This document contains notes on inverse trigonometric functions including:
- Definitions and graphs of arcsin, arccos, arctan, and arcsec functions
- Derivations of the derivatives of arcsin, arccos, and arctan using the Inverse Function Theorem
- Examples of composing inverse trig functions and finding their derivatives
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 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
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.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
Dynamic stiffness and eigenvalues of nonlocal nano beams - new methods for dynamic analysis of nano-scale structures. This lecture gives a review and proposed new techniques.
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 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
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.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
Dynamic stiffness and eigenvalues of nonlocal nano beams - new methods for dynamic analysis of nano-scale structures. This lecture gives a review and proposed new techniques.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Tensor Train (TT) decomposition [3] is a generalization of SVD decomposition from matrices to tensors (=multidimensional arrays).
It represents a tensor compactly in terms of factors and allows to work with the tensor via its factors without materializing the tensor itself.
For example, we can find the elementwise product of two TT-tensors of size 2^100 and get the result in the TT-format as well.
In the talk, we will show how Tensor Train decomposition can be used to represent parameters of neural networks [1] and polynomial models [2].
This parametrization allows exponentially many 'virtual' parameters while working only with small factors of the TT-format.
To train the model, i.e. optimize the objective subject to the constraint that the parameters are in the TT-format, [2] uses stochastic Riemannian optimization.
[1] Novikov, A., Podoprikhin, D., Osokin, A., & Vetrov, D. P. (2015). Tensorizing neural networks. In Advances in Neural Information Processing Systems.
[2] Novikov, A., Trofimov, M., & Oseledets, I. (2016). Tensor Train polynomial models via Riemannian optimization. arXiv:1605.03795.
[3] Oseledets, I. (2011). Tensor-train decomposition. SIAM Journal on Scientific Computing.
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
Numerical simulation on laminar convection flow and heat transfer over a non ...eSAT Journals
Abstract
A numerical algorithm is presented for studying laminar convection flow and heat transfer over a non-isothermal horizontal plate.
plate temperature Tw varies with x in the following prescribed manner:
T T Cx w
n 1
where C and n are constants. By means of similarity transformation, the original nonlinear partial differential equations of flow
are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and
integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity, and temperature profiles for various Prandtl number and n are illustrated graphically.
Flow and heat transfer parameters are derived. The results of the present simulation are then compared with experimental data in
literature with good agreement.
Keywords: Free Convection, Heat Transfer, Non-isothermal Horizontal Plate, Matlab, Numerical Simulation.
https://arxiv.org/abs/2011.04370
A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
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.
Optimization problems are just max/min problems with some additional reading comprehension.
Same content as the slide version, but laid out three to a page with space for notes.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Tensor Train (TT) decomposition [3] is a generalization of SVD decomposition from matrices to tensors (=multidimensional arrays).
It represents a tensor compactly in terms of factors and allows to work with the tensor via its factors without materializing the tensor itself.
For example, we can find the elementwise product of two TT-tensors of size 2^100 and get the result in the TT-format as well.
In the talk, we will show how Tensor Train decomposition can be used to represent parameters of neural networks [1] and polynomial models [2].
This parametrization allows exponentially many 'virtual' parameters while working only with small factors of the TT-format.
To train the model, i.e. optimize the objective subject to the constraint that the parameters are in the TT-format, [2] uses stochastic Riemannian optimization.
[1] Novikov, A., Podoprikhin, D., Osokin, A., & Vetrov, D. P. (2015). Tensorizing neural networks. In Advances in Neural Information Processing Systems.
[2] Novikov, A., Trofimov, M., & Oseledets, I. (2016). Tensor Train polynomial models via Riemannian optimization. arXiv:1605.03795.
[3] Oseledets, I. (2011). Tensor-train decomposition. SIAM Journal on Scientific Computing.
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
Numerical simulation on laminar convection flow and heat transfer over a non ...eSAT Journals
Abstract
A numerical algorithm is presented for studying laminar convection flow and heat transfer over a non-isothermal horizontal plate.
plate temperature Tw varies with x in the following prescribed manner:
T T Cx w
n 1
where C and n are constants. By means of similarity transformation, the original nonlinear partial differential equations of flow
are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and
integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity, and temperature profiles for various Prandtl number and n are illustrated graphically.
Flow and heat transfer parameters are derived. The results of the present simulation are then compared with experimental data in
literature with good agreement.
Keywords: Free Convection, Heat Transfer, Non-isothermal Horizontal Plate, Matlab, Numerical Simulation.
https://arxiv.org/abs/2011.04370
A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
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.
Optimization problems are just max/min problems with some additional reading comprehension.
Same content as the slide version, but laid out three to a page with space for notes.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...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.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...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.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...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.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...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.
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.
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.
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 15: Exponential Growth and Decay (Section 021 slides)Mel Anthony Pepito
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 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
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 15: Exponential Growth and Decay (Section 021 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.
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.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...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.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
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Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
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/
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
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Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
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Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
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In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
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Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
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In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
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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.
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GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
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1. Section 3.5
Inverse Trigonometric
Functions
V63.0121.021, Calculus I
New York University
November 2, 2010
Announcements
Midterm grades have been submitted
Quiz 3 this week in recitation on Section 2.6, 2.8, 3.1, 3.2
Thank you for the evaluations
Announcements
Midterm grades have been
submitted
Quiz 3 this week in
recitation on Section 2.6,
2.8, 3.1, 3.2
Thank you for the
evaluations
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 2 / 32
Objectives
Know the definitions,
domains, ranges, and other
properties of the inverse
trignometric functions:
arcsin, arccos, arctan,
arcsec, arccsc, arccot.
Know the derivatives of the
inverse trignometric
functions.
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 3 / 32
Notes
Notes
Notes
1
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
2. What is an inverse function?
Definition
Let f be a function with domain D and range E. The inverse of f is the
function 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
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 4 / 32
What functions are invertible?
In order for f −1
to be a function, there must be only one a in D
corresponding to each b in E.
Such a function is called one-to-one
The graph of such a function passes the horizontal line test: any
horizontal line intersects the graph in exactly one point if at all.
If f is continuous, then f −1
is continuous.
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 5 / 32
Outline
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Arcsine
Arccosine
Arctangent
Arcsecant
Applications
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 6 / 32
Notes
Notes
Notes
2
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
3. arcsin
Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].
x
y
sin
−
π
2
π
2
y = x
arcsin
The domain of arcsin is [−1, 1]
The range of arcsin is −
π
2
,
π
2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 7 / 32
arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
x
y
cos
0 π
y = x
arccos
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 8 / 32
arctan
Arctan is the inverse of the tangent function after restriction to [−π/2, π/2].
x
y
tan
−
3π
2
−
π
2
π
2
3π
2
y = x
arctan
−
π
2
π
2
The domain of arctan is (−∞, ∞)
The range of arctan is −
π
2
,
π
2
lim
x→∞
arctan x =
π
2
, lim
x→−∞
arctan x = −
π
2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 9 / 32
Notes
Notes
Notes
3
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
4. arcsec
Arcsecant is the inverse of secant after restriction to [0, π/2) ∪ (π, 3π/2].
x
y
sec
−
3π
2
−
π
2
π
2
3π
2
y = x
π
2
3π
2
The domain of arcsec is (−∞, −1] ∪ [1, ∞)
The range of arcsec is 0,
π
2
∪
π
2
, π
lim
x→∞
arcsec x =
π
2
, lim
x→−∞
arcsec x =
3π
2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 10 / 32
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
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 11 / 32
Check: Values of inverse trigonometric functions
Example
Find
arcsin(1/2)
arctan(−1)
arccos −
√
2
2
Solution
π
6
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 12 / 32
Notes
Notes
Notes
4
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
5. 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
.
I use csc x for
1
sin x
and arcsin x for the inverse of sin x.
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 15 / 32
Outline
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Arcsine
Arccosine
Arctangent
Arcsecant
Applications
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 16 / 32
The Inverse Function Theorem
Theorem (The Inverse Function Theorem)
Let f be differentiable at a, and f (a) = 0. Then f −1
is defined in an open
interval containing b = f (a), and
(f −1
) (b) =
1
f (f −1(b))
In Leibniz notation we have
dx
dy
=
1
dy/dx
Upshot: Many times the derivative of f −1
(x) can be found by implicit
differentiation and the derivative of f :
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 17 / 32
Notes
Notes
Notes
5
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
6. Illustrating the Inverse Function Theorem
Example
Use the inverse function theorem to find the derivative of the square root function.
Solution (Newtonian notation)
Let f (x) = x2
so that f −1
(y) =
√
y. Then f (u) = 2u so for any b > 0 we have
(f −1
) (b) =
1
2
√
b
Solution (Leibniz notation)
If the original function is y = x2
, then the inverse function is defined by x = y2
.
Differentiate implicitly:
1 = 2y
dy
dx
=⇒
dy
dx
=
1
2
√
x
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 18 / 32
Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
cos y
dy
dx
= 1 =⇒
dy
dx
=
1
cos y
=
1
cos(arcsin x)
To simplify, look at a right
triangle:
cos(arcsin x) = 1 − x2
So
d
dx
arcsin(x) =
1
√
1 − x2
1
x
y = arcsin x
1 − x2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 19 / 32
Graphing arcsin and its derivative
The domain of f is [−1, 1],
but the domain of f is
(−1, 1)
lim
x→1−
f (x) = +∞
lim
x→−1+
f (x) = +∞ |
−1
|
1
arcsin
1
√
1 − x2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 20 / 32
Notes
Notes
Notes
6
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
7. Composing with arcsin
Example
Let f (x) = arcsin(x3
+ 1). Find f (x).
Solution
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 21 / 32
Derivation: The derivative of arccos
Let y = arccos x, so x = cos y. Then
− sin y
dy
dx
= 1 =⇒
dy
dx
=
1
− sin y
=
1
− sin(arccos x)
To simplify, look at a right
triangle:
sin(arccos x) = 1 − x2
So
d
dx
arccos(x) = −
1
√
1 − x2
1
1 − x2
x
y = arccos x
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 22 / 32
Graphing arcsin and arccos
|
−1
|
1
arcsin
arccos
Note
cos θ = sin
π
2
− θ
=⇒ arccos x =
π
2
− arcsin x
So it’s not a surprise that their
derivatives are opposites.
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 23 / 32
Notes
Notes
Notes
7
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
8. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
sec2
y
dy
dx
= 1 =⇒
dy
dx
=
1
sec2 y
= cos2
(arctan x)
To simplify, look at a right
triangle:
cos(arctan x) =
1
√
1 + x2
So
d
dx
arctan(x) =
1
1 + x2
x
1
y = arctan x
1 + x2
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 24 / 32
Graphing arctan and its derivative
x
y
arctan
1
1 + x2
π/2
−π/2
The domain of f and f are both (−∞, ∞)
Because of the horizontal asymptotes, lim
x→±∞
f (x) = 0
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 25 / 32
Composing with arctan
Example
Let f (x) = arctan
√
x. Find f (x).
Solution
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 26 / 32
Notes
Notes
Notes
8
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
9. Derivation: The derivative of arcsec
Try this first.
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 27 / 32
Another Example
Example
Let f (x) = earcsec 3x
. Find f (x).
Solution
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 28 / 32
Outline
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Arcsine
Arccosine
Arctangent
Arcsecant
Applications
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 29 / 32
Notes
Notes
Notes
9
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010
10. Application
Example
One of the guiding principles of
most sports is to “keep your eye
on the ball.” In baseball, a batter
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
batter’s angle of gaze need to
change to follow the ball as it
crosses home plate?
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 30 / 32
Solution
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 31 / 32
Summary
y y
arcsin x
1
√
1 − x2
arccos x −
1
√
1 − x2
arctan x
1
1 + x2
arccot x −
1
1 + x2
arcsec x
1
x
√
x2 − 1
arccsc x −
1
x
√
x2 − 1
Remarkable that the
derivatives of these
transcendental functions are
algebraic (or even rational!)
V63.0121.021, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 2, 2010 32 / 32
Notes
Notes
Notes
10
Section 3.5 : Inverse Trigonometric FunctionsV63.0121.021, Calculus I November 2, 2010