A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
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 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
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.
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 discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
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.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
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 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
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.
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 discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
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.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document provides an overview of Math 5045: Advanced Algebra I (Module Theory). It begins with definitions of rings, including commutative and non-commutative rings, rings with unity, units, zero-divisors, integral domains, and fields. It then discusses modules, including left and right modules, submodules, module homomorphisms, kernels, images, and the module HomR(M,N). It also covers quotient modules and the composition of module homomorphisms.
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.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and describes special types of discontinuities. The IVT states that if a function is continuous on a closed interval and takes on intermediate values, there exists a number in the interval where the function value is equal to the intermediate value. Examples are provided to illustrate the IVT, including proving the existence of the square root of two.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document provides an overview of Math 5045: Advanced Algebra I (Module Theory). It begins with definitions of rings, including commutative and non-commutative rings, rings with unity, units, zero-divisors, integral domains, and fields. It then discusses modules, including left and right modules, submodules, module homomorphisms, kernels, images, and the module HomR(M,N). It also covers quotient modules and the composition of module homomorphisms.
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.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and describes special types of discontinuities. The IVT states that if a function is continuous on a closed interval and takes on intermediate values, there exists a number in the interval where the function value is equal to the intermediate value. Examples are provided to illustrate the IVT, including proving the existence of the square root of two.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand the derivative of constants, the constant multiple rule, the sum rule, the difference rule, and derivatives of sine and cosine. Examples are provided to find the derivatives of squaring, cubing, square root, and cube root functions using the definition of the derivative. Graphs and properties of these functions and their derivatives are also discussed.
This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.
This document contains lecture notes on continuity from a Calculus I class at New York University. It begins with announcements about office hours and homework grades. It then reviews the definition of a limit and introduces the definition of continuity as a function having a limit equal to its value at a point. Examples are provided to demonstrate showing a function is continuous. The document states that polynomials, rational functions, and trigonometric functions are continuous based on their definitions and limit properties. It concludes by explaining the continuity of inverse trigonometric functions.
The document discusses limits, continuity, and related concepts. Some key points:
1) It defines the concept of a limit and explains how to evaluate one-sided and two-sided limits. A limit exists only if the left and right-sided limits are equal.
2) Continuity is defined as a function being defined at a point, and the limit existing and being equal to the function value. Functions like tan(x) are only continuous where the denominator is not 0.
3) Theorems are presented for evaluating limits of polynomials, sums, products, quotients of continuous functions, and the squeeze theorem. Piecewise functions may or may not be continuous depending on behavior at points of discontin
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
The document discusses several key concepts regarding derivatives:
(1) It explains how to use the derivative to determine if a function is increasing, decreasing, or neither on an interval using the signs of the derivative.
(2) It provides theorems and rules for finding local extrema (maxima and minima) of functions using the first and second derivative tests.
(3) It also discusses absolute extrema, monotonic functions, and the Rolle's Theorem and Mean Value Theorem which relate the derivative of a function to values of the function.
This document discusses continuity of functions, including definitions of continuity at a point and on an interval. It introduces the intermediate value theorem and extreme value theorem for continuous functions on closed intervals. It also states that differentiability at a point implies continuity at that point.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
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 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
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 (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.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
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What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
1. Section 1.5
Continuity
V63.0121.027, Calculus I
September 17, 2009
Announcements
Please put your homework in the envelope by last name
Quiz next week in Recitation on §§1.1–1.3
WebAssignments 1 and 2 due Tuesday
. . . . . .
4. Hatsumon
Here are some discussion questions to start.
Were you ever exactly three feet tall?
Was your height (in inches) ever equal to your weight (in
pounds)?
Is there a pair of points on opposite sides of the world at the
same temperature at the same time?
. . . . . .
5. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
6. Recall: Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of
f, then
lim f(x) = f(a)
x→a
. . . . . .
7. Definition of Continuity
Definition
Let f be a function
defined near a. We say
that f is continuous at a
if
lim f(x) = f(a).
x→a
. . . . . .
8. Definition of Continuity
Definition y
.
Let f be a function
defined near a. We say
that f is continuous at a f
. (a ) .
if
lim f(x) = f(a).
x→a
A function f is
continuous if it is
continuous at every . x
.
point in its domain. a
.
There are three important parts to this definition.
The limit has to exist
the function has to be defined
and these values have to agree. . . . . . .
9. Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it is
continuous on R = (−∞, ∞).
(b) Any rational function is continuous wherever it is defined;
that is, it is continuous on its domain.
. . . . . .
11. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x→2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x→a x→2 x→2
Each step comes from the limit laws.
. . . . . .
12. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x→2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x→a x→2 x→2
Each step comes from the limit laws.
Question
At which other points is f continuous?
. . . . . .
13. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x→2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x→a x→2 x→2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞).
. . . . . .
14. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x→2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x→a x→2 x→2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞). It is right continuous
at the point −1/4 since lim f(x) = f(−1/4).
x→−1/4+
. . . . . .
15. The Limit Laws give Continuity Laws
Theorem
If f and g are continuous at a and c is a constant, then the
following functions are also continuous at a:
f+g
f−g
cf
fg
f
(if g(a) ̸= 0)
g
. . . . . .
16. Why a sum of continuous functions is continuous
We want to show that
lim (f + g)(x) = (f + g)(a).
x→a
We just follow our nose:
lim (f + g)(x) = lim [f(x) + g(x)]
x→a x→a
= lim f(x) + lim g(x) (if these limits exist)
x→a x→a
= f(a) + g(a) (they do; f and g are cts)
= (f + g)(a)
. . . . . .
19. Trigonometric functions are continuous
t
.an
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .
20. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .
21. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { kπ | k ∈ Z }.
c
. ot
. . . . . .
22. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { kπ | k ∈ Z }.
c
. ot . sc
c
. . . . . .
25. Exponential and Logarithmic functions are continuous
For any base a > 1, .x
a
the function x → ax is .oga x
l
continuous on R
the function loga is
continuous on its .
domain: (0, ∞)
In particular ex and
ln = loge are continuous
on their domains
. . . . . .
28. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. .
. in−1
s
.
−
. π/2
. . . . . .
29. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
30. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
31. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. ot−1
c
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
33. Pitfall #1
Example
Let {
x2 if 0 ≤ x ≤ 1
f (x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
. . . . . .
34. Pitfall #1: The limit does not exist
Example
Let {
x2 if 0 ≤ x ≤ 1
f (x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
Solution
At any point a in [0, 2] besides 1, lim f(x) = f(a) because f is
x→a
represented by a polynomial near a, and polynomials have the
direct substitution property. However,
lim f(x) = lim x2 = 12 = 1
x→1− x→1−
lim f(x) = lim 2x = 2(1) = 2
x→1+ x→1+
So f has no limit at 1. Therefore f is not continuous at 1.
. . . . . .
36. Pitfall #2
Example
Let
x2 + 2x + 1
f (x ) =
x+1
At which points is f continuous?
. . . . . .
37. Pitfall #2: The function has no value
Example
Let
x2 + 2x + 1
f (x ) =
x+1
At which points is f continuous?
Solution
Because f is rational, it is continuous on its whole domain. Note
that −1 is not in the domain of f, so f is not continuous there.
. . . . . .
39. Pitfall #3
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
. . . . . .
40. Pitfall #3: function value ̸= limit
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
Solution
f is not continuous at 1 because f(1) = π but lim f(x) = 7.
x→1
. . . . . .
42. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x→a
defined at a or its value at a is not equal to the limit
at a.
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x→a− x→a+
are different. f(a) is one of these limits.
. . . . . .
44. The greatest integer function
[[x]] is the greatest integer ≤ x.
y
.
. .
3
y
. = [[x]]
. .
2 . .
. .
1 . .
. . . . . . x
.
−
. 2 −
. 1 1
. 2
. 3
.
.. 1 .
−
. .. 2 .
−
. . . . . .
45. The greatest integer function
[[x]] is the greatest integer ≤ x.
y
.
. .
3
y
. = [[x]]
. .
2 . .
. .
1 . .
. . . . . . x
.
−
. 2 −
. 1 1
. 2
. 3
.
.. 1 .
−
. .. 2 .
−
This function has a jump discontinuity at each integer.
. . . . . .
46. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
47. A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
. . . . . .
49. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
. (x )
.
.
. x
.
. . . . . .
50. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
. (x )
f
. (b ) .
f
. (a ) .
. a
. x
.
b
.
. . . . . .
51. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b).
f
. (x )
f
. (b ) .
N
.
f
. (a ) .
. a
. x
.
b
.
. . . . . .
52. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
. (x )
f
. (b ) .
N
. .
f
. (a ) .
. a
. c
. x
.
b
.
. . . . . .
53. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
. (x )
f
. (b ) .
N
.
f
. (a ) .
. a
. x
.
b
.
. . . . . .
54. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
. (x )
f
. (b ) .
N
. . . .
f
. (a ) .
. a c
. .1 x
.
c
.2 c b
.3 .
. . . . . .
55. What the IVT does not say
The Intermediate Value Theorem is an “existence” theorem.
It does not say how many such c exist.
It also does not say how to find c.
Still, it can be used in iteration or in conjunction with other
theorems to answer these questions.
. . . . . .
56. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
. . . . . .
57. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2].
. . . . . .
58. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
. . . . . .
59. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the method
of bisections.
. . . . . .
66. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
. . . . . .
67. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
(More careful analysis yields 1.32472.)
. . . . . .
68. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
69. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
. . . . . .
70. Question 1: True!
Let h(t) be height, which varies continuously over time. Then
h(birth) < 3 ft and h(now) > 3 ft. So there is a point c in
(birth, now) where h(c) = 3.
. . . . . .
71. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
. . . . . .
72. Question 2: True!
Let h(t) be height in inches and w(t) be weight in pounds, both
varying continuously over time. Let f(t) = h(t) − w(t). For most
of us (call your mom), f(birth) > 0 and f(now) < 0. So there is a
point c in (birth, now) where f(c) = 0. In other words,
h(c) − w(c) = 0 ⇐⇒ h(c) = w(c).
. . . . . .
73. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
True or False
Right now there are two points on opposite sides of the Earth
with exactly the same temperature.
. . . . . .
74. Question 3
Let T(θ) be the temperature at the point on the equator at
longitude θ.
How can you express the statement that the temperature on
opposite sides is the same?
How can you ensure this is true?
. . . . . .
75. Question 3: True!
Let f(θ) = T(θ) − T(θ + 180◦ )
Then
f(0) = T(0) − T(180)
while
f(180) = T(180) − T(360) = −f(0)
So somewhere between 0 and 180 there is a point θ where
f(θ) = 0!
. . . . . .