This document contains lecture notes on inverse trigonometric functions. It begins with definitions of inverse functions and conditions for a function to have an inverse. It then defines the inverse trigonometric functions arcsin, arccos, arctan, and arcsec and gives their domains and ranges. Examples are provided to illustrate calculating values of the inverse trigonometric functions. The document concludes with a brief discussion of notational ambiguity and an outline mentioning that derivatives of the inverse trigonometric functions will be covered.
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.
1) The document discusses inverse trigonometric functions such as sin-1, cos-1, tan-1, cot-1, sec-1, and cosec-1.
2) It explains that the inverse functions are defined by restricting the domains of the original trigonometric functions (sin, cos, etc.) so that they become one-to-one mappings.
3) Graphs of the inverse functions are obtained by reflecting the graphs of the original functions about the line y=x.
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.
The document defines and discusses inverse trigonometric functions. It defines them as the inverses of trigonometric functions like sine, cosine, and tangent, with restricted domains. Some key properties discussed include identities, derivatives, and integrals of inverse trigonometric functions. Graphs of inverse sine and cosine are reflections of sine and cosine about the line y=x.
This document contains a practice test with multiple choice and written response questions about transformations of graphs of functions. Some key questions ask students to:
1) Identify which statement about transforming a graph is false.
2) Determine if reflecting a function's graph in the y-axis will produce its inverse.
3) Write equations of functions after transformations like translations, stretches, and reflections.
4) Sketch and describe transformations of a graph to satisfy a given equation.
5) Determine the domain and range of a transformed function.
6) Find the inverse of a function and restrict its domain to make the inverse a function.
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.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic 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.
1) The document discusses inverse trigonometric functions such as sin-1, cos-1, tan-1, cot-1, sec-1, and cosec-1.
2) It explains that the inverse functions are defined by restricting the domains of the original trigonometric functions (sin, cos, etc.) so that they become one-to-one mappings.
3) Graphs of the inverse functions are obtained by reflecting the graphs of the original functions about the line y=x.
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.
The document defines and discusses inverse trigonometric functions. It defines them as the inverses of trigonometric functions like sine, cosine, and tangent, with restricted domains. Some key properties discussed include identities, derivatives, and integrals of inverse trigonometric functions. Graphs of inverse sine and cosine are reflections of sine and cosine about the line y=x.
This document contains a practice test with multiple choice and written response questions about transformations of graphs of functions. Some key questions ask students to:
1) Identify which statement about transforming a graph is false.
2) Determine if reflecting a function's graph in the y-axis will produce its inverse.
3) Write equations of functions after transformations like translations, stretches, and reflections.
4) Sketch and describe transformations of a graph to satisfy a given equation.
5) Determine the domain and range of a transformed function.
6) Find the inverse of a function and restrict its domain to make the inverse a function.
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.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
This document discusses Fourier series and Parseval's theorem. It explains that Parseval's theorem gives the relationship between Fourier coefficients. Specifically, it states that if a Fourier series converges uniformly, the integral of the square of the original function over its domain is equal to the sum of the square of the Fourier coefficients. The document also provides an example of using Parseval's theorem to find the total square error of a Fourier approximation and proving an identity.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
The document defines and discusses various concepts related to functions, including:
- A function assigns exactly one output element to each input element. Functions can be represented graphically.
- Key properties of functions include being one-to-one (injective), onto (surjective), and bijective (both one-to-one and onto).
- Important functions discussed include the identity function I(x)=x, the floor function ⌊x⌋ which returns the largest integer less than or equal to x, and the ceiling function ⌈x⌉ which returns the smallest integer greater than or equal to x.
Matrix Models of 2D String Theory in Non-trivial BackgroundsUtrecht University
This thesis examines matrix models of 2D string theory in non-trivial backgrounds. It begins with an introduction to string theory partition functions and topological expansions. It then discusses critical strings in background fields and how non-critical strings relate to Liouville gravity. Matrix models are introduced as a way to discretize string worldsheets. The remainder of the thesis explores how matrix quantum mechanics can describe non-critical strings in particular backgrounds like the linear dilaton model.
The document discusses directional derivatives and the gradient. It defines the directional slope of a vector v as the ratio of the y-component to the x-component of v. It then gives an example of calculating the directional slopes of several vectors. It explains that the directional slope indicates the steepness and direction of v. For a differentiable function f(x), the directional derivative in the direction of the unit vector <1,0> is the partial derivative df/dx, while the opposite direction has derivative -df/dx.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
This document contains a tutorial on hyperbolic functions and inverse hyperbolic functions with various examples and exercises to evaluate. It begins by defining hyperbolic functions like sinh, cosh, tanh and evaluating expressions in terms of them. It then covers identities involving hyperbolic functions, using them to solve equations, and expressing hyperbolic functions in terms of exponents. The document also explores inverse hyperbolic functions, expressing them in terms of logarithms, and evaluating inverse function expressions. Finally, it covers inverse trigonometric functions, evaluating expressions and solving equations involving inverse trig and hyperbolic functions.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
The document discusses polar equations and their use in representing conic sections. It defines key terms like focus, directrix, and eccentricity used to describe ellipses, parabolas, and hyperbolas. Ellipses and hyperbolas are defined geometrically as all points where the distance to one focus (PF) divided by the distance to the corresponding directrix (PD) is a constant (the eccentricity). Examples are given of the polar forms of different conic sections for varying eccentricities.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
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.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
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 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
This document discusses Fourier series and Parseval's theorem. It explains that Parseval's theorem gives the relationship between Fourier coefficients. Specifically, it states that if a Fourier series converges uniformly, the integral of the square of the original function over its domain is equal to the sum of the square of the Fourier coefficients. The document also provides an example of using Parseval's theorem to find the total square error of a Fourier approximation and proving an identity.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
The document defines and discusses various concepts related to functions, including:
- A function assigns exactly one output element to each input element. Functions can be represented graphically.
- Key properties of functions include being one-to-one (injective), onto (surjective), and bijective (both one-to-one and onto).
- Important functions discussed include the identity function I(x)=x, the floor function ⌊x⌋ which returns the largest integer less than or equal to x, and the ceiling function ⌈x⌉ which returns the smallest integer greater than or equal to x.
Matrix Models of 2D String Theory in Non-trivial BackgroundsUtrecht University
This thesis examines matrix models of 2D string theory in non-trivial backgrounds. It begins with an introduction to string theory partition functions and topological expansions. It then discusses critical strings in background fields and how non-critical strings relate to Liouville gravity. Matrix models are introduced as a way to discretize string worldsheets. The remainder of the thesis explores how matrix quantum mechanics can describe non-critical strings in particular backgrounds like the linear dilaton model.
The document discusses directional derivatives and the gradient. It defines the directional slope of a vector v as the ratio of the y-component to the x-component of v. It then gives an example of calculating the directional slopes of several vectors. It explains that the directional slope indicates the steepness and direction of v. For a differentiable function f(x), the directional derivative in the direction of the unit vector <1,0> is the partial derivative df/dx, while the opposite direction has derivative -df/dx.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
This document contains a tutorial on hyperbolic functions and inverse hyperbolic functions with various examples and exercises to evaluate. It begins by defining hyperbolic functions like sinh, cosh, tanh and evaluating expressions in terms of them. It then covers identities involving hyperbolic functions, using them to solve equations, and expressing hyperbolic functions in terms of exponents. The document also explores inverse hyperbolic functions, expressing them in terms of logarithms, and evaluating inverse function expressions. Finally, it covers inverse trigonometric functions, evaluating expressions and solving equations involving inverse trig and hyperbolic functions.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
The document discusses polar equations and their use in representing conic sections. It defines key terms like focus, directrix, and eccentricity used to describe ellipses, parabolas, and hyperbolas. Ellipses and hyperbolas are defined geometrically as all points where the distance to one focus (PF) divided by the distance to the corresponding directrix (PD) is a constant (the eccentricity). Examples are given of the polar forms of different conic sections for varying eccentricities.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
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.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
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 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
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.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
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 document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
1. Section 3.5
Inverse Trigonometric
Functions
V63.0121.006/016, Calculus I
March 11, 2010
Announcements
Exams returned in recitation
There is WebAssign due Tuesday March 23 and written HW
due Thursday March 25 . . . . . .
2. Announcements
Exams returned in recitation
There is WebAssign due Tuesday March 23 and written HW
due Thursday March 25
next quiz is Friday April 2
. . . . . .
3. 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.
. . . . . .
4. 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
. . . . . .
5. 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.
. . . . . .
7. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
. . . x
.
π π s
. in
−
. .
2 2
. . . . . .
8. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
.
. . . x
.
π π s
. in
−
. . .
2 2
. . . . . .
9. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
y
. =x
.
. . . x
.
π π s
. in
−
. . .
2 2
. . . . . .
10. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
. . rcsin
a
.
. . . x
.
π π s
. in
−
. . .
2 2
.
The domain of arcsin is [−1, 1]
[ π π]
The range of arcsin is − ,
2 2
. . . . . .
11. arccos
Arccos is the inverse of the cosine function after restriction to
[0, π]
y
.
c
. os
. . x
.
0
. .
π
. . . . . .
12. arccos
Arccos is the inverse of the cosine function after restriction to
[0, π]
y
.
.
c
. os
. . x
.
0
. .
π
.
. . . . . .
13. arccos
Arccos is the inverse of the cosine function after restriction to
[0, π]
y
.
y
. =x
.
c
. os
. . x
.
0
. .
π
.
. . . . . .
14. arccos
Arccos is the inverse of the cosine function after restriction to
[0, π]
. . rccos
a
y
.
.
c
. os
. . . x
.
0
. .
π
.
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
. . . . . .
15. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
16. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
17. arctan
Arctan is the inverse of the tangent function after restriction to
y
. =x
[−π/2, π/2].
y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
18. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
π
. a
. rctan
2
. x
.
π
−
.
2
The domain of arctan is (−∞, ∞)
( π π)
The range of arctan is − ,
2 2
π π
lim arctan x = , lim arctan x = −
x→∞ 2 x→−∞ 2
. . . . . .
19. arcsec
Arcsecant is the inverse of secant after restriction to
[0, π/2) ∪ (π, 3π/2].
y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
s
. ec
. . . . . .
20. arcsec
Arcsecant is the inverse of secant after restriction to
[0, π/2) ∪ (π, 3π/2].
y
.
.
. x
.
3π π π 3π
−
. −
. . . .
2 2 2 2
s
. ec
. . . . . .
21. arcsec
Arcsecant is the inverse of secant after restriction to
y
. =x
[0, π/2) ∪ (π, 3π/2].
y
.
.
. x
.
3π π π 3π
−
. −
. . . .
2 2 2 2
s
. ec
. . . . . .
22. arcsec 3π
.
Arcsecant is the inverse of secant after restriction to
2
[0, π/2) ∪ (π, 3π/2].
. . y
π
.
2 .
. . x
.
.
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 =
x→∞ 2 x→−∞ 2
. . . . . .
23. Values of Trigonometric Functions
π π π π
x 0
6 4 3 2
√ √
1 2 3
sin x 0 1
2 2 2
√ √
3 2 1
cos x 1 0
2 2 2
1 √
tan x 0 √ 1 3 undef
3
√ 1
cot x undef 3 1 √ 0
3
2 2
sec x 1 √ √ 2 undef
3 2
2 2
csc x undef 2 √ √ 1
2 3
. . . . . .
27. What is arctan(−1)?
.
( )
3
. π/4 3π
. Yes, tan = −1
4
√
2
s
. in(3π/4) =
2
.
√ .
2
. os(3π/4) = −
c
2
.
−
. π/4
. . . . . .
28. What is arctan(−1)?
.
( )
3
. π/4 3π
. Yes, tan = −1
4
√ But, the range of arctan
( π π)
2
s
. in(3π/4) = is − ,
2 2 2
.
√ .
2
. os(3π/4) = −
c
2
.
−
. π/4
. . . . . .
29. What is arctan(−1)?
.
( )
3
. π/4 3π
. Yes, tan = −1
4
But, the range of arctan
( π π)
√ is − ,
2 2 2
c
. os(π/4) =
. 2 Another angle whose
. π
tangent is −1 is − , and
√ 4
2 this is in the right range.
. in(π/4) = −
s
2
.
−
. π/4
. . . . . .
30. What is arctan(−1)?
.
( )
3
. π/4 3π
. Yes, tan = −1
4
But, the range of arctan
( π π)
√ is − ,
2 2 2
c
. os(π/4) =
. 2 Another angle whose
. π
tangent is −1 is − , and
√ 4
2 this is in the right range.
. in(π/4) = −
s π
2 So arctan(−1) = −
4
.
−
. π/4
. . . . . .
33. Caution: Notational ambiguity
. in2 x =.(sin x)2
s . in−1 x = (sin x)−1
s
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
. . . . . .
36. 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
1
(f−1 )′ (b) = ′ −1
f (f (b))
“Proof”.
If y = f−1 (x), then
f(y ) = x ,
So by implicit differentiation
dy dy 1 1
f′ (y) = 1 =⇒ = ′ = ′ −1
dx dx f (y) f (f (x))
. . . . . .
37. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
. . . . . .
38. The derivative of arcsin
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:
.
. . . . . .
39. The derivative of arcsin
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:
1
.
x
.
.
. . . . . .
40. The derivative of arcsin
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:
1
.
x
.
y
. = arcsin x
.
. . . . . .
41. The derivative of arcsin
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:
1
.
x
.
y
. = arcsin x
. √
. 1 − x2
. . . . . .
42. The derivative of arcsin
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
.
x
.
y
. = arcsin x
. √
. 1 − x2
. . . . . .
43. The derivative of arcsin
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
.
x
.
So
d 1 y
. = arcsin x
arcsin(x) = √
dx 1 − x2 . √
. 1 − x2
. . . . . .
44. Graphing arcsin and its derivative
1
.√
1 − x2
The domain of f is
[−1, 1], but the domain . . rcsin
a
of f′ is (−1, 1)
lim f′ (x) = +∞
x →1 −
lim f′ (x) = +∞ .
| . .
|
x→−1+ −
. 1 1
.
.
. . . . . .
45. 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)
. . . . . .
46. 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
. √
. 1 − x2
So
d 1 y
. = arccos x
arccos(x) = − √ .
dx 1 − x2 x
.
. . . . . .
48. Graphing arcsin and arccos
. . rccos
a
Note
(π )
cos θ = sin −θ
. . rcsin
a 2
π
=⇒ arccos x = − arcsin x
2
.
| . |.
. So it’s not a surprise that their
−
. 1 1
. derivatives are opposites.
.
. . . . . .
49. 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
. . . . . .
50. 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:
.
. . . . . .
51. 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:
x
.
.
1
.
. . . . . .
52. 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:
x
.
y
. = arctan x
.
1
.
. . . . . .
53. 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 + x2 x
.
y
. = arctan x
.
1
.
. . . . . .
54. 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 √
. 1 + x2 x
.
y
. = arctan x
.
1
.
. . . . . .
55. 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 √
. 1 + x2 x
.
So
d 1 y
. = arctan x
arctan(x) = .
dx 1 + x2
1
.
. . . . . .
56. Graphing arctan and its derivative
y
.
. /2
π
a
. rctan
1
.
1 + x2
. x
.
−
. π/2
The domain of f and f′ are both (−∞, ∞)
Because of the horizontal asymptotes, lim f′ (x) = 0
x→±∞
. . . . . .
57. Example
√
Let f(x) = arctan x. Find f′ (x).
. . . . . .
58. Example
√
Let f(x) = arctan x. Find f′ (x).
Solution
d √ 1 d√ 1 1
arctan x = (√ )2 x= · √
dx 1+ x dx 1+x 2 x
1
= √ √
2 x + 2x x
. . . . . .
60. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
. . . . . .
61. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
.
. . . . . .
62. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
.
. . . . . .
63. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
x
.
.
1
.
. . . . . .
64. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
x
.
y
. = arcsec x
.
1
.
. . . . . .
65. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
√
x2 − 1
tan(arcsec x) = √
1 x
. . x2 − 1
y
. = arcsec x
.
1
.
. . . . . .
66. The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
√
x2 − 1
tan(arcsec x) = √
1 x
. . x2 − 1
So
d 1 y
. = arcsec x
arcsec(x) = √ .
dx x x2 − 1
1
.
. . . . . .
70. 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?
. . . . . .
71. Let y(t) be the distance from the ball to home plate, and θ the
angle the batter’s eyes make with home plate while following the
ball. We know y′ = −130 and we want θ′ at the moment that
y = 0.
y
.
1
. 30 ft/sec
.
θ
.
. 2
. ft
. . . . . .
72. Let y(t) be the distance from the ball to home plate, and θ the
angle the batter’s eyes make with home plate while following the
ball. We know y′ = −130 and we want θ′ at the moment that
y = 0.
We have θ = arctan(y/2).
Thus
dθ 1 1 dy
= ·
2 2 dt
dt 1 + ( y /2 )
y
.
1
. 30 ft/sec
.
θ
.
. 2
. ft
. . . . . .
73. Let y(t) be the distance from the ball to home plate, and θ the
angle the batter’s eyes make with home plate while following the
ball. We know y′ = −130 and we want θ′ at the moment that
y = 0.
We have θ = arctan(y/2).
Thus
dθ 1 1 dy
= ·
2 2 dt
dt 1 + ( y /2 )
When y = 0 and y′ = −130, y
.
then
dθ 1 1
= · (−130) = −65 rad/sec 1
. 30 ft/sec
dt y =0 1+0 2
.
θ
.
. 2
. ft
. . . . . .
74. Let y(t) be the distance from the ball to home plate, and θ the
angle the batter’s eyes make with home plate while following the
ball. We know y′ = −130 and we want θ′ at the moment that
y = 0.
We have θ = arctan(y/2).
Thus
dθ 1 1 dy
= ·
2 2 dt
dt 1 + ( y /2 )
When y = 0 and y′ = −130, y
.
then
dθ 1 1
= · (−130) = −65 rad/sec 1
. 30 ft/sec
dt y =0 1+0 2
.
θ
The human eye can only .
track at 3 rad/sec! . 2
. ft
. . . . . .
75. Recap
y y′
1
arcsin x √
1 − x2
1
arccos x − √ Remarkable that the
1 − x2
derivatives of these
1 transcendental functions
arctan x
1 + x2 are algebraic (or even
1 rational!)
arccot x −
1 + x2
1
arcsec x √
x x2 − 1
1
arccsc x − √
x x2 − 1
. . . . . .