The document provides information about an upcoming calculus class, including announcements of an upcoming quiz and that class will be held on November 24. It then outlines the objectives and importance of curve sketching functions by analyzing properties like critical points, maxima/minima, and inflection points. The remainder of the document provides examples and steps for sketching curves of specific cubic and quartic functions through analyzing monotonicity using the first derivative and concavity using the second derivative.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions.
- There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24.
- The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.
The document is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers:
- Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative
- Examples of applying this process to functions like x2 - 1 and x2/3(x + 2)
- Definitions of increasing, decreasing, and concavity
- How the second derivative test can determine concavity by examining the sign of the second derivative
The document is about implicit differentiation and contains the following:
- It introduces implicit differentiation using the example of finding the slope of the curve x^2 + y^2 = 1 at the point (3/5, -4/5).
- It shows solving this problem explicitly by isolating y and taking the derivative, as well as implicitly by treating y as a function f(x) and differentiating the equation x^2 + f(x)^2 = 1.
- The objectives, outline, and motivation for implicit differentiation are provided to set up the key concepts covered in the section.
Linear vs. quadratic classifier power pointAlaa Tharwat
- The document discusses linear and quadratic discriminant classifiers, which are used to classify patterns into categories.
- Linear discriminant classifiers use linear decision boundaries, while quadratic discriminant classifiers use quadratic decision boundaries defined by quadratic functions.
- The document provides equations to calculate the discriminant functions and decision boundaries for linear and quadratic classifiers. It also gives an example to illustrate the classification process for three classes of data.
This chapter introduces matrices and their basic arithmetic operations. Matrices allow linear equations to be written in a compact matrix form. The key operations covered are:
1) Matrix addition and subtraction are performed element-wise.
2) A matrix can be multiplied by a scalar by multiplying each element.
3) Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second.
4) For matrix multiplication to have an inverse, the matrices must satisfy a condition on their elements.
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 provides information about an upcoming calculus class, including announcements of an upcoming quiz and that class will be held on November 24. It then outlines the objectives and importance of curve sketching functions by analyzing properties like critical points, maxima/minima, and inflection points. The remainder of the document provides examples and steps for sketching curves of specific cubic and quartic functions through analyzing monotonicity using the first derivative and concavity using the second derivative.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions.
- There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24.
- The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.
The document is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers:
- Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative
- Examples of applying this process to functions like x2 - 1 and x2/3(x + 2)
- Definitions of increasing, decreasing, and concavity
- How the second derivative test can determine concavity by examining the sign of the second derivative
The document is about implicit differentiation and contains the following:
- It introduces implicit differentiation using the example of finding the slope of the curve x^2 + y^2 = 1 at the point (3/5, -4/5).
- It shows solving this problem explicitly by isolating y and taking the derivative, as well as implicitly by treating y as a function f(x) and differentiating the equation x^2 + f(x)^2 = 1.
- The objectives, outline, and motivation for implicit differentiation are provided to set up the key concepts covered in the section.
Linear vs. quadratic classifier power pointAlaa Tharwat
- The document discusses linear and quadratic discriminant classifiers, which are used to classify patterns into categories.
- Linear discriminant classifiers use linear decision boundaries, while quadratic discriminant classifiers use quadratic decision boundaries defined by quadratic functions.
- The document provides equations to calculate the discriminant functions and decision boundaries for linear and quadratic classifiers. It also gives an example to illustrate the classification process for three classes of data.
This chapter introduces matrices and their basic arithmetic operations. Matrices allow linear equations to be written in a compact matrix form. The key operations covered are:
1) Matrix addition and subtraction are performed element-wise.
2) A matrix can be multiplied by a scalar by multiplying each element.
3) Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second.
4) For matrix multiplication to have an inverse, the matrices must satisfy a condition on their elements.
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 is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
You are looking for someone with strong communication skills, a hard work ethic, great time management abilities, who can motivate staff and is organized. However, this individual wants to be paid appropriately for their skills and cannot accept very low pay. Check out their portfolio for more details on their qualifications and experience.
These four markets - Atlanta, Austin, San Francisco, and Washington DC - are consistently in the top ten markets for online media activities measured since 2002. However, the reasons behind their high rankings vary - the markets have different demographic profiles and their residents exhibit different motivations and behaviors online. While they share some qualities, each local market has its own complex influences that shape how its population engages with digital media.
Natural disasters are increasing in intensity and frequency. Preparing for disasters requires understanding how risks differently impact men and women. Disaster preparedness is most effective when communities have inclusive plans and communication with government. However, women's roles in mitigation and preparation are often overlooked. To ensure gender-inclusive preparedness, experts recommend conducting gender analysis, including women in assessments and response coordination, and developing common assessment tools.
THE 4 R’S – REASON, REDCAP, REVIEW AND RESEARCH - IN A LARGE HEALTHCARE ORGAN...hiij
The organization acquired the REDCap platform to facilitate clinical review and research data collection in a more sustainable way than previous methods. They found REDCap was easily installed and maintained, with rapid uptake across various stakeholders. Data from the organization's clinical data environment was successfully integrated into REDCap using its Dynamic Data Pull functionality. In summary, acquiring and installing REDCap has been hugely successful, providing a great facility for a large number of organizational stakeholders going forward.
Dokumen tersebut membahas tentang penyediaan dan pemanfaatan ruang terbuka hijau (RTH) di kawasan perkotaan. Beberapa poin penting yang diangkat antara lain:
1) Proporsi RTH pada wilayah perkotaan minimal 30% yang terdiri atas 20% RTH publik dan 10% RTH privat
2) Penyediaan RTH berdasarkan luas wilayah dan jumlah penduduk dengan ketentuan minimal untuk setiap unit lingkungan
3) Pemanfaatan R
The document provides notes on curve sketching from a Calculus I class at New York University. It includes announcements about an upcoming homework assignment and class, objectives of learning how to graph functions, and notes on techniques for curve sketching such as using the increasing/decreasing test and concavity test to determine properties of a function's graph. Examples are provided of graphing cubic and quartic functions as well as functions with absolute values.
This document provides an example of graphing a cubic function step-by-step. It begins by finding the derivative to determine where the function is increasing and decreasing. The derivative is factored to yield a sign chart showing the function is decreasing on (-∞,-1) and (2,∞) and increasing on (-1,2). This information will be used to sketch the graph of the cubic function.
The document provides an overview of curve sketching techniques for functions. It discusses objectives like graphing functions completely and indicating key features. It then covers topics like the increasing/decreasing test, concavity test, a checklist for graphing functions, and examples of graphing cubic, quartic, and other types of functions. Examples are worked through step-by-step to demonstrate the process of analyzing monotonicity, concavity, and asymptotes to fully sketch the graph of various functions.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
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.
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 more good examples.
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Mel Anthony Pepito
Describe the monotonicity of f(x) = arctan(x).
The derivative of arctan(x) is f'(x) = 1/(1+x^2), which is always positive. Therefore, f(x) = arctan(x) is an increasing function on (-∞,∞).
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesMel Anthony Pepito
f is decreasing on (−∞, −4/5] and increasing on [−4/5, ∞).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 4.2 The Shapes of Curves November 16, 2010 13 / 32
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Matthew Leingang
1) The document discusses using derivatives to determine the shapes of curves, including intervals of increasing/decreasing behavior, local extrema, and concavity.
2) Key tests covered are the increasing/decreasing test, first derivative test, concavity test, and second derivative test. These allow determining monotonicity, local extrema, and concavity from the signs of the first and second derivatives.
3) Examples demonstrate applying these tests to find the intervals of monotonicity, points of local extrema, and intervals of concavity for various functions.
Lesson 11: Implicit Differentiation (Section 41 slides)Mel Anthony Pepito
This document provides an overview of implicit differentiation. It begins with a motivating example of finding the slope of the tangent line to the curve x^2 + y^2 = 1 at the point (3/5, -4/5). It is shown that while y is not explicitly defined as a function of x for this curve, it can be treated as such locally using implicit differentiation. The key steps are to take the derivative of the equation with respect to x, which introduces a term for dy/dx, and then solve for dy/dx. This reveals that implicit differentiation allows the derivative of implicitly defined functions to be found.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
You are looking for someone with strong communication skills, a hard work ethic, great time management abilities, who can motivate staff and is organized. However, this individual wants to be paid appropriately for their skills and cannot accept very low pay. Check out their portfolio for more details on their qualifications and experience.
These four markets - Atlanta, Austin, San Francisco, and Washington DC - are consistently in the top ten markets for online media activities measured since 2002. However, the reasons behind their high rankings vary - the markets have different demographic profiles and their residents exhibit different motivations and behaviors online. While they share some qualities, each local market has its own complex influences that shape how its population engages with digital media.
Natural disasters are increasing in intensity and frequency. Preparing for disasters requires understanding how risks differently impact men and women. Disaster preparedness is most effective when communities have inclusive plans and communication with government. However, women's roles in mitigation and preparation are often overlooked. To ensure gender-inclusive preparedness, experts recommend conducting gender analysis, including women in assessments and response coordination, and developing common assessment tools.
THE 4 R’S – REASON, REDCAP, REVIEW AND RESEARCH - IN A LARGE HEALTHCARE ORGAN...hiij
The organization acquired the REDCap platform to facilitate clinical review and research data collection in a more sustainable way than previous methods. They found REDCap was easily installed and maintained, with rapid uptake across various stakeholders. Data from the organization's clinical data environment was successfully integrated into REDCap using its Dynamic Data Pull functionality. In summary, acquiring and installing REDCap has been hugely successful, providing a great facility for a large number of organizational stakeholders going forward.
Dokumen tersebut membahas tentang penyediaan dan pemanfaatan ruang terbuka hijau (RTH) di kawasan perkotaan. Beberapa poin penting yang diangkat antara lain:
1) Proporsi RTH pada wilayah perkotaan minimal 30% yang terdiri atas 20% RTH publik dan 10% RTH privat
2) Penyediaan RTH berdasarkan luas wilayah dan jumlah penduduk dengan ketentuan minimal untuk setiap unit lingkungan
3) Pemanfaatan R
The document provides notes on curve sketching from a Calculus I class at New York University. It includes announcements about an upcoming homework assignment and class, objectives of learning how to graph functions, and notes on techniques for curve sketching such as using the increasing/decreasing test and concavity test to determine properties of a function's graph. Examples are provided of graphing cubic and quartic functions as well as functions with absolute values.
This document provides an example of graphing a cubic function step-by-step. It begins by finding the derivative to determine where the function is increasing and decreasing. The derivative is factored to yield a sign chart showing the function is decreasing on (-∞,-1) and (2,∞) and increasing on (-1,2). This information will be used to sketch the graph of the cubic function.
The document provides an overview of curve sketching techniques for functions. It discusses objectives like graphing functions completely and indicating key features. It then covers topics like the increasing/decreasing test, concavity test, a checklist for graphing functions, and examples of graphing cubic, quartic, and other types of functions. Examples are worked through step-by-step to demonstrate the process of analyzing monotonicity, concavity, and asymptotes to fully sketch the graph of various functions.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
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.
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 more good examples.
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Mel Anthony Pepito
Describe the monotonicity of f(x) = arctan(x).
The derivative of arctan(x) is f'(x) = 1/(1+x^2), which is always positive. Therefore, f(x) = arctan(x) is an increasing function on (-∞,∞).
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesMel Anthony Pepito
f is decreasing on (−∞, −4/5] and increasing on [−4/5, ∞).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 4.2 The Shapes of Curves November 16, 2010 13 / 32
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Matthew Leingang
1) The document discusses using derivatives to determine the shapes of curves, including intervals of increasing/decreasing behavior, local extrema, and concavity.
2) Key tests covered are the increasing/decreasing test, first derivative test, concavity test, and second derivative test. These allow determining monotonicity, local extrema, and concavity from the signs of the first and second derivatives.
3) Examples demonstrate applying these tests to find the intervals of monotonicity, points of local extrema, and intervals of concavity for various functions.
Lesson 11: Implicit Differentiation (Section 41 slides)Mel Anthony Pepito
This document provides an overview of implicit differentiation. It begins with a motivating example of finding the slope of the tangent line to the curve x^2 + y^2 = 1 at the point (3/5, -4/5). It is shown that while y is not explicitly defined as a function of x for this curve, it can be treated as such locally using implicit differentiation. The key steps are to take the derivative of the equation with respect to x, which introduces a term for dy/dx, and then solve for dy/dx. This reveals that implicit differentiation allows the derivative of implicitly defined functions to be found.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Mel Anthony Pepito
The document summarizes key points from Section 5.3 of a calculus class at New York University on evaluating definite integrals. It outlines the objectives of the section, provides an example of using the midpoint rule to estimate a definite integral, and reviews properties and comparison properties of definite integrals, including how to interpret definite integrals as the net change of a function over an interval.
This document outlines a calculus lecture on evaluating definite integrals. The lecture will:
1) Use the Evaluation Theorem to evaluate definite integrals and write antiderivatives as indefinite integrals.
2) Interpret definite integrals as the "net change" of a function over an interval.
3) Provide examples of evaluating definite integrals and estimating integrals using the midpoint rule.
4) Discuss properties of integrals such as additivity and illustrate how a definite integral from a to c can be broken into integrals from a to b and b to c.
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.
Similar to Lesson 21: Curve Sketching (Section 021 handout) (20)
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
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 use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
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. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
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.
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 include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
1. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Notes
Section 4.4
Curve Sketching
V63.0121.021, Calculus I
New York University
November 18, 2010
Announcements
There is class on November 23. The homework is due on
November 24. Turn in homework to my mailbox or bring to class on
November 23.
Announcements
Notes
There is class on
November 23. The
homework is due on
November 24. Turn in
homework to my mailbox or
bring to class on
November 23.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 2 / 55
Objectives
Notes
given a function, graph it
completely, indicating
zeroes (if easy)
asymptotes if applicable
critical points
local/global max/min
inflection points
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 3 / 55
1
2. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Why?
Notes
Graphing functions is like
dissection . . . or diagramming
sentences
You can really know a lot about
a function when you know all of
its anatomy.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 4 / 55
The Increasing/Decreasing Test
Notes
Theorem (The Increasing/Decreasing Test)
If f > 0 on (a, b), then f is increasing on (a, b). If f < 0 on (a, b), then
f is decreasing on (a, b).
Example
Here f (x) = x 3 + x 2 , and f (x) = 3x 2 + 2x.
f (x)
f (x)
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 5 / 55
Testing for Concavity
Notes
Theorem (Concavity Test)
If f (x) > 0 for all x in (a, b), then the graph of f is concave upward on
(a, b) If f (x) < 0 for all x in (a, b), then the graph of f is concave
downward on (a, b).
Example
Here f (x) = x 3 + x 2 , f (x) = 3x 2 + 2x, and f (x) = 6x + 2.
f (x) f (x)
f (x)
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 6 / 55
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3. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Graphing Checklist
Notes
To graph a function f , follow this plan:
0. Find when f is positive, negative, zero, not
defined.
1. Find f and form its sign chart. Conclude
information about increasing/decreasing
and local max/min.
2. Find f and form its sign chart. Conclude
concave up/concave down and inflection.
3. Put together a big chart to assemble
monotonicity and concavity data
4. Graph!
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 7 / 55
Outline
Notes
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 8 / 55
Graphing a cubic
Notes
Example
Graph f (x) = 2x 3 − 3x 2 − 12x.
(Step 0) First, let’s find the zeros. We can at least factor out one power of
x:
f (x) = x(2x 2 − 3x − 12)
so f (0) = 0. The other factor is a quadratic, so we the other two roots are
√
3 ± 32 − 4(2)(−12) 3 ± 105
x= =
4 4
It’s OK to skip this step for now since the roots are so complicated.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 9 / 55
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4. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 1: Monotonicity
Notes
f (x) = 2x 3 − 3x 2 − 12x
=⇒ f (x) = 6x 2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− − +
x −2
2
− + +
x +1
−1
+ − + f (x)
−1 2 f (x)
max min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 10 / 55
Step 2: Concavity
Notes
f (x) = 6x 2 − 6x − 12
=⇒ f (x) = 12x − 6 = 6(2x − 1)
Another sign chart:
−− ++ f (x)
1/2 f (x)
IP
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 11 / 55
Step 3: One sign chart to rule them all
Notes
Remember, f (x) = 2x 3 − 3x 2 − 12x.
+ − − + f (x)
−1 2 monotonicity
−− −− ++ ++ f (x)
1/2 concavity
7 −61/2 −20 f (x)
−1 1/2 2 shape of f
max IP min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 12 / 55
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5. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Combinations of monotonicity and concavity
Notes
increasing, decreasing,
concave concave
down down
II I
III IV
decreasing, increasing,
concave up concave up
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 13 / 55
Step 3: One sign chart to rule them all
Notes
Remember, f (x) = 2x 3 − 3x 2 − 12x.
+ − − + f (x)
−1 2 monotonicity
−− −− ++ ++ f (x)
1/2 concavity
7 −61/2 −20 f (x)
−1 1/2 2 shape of f
max IP min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 14 / 55
Step 4: Graph
f (x) Notes
f (x) = 2x 3 − 3x 2 − 12x
√
(−1, 7)
3− 105
4 ,0 (0, 0)
√
x
(1/2, −61/2) 3+ 105
4 ,0
(2, −20)
7 −61/2 −20 f (x)
−1 1/2 2 shape of f
max IP min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 15 / 55
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6. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Graphing a quartic
Notes
Example
Graph f (x) = x 4 − 4x 3 + 10
(Step 0) We know f (0) = 10 and lim f (x) = +∞. Not too many other
x→±∞
points on the graph are evident.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 16 / 55
Step 1: Monotonicity
Notes
f (x) = x 4 − 4x 3 + 10
=⇒ f (x) = 4x 3 − 12x 2 = 4x 2 (x − 3)
We make its sign chart.
+ 0 + +
4x 2
0
− − 0 +
(x − 3)
3
− 0 − 0 + f (x)
0 3 f (x)
min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 17 / 55
Step 2: Concavity
Notes
f (x) = 4x 3 − 12x 2
=⇒ f (x) = 12x 2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0 + +
12x
0
− − 0 +
x −2
2
++ 0 −− 0 ++ f (x)
0 2 f (x)
IP IP
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 18 / 55
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7. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 3: Grand Unified Sign Chart
Notes
Remember, f (x) = x 4 − 4x 3 + 10.
− 0 − − 0 + f (x)
0 3 monotonicity
++ 0 −− 0 ++ ++ f (x)
0 2 concavity
10 −6 −17 f (x)
0 2 3 shape
IP IP min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 19 / 55
Step 4: Graph
y Notes
f (x) = x 4 − 4x 3 + 10
(0, 10)
x
(2, −6)
(3, −17)
10 −6 −17 f (x)
0 2 3 shape
IP IP min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 20 / 55
Outline
Notes
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 21 / 55
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8. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Graphing a function with a cusp
Notes
Example
Graph f (x) = x + |x|
This function looks strange because of the absolute value. But whenever
we become nervous, we can just take cases.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 22 / 55
Step 0: Finding Zeroes
Notes
f (x) = x + |x|
First, look at f by itself. We can tell that f (0) = 0 and that f (x) > 0
if x is positive.
Are there negative numbers which are zeroes for f ?
√
x + −x = 0
√
−x = −x
−x = x 2
2
x +x =0
The only solutions are x = 0 and x = −1.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 23 / 55
Step 0: Asymptotic behavior
Notes
f (x) = x + |x|
lim f (x) = ∞, because both terms tend to ∞.
x→∞
lim f (x) is indeterminate of the form −∞ + ∞. It’s the same as
x→−∞ √
lim (−y + y )
y →+∞
√
√ √ y +y
lim (−y + y ) = lim ( y − y ) · √
y →+∞ y →∞ y +y
y − y2
= lim √ = −∞
y →∞ y +y
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 24 / 55
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9. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 1: The derivative
Notes
Remember, f (x) = x + |x|.
To find f , first assume x > 0. Then
d √ 1
f (x) = x + x =1+ √
dx 2 x
Notice
f (x) > 0 when x > 0 (so no critical points here)
lim f (x) = ∞ (so 0 is a critical point)
x→0+
lim f (x) = 1 (so the graph is asymptotic to a line of slope 1)
x→∞
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 25 / 55
Step 1: The derivative
Notes
Remember, f (x) = x + |x|.
If x is negative, we have
d √ 1
f (x) = x + −x = 1 − √
dx 2 −x
Notice
lim f (x) = −∞ (other side of the critical point)
x→0−
lim f (x) = 1 (asymptotic to a line of slope 1)
x→−∞
f (x) = 0 when
1 √ 1 1 1
1− √ = 0 =⇒ −x = =⇒ −x = =⇒ x = −
2 −x 2 4 4
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 26 / 55
Step 1: Monotonicity
Notes
1
1 + √
if x > 0
f (x) = 2 x
1 − √1
if x < 0
2 −x
We can’t make a multi-factor sign chart because of the absolute value,
but we can test points in between critical points.
+ 0 − ∞ + f (x)
−1 0 f (x)
4
max min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 27 / 55
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10. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 2: Concavity
Notes
If x > 0, then
d 1 1
f (x) = 1 + x −1/2 = − x −3/2
dx 2 4
This is negative whenever x > 0.
If x < 0, then
d 1 1
f (x) = 1 − (−x)−1/2 = − (−x)−3/2
dx 2 4
which is also always negative for negative x.
1
In other words, f (x) = − |x|−3/2 .
4
Here is the sign chart:
−− −∞ −− f (x)
0 f (x)
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 28 / 55
Step 3: Synthesis
Notes
Now we can put these things together.
f (x) = x + |x|
+1 + 0 − ∞ + +1 (x)
f
−1 0 monotonicity
4
−∞ −− −−−∞ −− −∞ (x)
f
0 concavity
1
−∞ 0 4 0 +∞(x)
f
−1 −1 0 shape
4
zero max min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 29 / 55
Graph
Notes
f (x) = x + |x|
f (x)
(− 1 , 1 )
4 4
(−1, 0)
x
(0, 0)
1
−∞ 0 4 0 +∞ f (x)
−1 −1 0 shape
4
zero max min
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 30 / 55
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11. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Example with Horizontal Asymptotes
Notes
Example
2
Graph f (x) = xe −x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 31 / 55
Step 1: Monotonicity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 32 / 55
Step 2: Concavity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 33 / 55
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12. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 3: Synthesis
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 34 / 55
Step 4: Graph
Notes
f (x)
x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 35 / 55
Example with Vertical Asymptotes
Notes
Example
1 1
Graph f (x) = + 2
x x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 36 / 55
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13. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 0
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 37 / 55
Step 1: Monotonicity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 39 / 55
Step 2: Concavity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 40 / 55
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14. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 3: Synthesis
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 41 / 55
Step 4: Graph
Notes
y
x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 42 / 55
Trigonometric and polynomial together
Notes
Problem
Graph f (x) = cos x − x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 43 / 55
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15. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 0: intercepts and asymptotes
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 44 / 55
Step 1: Monotonicity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 45 / 55
Step 2: Concavity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 46 / 55
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16. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 3: Synthesis
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 47 / 55
Step 4: Graph
Notes
f (x) = cos x − x
y
x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 48 / 55
Logarithmic
Notes
Problem
Graph f (x) = x ln x 2
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 49 / 55
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17. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 0: Intercepts and Asymptotes
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 50 / 55
Step 1: Monotonicity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 51 / 55
Step 2: Concavity
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 52 / 55
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18. V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010
Step 3: Synthesis
Notes
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 53 / 55
Step 4: Graph
Notes
y
x
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 54 / 55
Summary
Notes
Graphing is a procedure that gets easier with practice.
Remember to follow the checklist.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 2010 55 / 55
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