The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
Lesson 26: The Fundamental Theorem of Calculus (Section 4 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
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.
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 4 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
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.
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
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.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
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.
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
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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
This document provides an introduction to basic definite integration. It defines definite integration as calculating the area under a curve between two limits using antiderivatives. It demonstrates how to calculate definite integrals of simple functions and interpret the results as areas. It also discusses how the sign of the integral depends on whether the function lies above or below the x-axis. Quizzes are included to assess understanding of these concepts.
The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
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.
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, 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 anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
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
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
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.
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
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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
This document provides an introduction to basic definite integration. It defines definite integration as calculating the area under a curve between two limits using antiderivatives. It demonstrates how to calculate definite integrals of simple functions and interpret the results as areas. It also discusses how the sign of the integral depends on whether the function lies above or below the x-axis. Quizzes are included to assess understanding of these concepts.
The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
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.
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, 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 anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
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 27: Integration by Substitution (Section 4 version)Matthew Leingang
The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
Lesson 27: Integration by Substitution (Section 10 version)Matthew Leingang
The method of substitution is the chain rule in reverse. At first it looks magical, then logical, and then you realize there's an art to choosing the right substitution. We try to demystify with many worked-out examples.
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 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 summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
Bai giang ham so kha vi va vi phan cua ham nhieu bienNhan Nguyen
This document introduces differentials in functions of several variables. It begins with a review of differentials in two variables using differentials dx and dy. It then extends the concept to functions of several variables, where the total differential dz is defined as the sum of its partial derivatives with respect to each variable times the differentials of those variables. Examples are provided to demonstrate calculating total differentials and comparing them to actual changes. The relationship between differentiability and continuity is also discussed.
Lesson 19: Double Integrals over General RegionsMatthew Leingang
The document is notes from a math class on double integrals over general regions. It includes announcements about office hours and problem sessions. It defines double integrals over general regions as limits of integrals over unions of rectangles approximating the region. It discusses properties of double integrals and iterated integrals over curved regions of Type I and Type II. It provides examples and worksheets for students to practice evaluating double integrals.
The document discusses several key topics:
1) The First Fundamental Theorem of Calculus, which states that if f is continuous on [a,b] and F is an antiderivative of f, then the integral of f from a to x is equal to F(x) - F(a).
2) Examples of differentiating functions defined by integrals, including area functions and the error function (Erf).
3) The Second Fundamental Theorem of Calculus (weak form), which relates the integral of a continuous function f to antiderivatives F of f, stating that the integral of f from a to b is equal to F(b) - F(a).
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through elimination, graphical, and symbolic approaches.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the theorem on Lagrange multipliers and examples of its application to problems with more than two variables or multiple constraints.
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through symbolic, graphical, and other perspectives.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the definition of compact sets and stating the compact set method for finding extreme values of a function over a compact domain.
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.
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 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 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 Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
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.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
2. 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
∆x→0
a i=1
. . . . . .
4. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
. . . . . .
5. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If v(t) represents the velocity of a particle moving rectilinearly,
then ∫ t1
v(t) dt = s(t1 ) − s(t0 ).
t0
. . . . . .
6. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If MC(x) represents the marginal cost of making x units of a
product, then
∫x
C(x) = C(0) + MC(q) dq.
0
. . . . . .
7. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If ρ(x) represents the density of a thin rod at a distance of x from
its end, then the mass of the rod up to x is
∫x
m(x) = ρ(s) ds.
0
. . . . . .
8. My first table of integrals
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
∫ ∫
∫
xn+1
xn dx = cf(x) dx = c f(x) dx
+ C (n ̸= −1)
n+1 ∫
∫
1
ex dx = ex + C dx = ln |x| + C
x
∫
∫
ax
ax dx = +C
sin x dx = − cos x + C
ln a
∫
∫
csc2 x dx = − cot x + C
cos x dx = sin x + C
∫
∫
sec2 x dx = tan x + C csc x cot x dx = − csc x + C
∫
∫
1
√ dx = arcsin x + C
sec x tan x dx = sec x + C
1 − x2
∫
1
dx = arctan x + C
1 + x2
. . . . . .
9. Outline
My first table of integrals
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
Facts about g from f
A problem
. . . . . .
10. An area function
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Can we evaluate the
0
integral in g(x)?
.
x
.
0
.
. . . . . .
11. An area function
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Can we evaluate the
0
integral in g(x)?
Dividing the interval [0, x] into n pieces
x ix
gives ∆x = and xi = 0 + i∆x = .
n n
So
x x3 x (2x)3 x (nx)3
· 3 + · 3 + ··· + · 3
Rn =
nn n n n n
4( )
x
= 4 1 3 + 2 3 + 3 3 + · · · + n3
n
x4 [ 1 ]2
= 4 2 n(n + 1)
.
n
x
.
0
.
x4 n2 (n + 1)2 x4
→
=
4n4 4
as n → ∞.
. . . . . .
14. The area function
Let f be a function which is integrable (i.e., continuous or with
finitely many jump discontinuities) on [a, b]. Define
∫x
g(x) = f(t) dt.
a
When is g increasing?
. . . . . .
15. The area function
Let f be a function which is integrable (i.e., continuous or with
finitely many jump discontinuities) on [a, b]. Define
∫x
g(x) = f(t) dt.
a
When is g increasing?
When is g decreasing?
. . . . . .
16. The area function
Let f be a function which is integrable (i.e., continuous or with
finitely many jump discontinuities) on [a, b]. Define
∫x
g(x) = f(t) dt.
a
When is g increasing?
When is g decreasing?
Over a small interval, what’s the average rate of change of g?
. . . . . .
18. Proof.
Let h > 0 be given so that x + h < b. We have
g(x + h) − g(x)
=
h
. . . . . .
19. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
. . . . . .
20. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
Let Mh be the maximum value of f on [x, x + h], and mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
f(t) dt
x
. . . . . .
21. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
Let Mh be the maximum value of f on [x, x + h], and mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
f(t) dt ≤ Mh · h
x
. . . . . .
22. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
Let Mh be the maximum value of f on [x, x + h], and mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
. . . . . .
23. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
Let Mh be the maximum value of f on [x, x + h], and mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
So
g(x + h) − g(x)
mh ≤ ≤ Mh .
h
. . . . . .
24. Proof.
Let h > 0 be given so that x + h < b. We have
∫ x+h
g(x + h) − g(x) 1
f(t) dt.
=
h h x
Let Mh be the maximum value of f on [x, x + h], and mh the
minimum value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
So
g(x + h) − g(x)
mh ≤ ≤ Mh .
h
As h → 0, both mh and Mh tend to f(x). Zappa-dappa.
. . . . . .
25. Meet the Mathematician: James Gregory
Scottish, 1638-1675
Astronomer and
Geometer
Conceived
transcendental numbers
and found evidence that
π was transcendental
Proved a geometric
version of 1FTC as a
lemma but didn’t take it
further
. . . . . .
26. Meet the Mathematician: Isaac Barrow
English, 1630-1677
Professor of Greek,
theology, and
mathematics at
Cambridge
Had a famous student
. . . . . .
27. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
. . . . . .
28. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily
disgraced by the
calculus priority dispute
. . . . . .
30. Differentiation and Integration as reverse processes
Putting together 1FTC and 2FTC, we get a beautiful relationship
between the two fundamental concepts in calculus.
∫ x
d
f(t) dt = f(x)
dx a
∫ b
F′ (x) dx = F(b) − F(a).
a
. . . . . .
31. Outline
My first table of integrals
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
Facts about g from f
A problem
. . . . . .
32. Differentiation of area functions
Example ∫
x
t3 dt. We know g′ (x) = x3 . What if instead we had
Let g(x) =
0
∫ 3x
t3 dt.
h(x) =
0
What is h′ (x)?
. . . . . .
33. Differentiation of area functions
Example ∫
x
t3 dt. We know g′ (x) = x3 . What if instead we had
Let g(x) =
0
∫ 3x
t3 dt.
h(x) =
0
What is h′ (x)?
Solution ∫ u
t3 dt
We can think of h as the composition g k, where g(u) =
◦
0
and k(x) = 3x. Then
h′ (x) = g′ (k(x))k′ (x) = 3(k(x))3 = 3(3x)3 = 81x3 .
. . . . . .
34. Example
∫ sin2 x
(17t2 + 4t − 4) dt. What is h′ (x)?
Let h(x) =
0
. . . . . .
35. Example
∫ sin2 x
(17t2 + 4t − 4) dt. What is h′ (x)?
Let h(x) =
0
Solution
We have
∫ sin2 x
d
(17t2 + 4t − 4) dt
dx 0
( )d
= 17(sin2 x)2 + 4(sin2 x) − 4 · sin2 x
dx
( )
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
. . . . . .
36. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
. . . . . .
37. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
It turns out erf is the shape of the bell curve.
. . . . . .
38. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative.
erf′ (x) =
. . . . . .
39. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative.
2 2
erf′ (x) = √ e−x .
π
. . . . . .
40. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative.
2 2
erf′ (x) = √ e−x .
π
Example
d
erf(x2 ).
Find
dx
. . . . . .
41. Erf
Here’s a function with a funny name but an important role:
∫x
2 2
e−t dt.
√
erf(x) =
π0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative.
2 2
erf′ (x) = √ e−x .
π
Example
d
erf(x2 ).
Find
dx
Solution
By the chain rule we have
d d 2 4
22 4
erf(x2 ) = erf′ (x2 ) x2 = √ e−(x ) 2x = √ xe−x .
dx dx π π
. . . . . .
42. Other functions defined by integrals
The future value of an asset:
∫∞
π(τ )e−rτ dτ
FV(t) =
t
where π(τ ) is the profitability at time τ and r is the discount
rate.
The consumer surplus of a good:
∫ q∗
CS(q∗ ) = (f(q) − p∗ ) dq
0
where f(q) is the demand function and p∗ and q∗ the
equilibrium price and quantity.
. . . . . .
43. Outline
My first table of integrals
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
Facts about g from f
A problem
. . . . . .
44. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
.(
• .3,3)
3
.
.( .(
• .2,2) • .5,2)
2
.
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
45. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
What is the particle’s velocity
.(
• .3,3)
3
. at time t = 5?
.( .(
• .2,2) • .5,2)
2
.
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
46. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
What is the particle’s velocity
.(
• .3,3)
3
. at time t = 5?
.( .(
• .2,2) • .5,2)
2
.
Solution
. .1,1)
(
1
. •
Recall that by the FTC we
. ... . . . . . . have
4
. 7
.
123
... 5
. 8
. 9
.
6
.
s′ (t) = f(t).
So s′ (5) = f(5) = 2.
. . . . . .
47. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
Is the acceleration of the par-
.(
• .3,3)
3
. ticle at time t = 5 positive or
.( .(
• .2,2) • .5,2)
2
. negative?
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
48. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
Is the acceleration of the par-
.(
• .3,3)
3
. ticle at time t = 5 positive or
.( .(
• .2,2) • .5,2)
2
. negative?
. .1,1)
(
1
. •
Solution
. ... . . . . . . We have s′′ (5) = f′ (5), which
4
. 7
.
123
... 5
. 8
. 9
.
6
.
looks negative from the
graph.
. . . . . .
49. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
What is the particle’s position
.(
• .3,3)
3
. at time t = 3?
.( .(
• .2,2) • .5,2)
2
.
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
50. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
What is the particle’s position
.(
• .3,3)
3
. at time t = 3?
.( .(
• .2,2) • .5,2)
2
.
Solution
. .1,1)
(
1
. •
Since on [0, 3], f(x) = x, we
. ... . . . . . . have
4
. 7
.
123
... 5
. 8
. 9
.
6
.
∫3
9
s(3) = x dx = .
2
0
. . . . . .
51. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
At what time during the first 9
.(
• .3,3)
3
. seconds does s have its largest
.( .(
• .2,2) • .5,2)
2
. value?
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
52. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
At what time during the first 9
.(
• .3,3)
3
. seconds does s have its largest
.( .(
• .2,2) • .5,2)
2
. value?
. .1,1)
(
1
. •
Solution
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
53. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
At what time during the first 9
.(
• .3,3)
3
. seconds does s have its largest
.( .(
• .2,2) • .5,2)
2
. value?
. .1,1)
(
1
. •
Solution
. ... . . . . . . The critical points of s are the
4
. 7
.
123
... 5
. 8
. 9
.
6
.
zeros of s′ = f.
. . . . . .
54. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
At what time during the first 9
.(
• .3,3)
3
. seconds does s have its largest
.( .(
• .2,2) • .5,2)
2
. value?
. .1,1)
(
1
. •
Solution
. ... . . . . . . By looking at the graph, we
4
. 7
.
123
... 5
. 8
. 9
.
6
.
see that f is positive from
t = 0 to t = 6, then negative
from t = 6 to t = 9.
. . . . . .
55. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
At what time during the first 9
.(
• .3,3)
3
. seconds does s have its largest
.( .(
• .2,2) • .5,2)
2
. value?
. .1,1)
(
1
. •
Solution
. ... . . . . . . Therefore s is increasing on
4
. 7
.
123
... 5
. 8
. 9
.
6
.
[0, 6], then decreasing on
[6, 9]. So its largest value is at
t = 6.
. . . . . .
56. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
Approximately when is the
.(
• .3,3)
3
. acceleration zero?
.( .(
• .2,2) • .5,2)
2
.
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
57. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
Approximately when is the
.(
• .3,3)
3
. acceleration zero?
.( .(
• .2,2) • .5,2)
2
.
Solution
. .1,1)
(
1
. s′′ = 0 when f′ = 0, which
•
. ... . . . . . . happens at t = 4 and t = 7.5
4
. 7
.
123
... 5
. 8
. 9
.
6
.
(approximately)
. . . . . .
58. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
When is the particle moving
.(
• .3,3)
3
. toward the origin? Away from
.( .(
• .2,2) • .5,2)
2
. the origin?
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
59. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
When is the particle moving
.(
• .3,3)
3
. toward the origin? Away from
.( .(
• .2,2) • .5,2)
2
. the origin?
. .1,1)
(
1
. •
Solution
. ... . . . . . . The particle is moving away
4
. 7
.
123
... 5
. 8
. 9
.
6
.
from the origin when s > 0
and s′ > 0.
. . . . . .
60. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
When is the particle moving
.(
• .3,3)
3
. toward the origin? Away from
.( .(
• .2,2) • .5,2)
2
. the origin?
. .1,1)
(
1
. •
Solution
. ... . . . . . . Since s(0) = 0 and s′ > 0 on
4
. 7
.
123
... 5
. 8
. 9
.
6
.
(0, 6), we know the particle
is moving away from the
origin then.
. . . . . .
61. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
When is the particle moving
.(
• .3,3)
3
. toward the origin? Away from
.( .(
• .2,2) • .5,2)
2
. the origin?
. .1,1)
(
1
. •
Solution
. ... . . . . . . After t = 6, s′ < 0, so the
4
. 7
.
123
... 5
. 8
. 9
.
6
.
particle is moving toward the
origin.
. . . . . .
62. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
On which side (positive or
.(
• .3,3)
3
. negative) of the origin does
.( .(
• .2,2) • .5,2)
2
. the particle lie at time t = 9?
. .1,1)
(
1
. •
. ... . . . . . .
4
. 7
.
123
... 5
. 8
. 9
.
6
.
. . . . . .
63. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
On which side (positive or
.(
• .3,3)
3
. negative) of the origin does
.( .(
• .2,2) • .5,2)
2
. the particle lie at time t = 9?
. .1,1)
(
1
. •
Solution
. ... . . . . . . We have s(9) =
∫6 ∫9
4
. 7
.
123
... 5
. 8
. 9
.
6
.
f(x) dx + f(x) dx,
0 6
where the left integral is
positive and the right integral
is negative.
. . . . . .
64. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
On which side (positive or
.(
• .3,3)
3
. negative) of the origin does
.( .(
• .2,2) • .5,2)
2
. the particle lie at time t = 9?
. .1,1)
(
1
. •
Solution
. ... . . . . . . In order to decide whether
4
. 7
.
123
... 5
. 8
. 9
.
6
.
s(9) is positive or negative,
we need to decide if the first
area is more positive than the
second area is negative.
. . . . . .
65. Facts about g from f
Let f be the function whose graph is given below.
Suppose the the position at time t seconds of a particle moving
∫t
along a coordinate axis is s(t) = f(x) dx meters. Use the graph
0
to answer the following questions.
4
.
On which side (positive or
.(
• .3,3)
3
. negative) of the origin does
.( .(
• .2,2) • .5,2)
2
. the particle lie at time t = 9?
. .1,1)
(
1
. •
Solution
. ... . . . . . . This appears to be the case,
4
. 7
.
123
... 5
. 8
. 9
.
6
.
so s(9) is positive.
. . . . . .