Lesson 31: The Divergence Theorem
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Gauss's divergence theorem, the last of the big three theorems in multivariable calculus, links the integral of the divergence of a vector field over a region with the flux integral of the vector field over the boundary surface.
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The document summarizes the Divergence Theorem. It states that the theorem relates the integral of the divergence of a vector field F over a region E to the surface integral of F over the boundary S of E. Specifically, the theorem states that the flux of F across S is equal to the triple integral of the divergence of F over E, for a region E that is a simple solid region or a finite union of such regions, and when F has continuous partial derivatives on a region containing E. An example application to computing the flux of a vector field over a unit sphere is also provided.
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The document announces that the midterm exam is on Wednesday April 30th, that Friday May 2nd is movie day, lists times for problem sessions on Sunday and Thursday at 7pm in room SC 310, and office hours on Tuesday and Wednesday from 2-4pm in room SC 323. It also tentatively lists the final exam date as May 23rd at 9:15am.
Lesson 25: Areas
The general area problem needs some kind of infinite process, whether an infinite series or a limit of finite sums. Once we define the definite integral, we examine its properties.
Lesson 23: Newton's Method
This document provides an outline for a lecture on Newton's Method for finding the zeros of functions or the roots of equations. It includes announcements about upcoming problem sessions, office hours, and an upcoming midterm exam. The outline also lists topics to be covered such as an introduction to Newton's Method, how it works graphically and symbolically, applications for finding zeros and roots, and potential flaws in the method like lack of convergence or convergence to incorrect values.
Lesson 22: Triple Integrals
You knew this was coming. From double integrals over plane regions we move onward to triple integrals over solid regions. The visualization is a little harder, but the calculus not that much.
vectors
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- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
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First order non-linear partial differential equation & its applications
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Making Lesson Plans
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-choice
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.
Electronic Grading of Paper Assessments
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.
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Lesson 27: Integration by Substitution (slides)
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)
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)
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)
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.
Lesson 25: Evaluating Definite Integrals (handout)
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)
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)
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 23: Antiderivatives (slides)
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.
Lesson 23: Antiderivatives (slides)
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.
Lesson 22: Optimization Problems (slides)
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.
Lesson 22: Optimization Problems (handout)
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.
Lesson 21: Curve Sketching (slides)
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.
Lesson 21: Curve Sketching (handout)
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)
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.
Lesson 19: The Mean Value Theorem (slides)
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.
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Streamlining assessment, feedback, and archival with auto-multiple-choice
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Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (slides)
Lesson 26: The Fundamental Theorem of Calculus (handout)
Lesson 26: The Fundamental Theorem of Calculus (handout)
Lesson 25: Evaluating Definite Integrals (handout)
Lesson 25: Evaluating Definite Integrals (handout)
Lesson 24: Areas and Distances, The Definite Integral (handout)
Lesson 24: Areas and Distances, The Definite Integral (handout)
Lesson 24: Areas and Distances, The Definite Integral (slides)
Lesson 24: Areas and Distances, The Definite Integral (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (handout)
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