defini on
If a ̸= 0, we define a0 = 1.
No ce 00 remains undefined (as a limit form, it’s
indeterminate).
21. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
= n
an a
22. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
= n
an a
Defini on
1
If n is a posi ve integer, we define a−n = .
an
23. Defini on
1
If n is a posi ve integer, we define a−n = .
an
24. Defini on
1
If n is a posi ve integer, we define a−n = .
an
Fact
1
The conven on that a−n = “works” for nega ve n as well.
an
m−n am
If m and n are any integers, then a = n.
a
26. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
27. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Defini on
√
If q is a posi ve integer, we define a1/q = q
a. We must have a ≥ 0
if q is even.
28. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Defini on
√
If q is a posi ve integer, we define a1/q = q a. We must have a ≥ 0
if q is even.
√q
(√ )p
No ce that ap = q a . So we can unambiguously say
ap/q = (ap )1/q = (a1/q )p
30. Conventions for irrational
exponents
So ax is well-defined if a is posi ve and x is ra onal.
What about irra onal powers?
Defini on
Let a > 0. Then
ax = lim ar
r→x
r ra onal
31. Conventions for irrational
exponents
So ax is well-defined if a is posi ve and x is ra onal.
What about irra onal powers?
Defini on
Let a > 0. Then
ax = lim ar
r→x
r ra onal
In other words, to approximate ax for irra onal x, take r close to x
but ra onal and compute ar .
32. Approximating a power with an
irrational exponent
r 2r
3 23
√=8
10
3.1 231/10 = √ 31 ≈ 8.57419
2
100
3.14 2314/100 = √ 314 ≈ 8.81524
2
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
39. Graphs of exponential functions
y
y = (1/2)x y = 10x 3x = 2x
y= y y = 1.5x
y = 1x
. x
40. Graphs of exponential functions
y
y = (y/= x(1/3)x
1 2) y = 10x 3x = 2x
y= y y = 1.5x
y = 1x
. x
41. Graphs of exponential functions
y
y = (y/= x(1/3)x
1 2) y = (1/10y x= 10x 3x = 2x
) y= y y = 1.5x
y = 1x
. x
42. Graphs of exponential functions
y
y y =y/=3(1/3)x
= (1(2/x)x
2) y = (1/10y x= 10x 3x = 2x
) y= y y = 1.5x
y = 1x
. x
43. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
44. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y
a
(a ) = axy
x y
(ab)x = ax bx
45. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y (nega ve exponents mean reciprocals)
a
(a ) = axy
x y
(ab)x = ax bx
46. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y (nega ve exponents mean reciprocals)
a
(a ) = axy (frac onal exponents mean roots)
x y
(ab)x = ax bx
47. Proof.
This is true for posi ve integer exponents by natural defini on
Our conven onal defini ons make these true for ra onal
exponents
Our limit defini on make these for irra onal exponents, too
53. Limits of exponential functions
Fact (Limits of exponen al
func ons) y
y (1 y )/3 x
y = =/(1= )(2/3)x y = y1/1010= 2x = 1.5
2
x
( = =x 3x y
y ) x
y
If a > 1, then
lim ax = ∞ and
x→∞
lim ax = 0
x→−∞
If 0 < a < 1, then y = 1x
lim ax = 0 and . x
x→∞
lim ax = ∞
x→−∞
54. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
55. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
56. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
57. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
58. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t .
59. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
60. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38,
61. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
62. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
63. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
$100(1.025)4t .
64. Compounded Interest: monthly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded twelve mes a year. How much do you have a er t
years?
65. Compounded Interest: monthly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded twelve mes a year. How much do you have a er t
years?
Answer
$100(1 + 10%/12)12t
66. Compounded Interest: general
Ques on
Suppose you save P at interest rate r, with interest compounded n
mes a year. How much do you have a er t years?
67. Compounded Interest: general
Ques on
Suppose you save P at interest rate r, with interest compounded n
mes a year. How much do you have a er t years?
Answer
( r )nt
B(t) = P 1 +
n
68. Compounded Interest: continuous
Ques on
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have a er t years?
69. Compounded Interest: continuous
Ques on
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have a er t years?
Answer
( ( )rnt
r )nt 1
B(t) = lim P 1 + = lim P 1 +
n→∞ n n→∞ n
[ ( )n ]rt
1
= P lim 1 +
n→∞ n
independent of P, r, or t
74. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
75. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
76. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
77. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
78. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
79. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irra onal 100 2.70481
1000 2.71692
106 2.71828
80. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irra onal 100 2.70481
1000 2.71692
e is transcendental
106 2.71828
81. Meet the Mathematician: Leonhard Euler
Born in Switzerland, lived
in Prussia (Germany) and
Russia
Eyesight trouble all his
life, blind from 1766
onward
Hundreds of
contribu ons to calculus,
number theory, graph
theory, fluid mechanics, Leonhard Paul Euler
op cs, and astronomy Swiss, 1707–1783
83. A limit
Ques on
eh − 1
What is lim ?
h→0 h
Answer
e = lim (1 + 1/n)n = lim (1 + h)1/h . So for a small h,
n→∞ h→0
e ≈ (1 + h) 1/h
. So
[ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
84. A limit
eh − 1
It follows that lim = 1.
h→0 h
2h − 1
This can be used to characterize e: lim = 0.693 · · · < 1
h→0 h
3h − 1
and lim = 1.099 · · · > 1
h→0 h
85. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
86. Logarithms
Defini on
The base a logarithm loga x is the inverse of the func on ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
96. Graphs of logarithmic functions
y
y =y3= 2x
x
y = log2 x
y = log3 x
(0, 1)
.
(1, 0) x
97. Graphs of logarithmic functions
y
y =y10y3= 2x
=x x
y = log2 x
y = log3 x
(0, 1)
y = log10 x
.
(1, 0) x
98. Graphs of logarithmic functions
y
y =x ex
y =y10y3= 2x
= x
y = log2 x
yy= log3 x
= ln x
(0, 1)
y = log10 x
.
(1, 0) x
99. Change of base formula for logarithms
Fact
logb x
If a > 0 and a ̸= 1, and the same for b, then loga x =
logb a
100. Change of base formula for logarithms
Fact
logb x
If a > 0 and a ̸= 1, and the same for b, then loga x =
logb a
Proof.
If y = loga x, then x = ay
So logb x = logb (ay ) = y logb a
Therefore
logb x
y = loga x =
logb a
102. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
103. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised?
104. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.
105. Upshot of changing base
The point of the change of base formula
logb x 1
loga x = = · logb x = constant · logb x
logb a logb a
is that all the logarithmic func ons are mul ples of each other. So
just pick one and call it your favorite.
Engineers like the common logarithm log = log10
Computer scien sts like the binary logarithm lg = log2
Mathema cians like natural logarithm ln = loge
Naturally, we will follow the mathema cians. Just don’t pronounce
it “lawn.”