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.
Lesson 12: Linear Approximations and Differentials (handout)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.
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.
Lesson 12: Linear Approximations and Differentials (handout)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.
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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
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.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the handout version to take notes on.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
A presentation slide of the paper "Doubly Decomposing Nonparametric Tensor Regression" (Imaizumi & Hayashi 2016 ICML).
Full paper
http://www.jmlr.org/proceedings/papers/v48/imaizumi16.html
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 15: Exponential Growth and Decay (handout)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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
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.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the handout version to take notes on.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
A presentation slide of the paper "Doubly Decomposing Nonparametric Tensor Regression" (Imaizumi & Hayashi 2016 ICML).
Full paper
http://www.jmlr.org/proceedings/papers/v48/imaizumi16.html
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 15: Exponential Growth and Decay (handout)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.
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.
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.
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.
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.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
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.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...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.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...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.
Similar to Lesson 16: Inverse Trigonometric Functions (handout) (20)
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 (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.
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.
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.
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.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
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 – USA
Expansion of bot farms – 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 – 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/
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
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.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
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
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
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.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
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/
1. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Sec on 3.6
Inverse Trigonometric Func ons
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
March 28, 2011
.
.
Notes
Announcements
Midterm has been returned. Please see
FAQ on Blackboard (under ”Exams and
Quizzes”)
Quiz 3 this week in recita on on
Sec on 2.6, 2.8, 3.1, 3.2
Quiz 4 April 14–15 on Sec ons 3.3, 3.4,
3.5, and 3.7
Quiz 5 April 28–29 on Sec ons 4.1, 4.2,
4.3, and 4.4
.
.
Notes
Objectives
Know the defini ons, domains, ranges,
and other proper es of the inverse
trignometric func ons: arcsin, arccos,
arctan, arcsec, arccsc, arccot.
Know the deriva ves of the inverse
trignometric func ons.
.
.
. 1
.
2. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
.
.
Notes
What is an inverse function?
Defini on
Let f be a func on with domain D and range E. The inverse of f is the
func on f−1 defined by:
f−1 (b) = a,
where a is chosen so that f(a) = b.
So
f−1 (f(x)) = x, f(f−1 (x)) = x
.
.
Notes
What functions are invertible?
In order for f−1 to be a func on, there must be only one a in D
corresponding to each b in E.
Such a func on is called one-to-one
The graph of such a func on passes the horizontal line test:
any horizontal line intersects the graph in exactly one point if at
all.
If f is con nuous, then f−1 is con nuous.
.
.
. 2
.
3. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Graphing the inverse function
y y=x
If b = f(a), then f−1 (b) = a.
So if (a, b) is on the graph of f,
then (b, a) is on the graph of f−1 .
On the xy-plane, the point (b, a) (b, a)
is the reflec on of (a, b) in the
line y = x. (a, b)
Therefore:
.
x
Fact
The graph of f−1 is the reflec on of the graph of f in the line y = x.
.
.
Notes
arcsin
Arcsin is the inverse of the sine func on a er restric on to
[−π/2, π/2].
y
y=x
arcsin
. x
π π sin
−
2 2
The domain of arcsin is [−1, 1]
[ π π]
The range of arcsin is − ,
2 2
.
.
Notes
arccos
Arccos is the inverse of the cosine func on a er restric on to [0, π]
arccos
y
y=x
cos
. x
0 π
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
.
.
. 3
.
4. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
arctan
y=x
Arctan is the inverse of the tangent func on a er restric on to
y
(−π/2, π/2).
π
2 arctan
. x
3π π π 3π
− −
2 2− π 2 2
2
( π π ∞)
The domain of arctan is (−∞, )
The range of arctan is − ,
2 2 tan
π π
lim arctan x = , lim arctan x = −
. x→∞ 2 x→−∞ 2
.
Notes
arcsec 3π
2
Arcsecant is the inverse of secant a er restric on to x
y=
[0, π/2) ∪ [π, 3π/2). y
π
2
. x
3π π π 3π
− −
2 2 2 2
The domain of arcsec is (−∞, −1] ∪ [1, ∞)
[ π ) (π ]
The range of arcsec is 0, ∪ ,π
2 2
π 3π
lim arcsec x = , lim arcsec x = sec
x→∞ 2 x→−∞ 2
.
.
Notes
Values of Trigonometric Functions
x 0 π/6 π/4 π/3 π/2
√ √
sin x 0 1/2 2/2 3/2 1
√ √
cos x 1 3/2 2/2 1/2 0
√ √
tan x 0 1/ 3 1 3 undef
√ √
cot x undef 3 1 1/ 3 0
√ √
sec x 1 2/ 3 2/ 2 2 undef
√ √
csc x undef 2 2/ 2 2/ 3 1
.
.
. 4
.
5. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Check: Values of inverse trigonometric functions
Example Solu on
Find π
arcsin(1/2) 6
arctan(−1)
( √ )
2
arccos −
2
.
.
Notes
Caution: Notational ambiguity
sin2 x =.(sin x)2 sin−1 x = (sin x)−1
sinn x means the nth power of sin x, except when n = −1!
The book uses sin−1 x for the inverse of sin x, and never for
(sin x)−1 .
1
I use csc x for and arcsin x for the inverse of sin x.
sin x
.
.
Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
.
.
. 5
.
6. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The Inverse Function Theorem
Theorem (The Inverse Func on Theorem)
Let f be differen able at a, and f′ (a) ̸= 0. Then f−1 is defined in an
open interval containing b = f(a), and
1
(f−1 )′ (b) =
f′ (f−1 (b))
In Leibniz nota on we have
dx 1
=
dy dy/dx
.
.
Notes
Illustrating the IFT
Example
Use the inverse func on theorem to find the deriva ve of the
square root func on.
Solu on (Newtonian nota on)
√
Let f(x) = x2 so that f−1 (y) = y. Then f′ (u) = 2u so for any b > 0
we have
1
(f−1 )′ (b) = √
2 b
.
.
Notes
Illustrating the IFT
Example
Use the inverse func on theorem to find the deriva ve of the
square root func on.
Solu on (Leibniz nota on)
If the original func on is y = x2 , then the inverse func on is defined
by x = y2 . Differen ate implicitly:
dy dy 1
1 = 2y =⇒ = √
dx dx 2 x
.
.
. 6
.
7. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The derivative of arcsine
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right triangle:
√
cos(arcsin x) = 1 − x2
1
So x
Fact
d 1 y = arcsin x
arcsin(x) = √ .√
dx 1 − x2 1 − x2
.
.
Notes
Graphing arcsin and its derivative
1
√
The domain of f is [−1, 1], 1 − x2
but the domain of f′ is
(−1, 1) arcsin
lim− f′ (x) = +∞
x→1
| . |
lim f′ (x) = +∞ −1 1
x→−1+
.
.
Notes
Composing with arcsin
Example
Let f(x) = arcsin(x3 + 1). Find f′ (x).
Solu on
.
.
. 7
.
8. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
dx dx − sin y − sin(arccos x)
To simplify, look at a right triangle:
√
sin(arccos x) = 1 − x2
1 √
So 1 − x2
Fact
d 1 y = arccos x
arccos(x) = − √ .
dx 1 − x2 x
.
.
Notes
Graphing arcsin and arccos
arccos Note
(π )
cos θ = sin −θ
arcsin 2
π
=⇒ arccos x = − arcsin x
2
| . |
−1 1 So it’s not a surprise that their
deriva ves are opposites.
.
.
Notes
The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right triangle:
1
cos(arctan x) = √
1 + x2
√
So 1 + x2 x
Fact
d 1 y = arctan x
.
arctan(x) = 1
dx 1 + x2
.
.
. 8
.
9. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Graphing arctan and its derivative
y
π/2
arctan
1
1 + x2
. x
−π/2
The domain of f and f′ are both (−∞, ∞)
Because of the horizontal asymptotes, lim f′ (x) = 0
x→±∞
.
.
Notes
Composing with arctan
Example
√
Let f(x) = arctan x. Find f′ (x).
Solu on
.
.
Notes
The derivative of arcsec
Try this first.
.
.
. 9
.
10. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Another Example
Example
Let f(x) = earcsec 3x . Find f′ (x).
Solu on
.
.
Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
.
.
Notes
Application
Example
One of the guiding principles of most
sports is to “keep your eye on the
ball.” In baseball, a ba er stands 2 ft
away from home plate as a pitch is
thrown with a velocity of 130 ft/sec
(about 90 mph). At what rate does
the ba er’s angle of gaze need to
change to follow the ball as it crosses
home plate?
.
.
. 10
.
11. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Solu on Notes
.
.
Notes
Summary
y y′ y y′
1 1
arcsin x √ arccos x − √
1−x 2 1 − x2
1 1
arctan x arccot x −
1 + x2 1 + x2
1 1
arcsec x √ arccsc x − √
x x2 − 1 x x2 − 1
Remarkable that the deriva ves of these transcendental
func ons are algebraic (or even ra onal!)
.
.
Notes
.
.
. 11
.