Thus begins the second "half" of calculus—in which we attempt to find areas of curved regions. Like with derivatives, we use a limiting process starting from things we know (areas of rectangles) and finer and finer approximations.
Lesson 25: Areas and Distances; The Definite IntegralMatthew Leingang
Thus begins the second "half" of calculus—in which we attempt to find areas of curved regions. Like with derivatives, we use a limiting process starting from things we know (areas of rectangles) and finer and finer approximations.
Extra ways to see: An Artist's Guide to Map Operationsjmallos
This document discusses map operations, which convert one subdivision of a surface into another subdivision with certain properties. It provides examples of map operations that yield bipartite, chess-colorable, quadrilateral-faced, triangle-faced, and other properties. It shows how map operations can be represented by Truchet tiles and correspond to unit weaving patterns, tensegrities, and plain weaving. Finally, it notes that spherical designs may work on other surfaces if they tolerate conformal distortions and variations in extrinsic and intrinsic curvature.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
Tarefa 1 áreas e perímetros de poligonos regularesnatalialeao5
This document provides a math word problem for 6th grade students involving areas and perimeters of regular polygons and figures. It presents a story about a student named Joao going on a journey through the world of polygons. To reach his final destination, he must solve various puzzles by calculating perimeters and areas of different shapes on clues or tickets, using a grid square as the unit of measurement. The document then provides a series of clues and problems for students to solve step-by-step to complete Joao's journey. It concludes with an exercise section providing additional practice calculating areas of different shapes.
This document contains a 33 question test on polygons and their properties. The test covers identifying polygons by name and type, finding missing side lengths and angle measures of polygons, calculating sums of interior and exterior angles, determining which statements about polygon properties are true or false, and identifying polygons based on given properties. The test includes multiple choice, true/false, and short answer questions.
OLIMPIADAS DE MATEMATICA BULGARIA 1960 2008Armando Cavero
This document contains problems from the Bulgarian Mathematical Olympiad from 1960-1964. It includes 6 multi-part problems from each year's third round and 4 multi-part problems from the 1962 and 1964 fourth rounds. The problems cover a variety of mathematical topics including algebra, geometry, trigonometry, and inequalities.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will cover fundamentals of calculus including limits, derivatives, integrals, and optimization. It will meet twice a week for lectures and recitations. Assessment will include weekly homework, biweekly quizzes, a midterm exam, and a final exam. Grades will be calculated based on scores on these assessments. The required textbook can be purchased in hardcover, looseleaf, or online formats. Students are encouraged to contact the professor or TAs with any questions.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
Lesson 25: Areas and Distances; The Definite IntegralMatthew Leingang
Thus begins the second "half" of calculus—in which we attempt to find areas of curved regions. Like with derivatives, we use a limiting process starting from things we know (areas of rectangles) and finer and finer approximations.
Extra ways to see: An Artist's Guide to Map Operationsjmallos
This document discusses map operations, which convert one subdivision of a surface into another subdivision with certain properties. It provides examples of map operations that yield bipartite, chess-colorable, quadrilateral-faced, triangle-faced, and other properties. It shows how map operations can be represented by Truchet tiles and correspond to unit weaving patterns, tensegrities, and plain weaving. Finally, it notes that spherical designs may work on other surfaces if they tolerate conformal distortions and variations in extrinsic and intrinsic curvature.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
Tarefa 1 áreas e perímetros de poligonos regularesnatalialeao5
This document provides a math word problem for 6th grade students involving areas and perimeters of regular polygons and figures. It presents a story about a student named Joao going on a journey through the world of polygons. To reach his final destination, he must solve various puzzles by calculating perimeters and areas of different shapes on clues or tickets, using a grid square as the unit of measurement. The document then provides a series of clues and problems for students to solve step-by-step to complete Joao's journey. It concludes with an exercise section providing additional practice calculating areas of different shapes.
This document contains a 33 question test on polygons and their properties. The test covers identifying polygons by name and type, finding missing side lengths and angle measures of polygons, calculating sums of interior and exterior angles, determining which statements about polygon properties are true or false, and identifying polygons based on given properties. The test includes multiple choice, true/false, and short answer questions.
OLIMPIADAS DE MATEMATICA BULGARIA 1960 2008Armando Cavero
This document contains problems from the Bulgarian Mathematical Olympiad from 1960-1964. It includes 6 multi-part problems from each year's third round and 4 multi-part problems from the 1962 and 1964 fourth rounds. The problems cover a variety of mathematical topics including algebra, geometry, trigonometry, and inequalities.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will cover fundamentals of calculus including limits, derivatives, integrals, and optimization. It will meet twice a week for lectures and recitations. Assessment will include weekly homework, biweekly quizzes, a midterm exam, and a final exam. Grades will be calculated based on scores on these assessments. The required textbook can be purchased in hardcover, looseleaf, or online formats. Students are encouraged to contact the professor or TAs with any questions.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
- The document is a section from a calculus course at NYU that discusses using derivatives to determine the shapes of curves.
- It covers using the first derivative to determine if a function is increasing or decreasing over an interval using the Increasing/Decreasing Test. It also discusses using the second derivative to determine if a function is concave up or down over an interval using the Second Derivative Test.
- Examples are provided to demonstrate finding intervals of monotonicity for functions and classifying critical points as local maxima, minima or neither using the First Derivative Test.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document summarizes a calculus lecture on the Mean Value Theorem. It begins with announcements about exams and assignments. It then outlines the topics to be covered: Rolle's Theorem and the Mean Value Theorem, applications of the MVT, and why the MVT is important. It provides heuristic motivations and mathematical statements of Rolle's Theorem and the Mean Value Theorem. It also includes a proof of the Mean Value Theorem using Rolle's Theorem.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
The document provides steps for graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x. Step 1 analyzes the monotonicity of the function by examining the sign chart of its derivative f'(x). Step 2 analyzes concavity by examining the sign chart of the second derivative f''(x). Step 3 combines these analyses into a single sign chart summarizing the function's properties over its domain. The goal is to completely graph the function, indicating zeros, asymptotes, critical points, maxima/minima, and inflection points.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Optimization is a killer feature of the derivative. Not only do we often want to optimize some system, nature does as well. We give a procedure and many examples.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
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 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.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
- The document is a section from a calculus course at NYU that discusses using derivatives to determine the shapes of curves.
- It covers using the first derivative to determine if a function is increasing or decreasing over an interval using the Increasing/Decreasing Test. It also discusses using the second derivative to determine if a function is concave up or down over an interval using the Second Derivative Test.
- Examples are provided to demonstrate finding intervals of monotonicity for functions and classifying critical points as local maxima, minima or neither using the First Derivative Test.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document summarizes a calculus lecture on the Mean Value Theorem. It begins with announcements about exams and assignments. It then outlines the topics to be covered: Rolle's Theorem and the Mean Value Theorem, applications of the MVT, and why the MVT is important. It provides heuristic motivations and mathematical statements of Rolle's Theorem and the Mean Value Theorem. It also includes a proof of the Mean Value Theorem using Rolle's Theorem.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
The document provides steps for graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x. Step 1 analyzes the monotonicity of the function by examining the sign chart of its derivative f'(x). Step 2 analyzes concavity by examining the sign chart of the second derivative f''(x). Step 3 combines these analyses into a single sign chart summarizing the function's properties over its domain. The goal is to completely graph the function, indicating zeros, asymptotes, critical points, maxima/minima, and inflection points.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Optimization is a killer feature of the derivative. Not only do we often want to optimize some system, nature does as well. We give a procedure and many examples.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
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 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.
The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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Lesson 25: Areas and Distances; The Definite Integral
1. Section 5.1–5.2
Areas and Distances
The Definite Integral
V63.0121.034, Calculus I
November 30, 2009
Announcements
Quiz 5 this week in recitation on 4.1–4.4, 4.7
Final Exam, December 18, 2:00–3:50pm
. . . . . .
2. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
The definite integral as a limit
Properties of the integral
. . . . . .
3. Easy Areas: Rectangle
Definition
The area of a rectangle with dimensions ℓ and w is the product
A = ℓw.
w
.
.
.
ℓ
It may seem strange that this is a definition and not a theorem but
we have to start somewhere.
. . . . . .
4. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a
rectangle.
.
b
.
. . . . . .
5. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a
rectangle.
h
.
.
b
.
. . . . . .
6. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a
rectangle.
h
.
.
. . . . . .
7. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a
rectangle.
h
.
.
b
.
. . . . . .
8. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a
rectangle.
h
.
.
b
.
So
A = bh
. . . . . .
9. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a
parallelogram.
.
b
.
. . . . . .
10. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a
parallelogram.
h
.
.
b
.
. . . . . .
11. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a
parallelogram.
h
.
.
b
.
So
1
A= bh
2
. . . . . .
12. Easy Areas: Polygons
Any polygon can be triangulated, so its area can be found by
summing the areas of the triangles:
.
. . . . . .
23. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r)(1 + r + · · · + rn ) = 1 − rn+1
So
1 − rn+1
1 + r + · · · + rn =
1−r
. . . . . .
24. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r)(1 + r + · · · + rn ) = 1 − rn+1
So
1 − rn+1
1 + r + · · · + rn =
1−r
Therefore
1 1 1 1 − (1/4)n+1
1+ + + ··· + n =
4 16 4 1 − 1/4
. . . . . .
25. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r)(1 + r + · · · + rn ) = 1 − rn+1
So
1 − rn+1
1 + r + · · · + rn =
1−r
Therefore
1 1 1 1 − (1/4)n+1 1 4
1+ + + ··· + n = → =
4 16 4 1− 1/4 3/4 3
as n → ∞.
. . . . . .
26. Cavalieri
Italian,
1598–1647
Revisited
the area
problem
with a
different
perspective
. . . . . .
27. Cavalieri’s method
Divide up the interval into
pieces and measure the area
. = x2
y
of the inscribed rectangles:
. .
0
. 1
.
. . . . . .
28. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
. . .
0
. 1 1
.
.
2
. . . . . .
29. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
L3 =
. . . .
0
. 1 2 1
.
. .
3 3
. . . . . .
30. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
. . . .
0
. 1 2 1
.
. .
3 3
. . . . . .
31. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
L4 =
. . . . .
0
. 1 2 3 1
.
. . .
4 4 4
. . . . . .
32. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. . . . . 64 64 64 64
0
. 1 2 3 1
.
. . .
4 4 4
. . . . . .
33. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. . . . . . 64 64 64 64
0
. 1 2 3 4 1
. L5 =
. . . .
5 5 5 5
. . . . . .
35. Cavalieri’s method
Divide up the interval into
. = x2
y pieces and measure the area
of the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
. . 64 64 64 64
1 4 9 16 30
0
. 1
. L5 = + + + =
. 125 125 125 125 125
Ln =?
. . . . . .
36. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
. . . . . .
37. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
The rectangle over the ith interval and under the parabola has
area ( )
1 i − 1 2 (i − 1)2
· = .
n n n3
. . . . . .
38. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
The rectangle over the ith interval and under the parabola has
area ( )
1 i − 1 2 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
. . . . . .
39. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
The rectangle over the ith interval and under the parabola has
area ( )
1 i − 1 2 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
The Arabs knew that
n(n − 1)(2n − 1)
1 + 22 + 32 + · · · + (n − 1)2 =
6
So
n(n − 1)(2n − 1)
Ln =
6n3
. . . . . .
40. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
The rectangle over the ith interval and under the parabola has
area ( )
1 i − 1 2 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
The Arabs knew that
n(n − 1)(2n − 1)
1 + 22 + 32 + · · · + (n − 1)2 =
6
So
n(n − 1)(2n − 1) 1
Ln = →
6n3 3
as n → ∞. . . . . . .
42. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
. . . . . .
43. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3 3
1 + 2 + 3 + · · · + (n − 1 )
=
n4
. . . . . .
44. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3 3
1 + 2 + 3 + · · · + (n − 1 )
=
n4
The formula out of the hat is
[1 ]2
1 + 23 + 33 + · · · + (n − 1)3 = 2 n(n − 1)
. . . . . .
45. Cavalieri’s method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3 3
1 + 2 + 3 + · · · + (n − 1 )
=
n4
The formula out of the hat is
[1 ]2
1 + 23 + 33 + · · · + (n − 1)3 = 2 n(n − 1)
So
n2 (n − 1)2 1
Ln = →
4n4 4
as n → ∞.
. . . . . .
46. Cavalieri’s method with different heights
1 13 1 2 3 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
1 3 + 23 + 33 + · · · + n3
=
n4
1 [1 ]2
= 4 2 n (n + 1 )
n
n2 (n + 1)2 1
= 4
→
4n 4
.
as n → ∞.
. . . . . .
47. Cavalieri’s method with different heights
1 13 1 2 3 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
1 3 + 23 + 33 + · · · + n3
=
n4
1 [1 ]2
= 4 2 n (n + 1 )
n
n2 (n + 1)2 1
= 4
→
4n 4
.
as n → ∞.
So even though the rectangles overlap, we still get the same
answer.
. . . . . .
48. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
The definite integral as a limit
Properties of the integral
. . . . . .
49. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want
to find the area between x = a, x = b, y = 0, and y = f(x).
For each positive integer n, divide up the interval into n pieces.
b−a
Then ∆x = . For each i between 1 and n, let xi be the nth
n
step between a and b. So
x0 = a
b−a
x 1 = x 0 + ∆x = a +
n
b−a
x 2 = x 1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
x x x
.0 .1 .2 xx x
. i . n−1. n ······
. . . . . . . .
a
. b
. b−a
xn = a + n ·
. . .
=b . . .
51. Forming Riemann sums
We have many choices of how to approximate the area:
Ln = f(x0 )∆x + f(x1 )∆x + · · · + f(xn−1 )∆x
Rn = f(x1 )∆x + f(x2 )∆x + · · · + f(xn )∆x
( ) ( ) ( )
x0 + x 1 x 1 + x2 xn−1 + xn
Mn = f ∆x + f ∆x + · · · + f ∆x
2 2 2
In general, choose ci to be a point in the ith interval [xi−1 , xi ].
Form the Riemann sum
Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x
∑ n
= f(ci )∆x
i =1
. . . . . .
52. Theorem of the Day
Theorem
If f is a continuous function on [a, b] or has finitely many jump
discontinuities, then
lim Sn = lim {f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x}
n→∞ n→∞
exists and is the same value no matter what choice of ci we made.
. . . . . .
55. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
. . . . . .
56. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
Only know the slope of
lines
. . . . . .
57. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
Only know the area of
Only know the slope of
polygons
lines
. . . . . .
58. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
Only know the area of
Only know the slope of
polygons
lines
Approximate curve with
a line
. . . . . .
59. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
Only know the area of
Only know the slope of
polygons
lines
Approximate region
Approximate curve with
with polygons
a line
. . . . . .
60. Analogies
The Tangent Problem
The Area Problem (Ch. 5)
(Ch. 2–4)
Want the area of a
Want the slope of a
curved region
curve
Only know the area of
Only know the slope of
polygons
lines
Approximate region
Approximate curve with
with polygons
a line
Take limit over better
Take limit over better
and better
and better
approximations
approximations
. . . . . .
61. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
The definite integral as a limit
Properties of the integral
. . . . . .
62. Distances
Just like area = length × width, we have
distance = rate × time.
So here is another use for Riemann sums.
. . . . . .
65. Analysis
This method of measuring position by recording velocity is
known as dead reckoning.
If we had velocity estimates at finer intervals, we’d get better
estimates.
If we had velocity at every instant, a limit would tell us our
exact position relative to the last time we measured it.
. . . . . .
66. Other uses of Riemann sums
Anything with a product!
Area, volume
Anything with a density: Population, mass
Anything with a “speed:” distance, throughput, power
Consumer surplus
Expected value of a random variable
. . . . . .
67. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
The definite integral as a limit
Properties of the integral
. . . . . .
68. The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a
to b is the number
∫ b ∑n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i =1
. . . . . .
71. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
. . . . . .
72. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
. . . . . .
73. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
. . . . . .
74. Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of computing an integral is called integration or
quadrature
. . . . . .
75. The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite
∫ b
integral f(x) dx exists.
a
. . . . . .
76. The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite
∫ b
integral f(x) dx exists.
a
Theorem
If f is integrable on [a, b] then
∫ b n
∑
f(x) dx = lim f(xi )∆x,
a n→∞
i=1
where
b−a
∆x = and xi = a + i ∆x
n
. . . . . .
77. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
The definite integral as a limit
Properties of the integral
. . . . . .
78. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫ b
1. c dx = c(b − a)
a
. . . . . .
79. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
. . . . . .
80. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
∫ b ∫ b
3. cf(x) dx = c f(x) dx.
a a
. . . . . .
81. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
∫ b ∫ b
3. cf(x) dx = c f(x) dx.
a a
∫ b ∫ b ∫ b
4. [f(x) − g(x)] dx = f(x) dx − g(x) dx.
a a a
. . . . . .
84. More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
This allows us to have
∫ c ∫ b ∫ c
5. f(x) dx = f(x) dx + f(x) dx for all a, b, and c.
a a b
. . . . . .
85. Example
Suppose f and g are functions with
∫ 4
f(x) dx = 4
0
∫ 5
f(x) dx = 7
0
∫ 5
g(x) dx = 3.
0
Find
∫ 5
(a) [2f(x) − g(x)] dx
0
∫ 5
(b) f(x) dx.
4
. . . . . .