## Calculus 3 Notes Pdf

The Total Derivative Recall, from calculus I, that if f : R → R is a function then. 510 Grade 12 pre-calculus mathematics (40S) : a course for independent study Previously published as : Senior 4 pre-calculus mathematics (40S) : a course for distance learning. Seifert Manifolds. is a real number have limits as x → c. Click to easily compare Larson Calculus to other calculus books available. Wenyu Zhai. pdf (1/20) Calculus I: Chapter 3 Notes Fall 2011 These are blank lecture notes for the week of 10 – 14 Oct 2011. In multivariable calculus, we progress from working with numbers on a line to points in space. Multivariate Calculus; Fall 2013 S. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Introduction to integration using area under velocity-time graphs. I do this for two reasons; rstly, it greatly increases the speed at which I can write mathematics,. ADVANCED CALCULUS Lecture notes for MA 440/540 & 441/541 2015/16 Rudi Weikard 1 2 3 4 5 x - 2 - 1 1 logHxL Based on lecture notes by G. Many of the sections not covered in Calculus III will be used on occasion there anyway and so they serve as a quick reference for when we need them. 10 Scalar Triple Product The volume of the parallelpiped determined by vectors ~a, ~b, and ~cis the magnitude of their scalar triple product:. Math 1950-3: Calculus III (Spring 2002) DU Department of Mathematics , PDF, Postscript. The derivative of f has several notations: ( ) [ ]( )f x dx d dx dy f , f x , y′ , ,. Syllabus Math 241 syllabus. edu/courses/mathematics/18-01-single-variable. Yusuf and Prof. You can use the review notes as a reference, to understand the subject better and improve your grades. He taught high school mathematics, including AB and BC Calculus for 34 years. These are special cases of moments of a probability distribution. 6 of our textbook. 6 Day 11 Chapter 1 Review Day 12 Chapter 1 Test. Core 40 Pre-Calculus Reference Sheet Standard Form of a Quadratic Equation. Tutorials for the Calculus Phobe: A collection of animated calculus tutorials in Flash format. What is the most likely sum? Outcomes are ordered pairs (i,j), 1 ≤ i ≤ 6, 1 ≤ j ≤ 6. s Eric and Erin Bancroft, Grove City College Fall 2018. Mean Value Theorem for integrals; average value. Use the chain rule to ﬁnd @z/@sfor z = x2y2 where x = scost and y = ssint As we saw in the previous example, these problems can get tricky because we need to keep all the information organized. AB Calculus BC Calculus Algebra II / Pre-Calc Review: Summer Packet. Michael Kelley Mark Wilding, Contributing Author. You need to know how to solve each of them without the use of the book, formula sheets and/or calculator. Similarly, moving across the root 3 to the interval ( 4; 3), we see that the factor x+ 3 = x ( 3) changes sign from positive to negative, while all the other factors x 2, x 1, and x+ 4 do not change sign. AP Calculus AB Notes, Worksheets and Classroom Policies. Probability Space Let (;F;P) be a probability space. These are notes for a one semester course in the diﬀerential calculus of several variables. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. From basic equations to advanced calculus, we explain mathematical concepts and help you ace your next test. In addition to most traditional areas of Pure and Applied Mathematics, the department contains all UWM programs in Actuarial Science and Atmospheric Science. Identify whether they are local minima, local maxima or saddle points. Integration by Parts. Existence of an additive inverse (!ידגנ רביא): for all a 2F there exists an element b 2F, such that. Calculus Tutoring Center Free Walk-in Tutoring - DU 326 Lecture Notes. Calculus 3 Course Notes for MATH 237 Edition 4. Bring whatever supplies (loose leaf paper, notebook, pen, pencil, etc) you personally like to use to take notes. Math 12 Pre-Calculus Chapter 2 & 3: Radical & Polynomial Functions 7 3. Introduction. com @stevemuench Please feel free to share the link to these notes. At times during this course, the topics may seem disjointed. 1: “Let a be a number in the domain of the function f. The book is in use at Whitman College and is occasionally updated to correct errors and add new material. 2 CALCULUS 3: NOTES AND EXERCISES ON CRAMER'S RULE In class we proved Cramer's rule for n = 3. ideas from algebra and calculus. This problem requires a geometrical argument: Solution 1: By similar triangles, f( x) 6 = −a 0−a = g(x) 3, and therefore f(x) g(x) = 6 3 = 2. Pre-Calc 11 Preview provides a course outline, including a general timeline. Enrolling in AP Calculus comes with the understanding that you will take the AP exam in May. Calculus 3 Course Notes for MATH 237 Edition 4. With CALCULUS, Eighth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. A scalar is written as an ordinary math symbol, as in a. The distinct feature of this part of the course is its focus on the multi-dimensional analysis, as opposed to one-dimensional analysis that you learned in Math 180 (Calculus I) and Math 181 (Calculus II). Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts These notes are copyright-protected but may be freely distributed for instructional nonproﬁt pruposes. We might want to keep track of our various drafts as the typos change. Definition 3. pdf form; less expensive than traditional textbooks. Putting these two things together, can we produce a graph of a function that does not. Let f(x) = √1−x3 x+2xFind f′(x). functions, identities and formulas, graphs: domain, range and transformations. This notes contain Integrals, Applications of Integration, Differential Equations, Infinite sequences and series and Application of Taylor polynomials. It should serve as a good review of the material covered in class during the last week or so. Angle Measure Angles can be measured in 2 ways, in degrees or in radians. The argument. 752 Chapter 11 Limits and an Introduction to Calculus In Example 3, note that has a limit as even though the function is not defined at This often happens, and it is important to realize that the existence or nonexistence of at has no bearing on the existence of the limit of as approaches Example 5 Using a Graph to Find a Limit. We select an increasing sequence of real numbers (x values) approaching 3 such that all the numbers are close to (but less than) 3. The most general advice is to watch what your professor writes. Preface These notes are written for a one-semester calculus course which meets three times a week and is, preferably, supported by a computer lab. Chapter 3 The Fundamental Theorem of Calculus In this chapter we will formulate one of the most important results of calculus, the Funda-mental Theorem. Calculus II notes and also put a copy in the Calculus III notes. Calculus III Class Notes. 100% Free AP Test Prep website that offers study material to high school students seeking to prepare for AP exams. Wenyu Zhai. 61 Diﬀerentiation of Processes Let E be a ﬂat space with translation space V. Chapters 3 and 4 add the details and rigor. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for. This Calculus Handbook was developed primarily through work with a number of AP Calculus classes, so it contains what most students need to prepare for the AP Calculus Exam (AB or BC) or a first‐year college Calculus course. of Revenue 3. † Calculus II. Calculus The traditional mathematical entry point for college and university students and a subject that still has the potential to provide important insights to today’s students. A BRIEF SUMMARY OF CALCULUS Calculus is one of the greatest intellectual achievements of humankind. 1: Graphing Quadratic Functions in Vertex Form • Section 3. Drawing on support from the US National Science Foundation (NSF DUE-0920350), Project MOSAIC. Math 2210 (Calculus 3) Lecture Videos. com Loading…. Week 3: Calculus III Notes While my previous notes attempted to give a fairly comprehensive view of Calculus I and Calculus II, it as at this point that I give up on that approach simply because there would be too much material to cover. Lecture Description File; Lecture 1: Introduction, Coordinates, & Vectors: PDF: Lecture 2: Dot and Cross Products: PDF: Lecture 3: Lines and Planes: PDF: Lecture 4. They cover sections 3. Whoops! There was a problem previewing Calculus Notes BC8. The graph of a function. 2 Reflections and Stretches/Compressions R2,R5 Definitions. 4_notes_back_complete 5. Calculus—Programmed instruction. Students who do not meet the placement requirements will be dropped. Click to easily compare Larson Calculus to other calculus books available. DepartmentofPhysics, SouthernIllinois University–Carbondale, Carbondale, Illinois 62901, USA. We select an increasing sequence of real numbers (x values) approaching 3 such that all the numbers are close to (but less than) 3. He served as an AP Calculus exam reader and table leader for 14 years. The website has been designed to give you easy access to study materials, book supplements and challenge problems that will help you with your study of calculus. Griffin's Math Classes. Vector Calculus - GATE Study Material in PDF In previous articles, we have already seen the basics of Calculus – Differentiation and Integration and applications. Math 263 Calculus III 2 Lecture 10 Sections 16. Sixth Edition. Use the drop menus below to access exercises in other Chapters and Sections Chapter P Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15. Mathematics—Programmed instruction. This university professor provides free and downloadable notes and tutorials for Mathematics classes he teaches, notably Calculus III and Differential Equations. We revisit all of the amazing theory we learned in Calculus I and II, but now we just generalize it to the multivariate setting. There's no signup, and no start or end dates. The absolute value function. jxyj jxjjyj. that the topics discussed in a beginning calculus course can be covered in virtually any order. click the image. Design and Images © MathNotes. Stochastic Processes The following notes are a summary of important de nitions and results from the theory of stochastic processes, proofs may be found in the usual books for example [Durrett, 1996]. These lecture notes should be accessible to anyone wanting to learn Calculus III or needing a refresher in some of the topics from the class. PowerPoint Section 7-4. Calculus I and II). You need to know how to solve each of them without the use of the book, formula sheets and/or calculator. The calculus of di erential forms give an alternative to vector calculus which is ultimately simpler and more exible. Multivariate Calculus; Fall 2013 S. We now recall the deﬁnition of nth order excision, from section 3 of [10]. The argument. Be sure to choose the correct chapter along with either a pre-test or post-test. Calculus—Programmed instruction. Here you can download the free Database Management System Pdf Notes - DBMS Notes Pdf latest and Old materials with multiple file links. 3B Limit Theorems 3. The following picture shows the. 6 THE CHAIN RULE. They cover sections 3. History of calculus; Generality of algebra; Nonstandard calculus. Not every topic will be hit, but the majority of them will be covered. Integration is treated before differentiation--this is a departure from most modern texts, but it is historically correct, and it is the best way to establish the true connection between the integral and the derivative. com/precalculus-pr-of-day. 8 Exponential Growth and Decay Math 1271, TA: Amy DeCelles 1. It will be updated periodically, and will evolve during the semester. The 2019 test will be given May 5, 2020If you do not plan on taking the AP Exam, we must have a conversation about it first. The most general advice is to watch what your professor writes. Functional notation. Lecture Notes on Integral Calculus UBC Math 103 Lecture Notes by Yue-Xian Li (Spring, 2004) 1 Introduction and highlights Di erential calculus you learned in the past term was about di erentiation. For this reason, when we integrate, we have to add a constant. 3: Completing the Square • Ch 3 Review: Printable Notes • Ti 83 • Section 3a • Section 3. Diﬀerential and Integral Calculus Review and Tutorial AnthonyA. and covariant rank 3 (i. 752 Chapter 11 Limits and an Introduction to Calculus In Example 3, note that has a limit as even though the function is not defined at This often happens, and it is important to realize that the existence or nonexistence of at has no bearing on the existence of the limit of as approaches Example 5 Using a Graph to Find a Limit. pdf from COMPUTERSC 51 at Harvard University. (f x )= xn where n is a positive integer. 100% Free AP Test Prep website that offers study material to high school students seeking to prepare for AP exams. com/math/calculus-intuitive-explanations/ Intuitive Explanations is a collection of. Then f is continuous at every point a 2 D if and only if the following holds: For every open set W of Rm, its inverse image f¡1(W) :=. Using this result will allow us to replace the technical calculations of Chapter 2 by much. Stochastic Calculus, Filtering, and Stochastic Control Lecture Notes (This version: May 29, 2007) Ramon van Handel Spring 2007. The website has been designed to give you easy access to study materials, book supplements and challenge problems that will help you with your study of calculus. 1 Matrix algebra An mby nmatrix Ais an array of complex numbers Aij for 1 i mand 1 j n. 2 - Integrals In single variable calculus we presented the integral of a continuous function as the 'opposite' of differentiation and showed how it was related to the area under the curve defined by the graph of the function. 6 Day 11 Chapter 1 Review Day 12 Chapter 1 Test. pdf format dvi format ps format Multivariable Calculus. frames and PDF formats. We have added the link. the matrix calculus is relatively simply while the matrix algebra and matrix arithmetic is messy and more involved. 1 Fundamental theorem of calculus: Part I Let f(x) be a boundedand continuousfunctionon an interval [a,b]. You’ll work to understand the theoretical basis and solve problems by applying your knowledge and skills. From sections 2. wrote these notes all of these topics were covered in Calculus II however, we have since moved several of them into Calculus III. Examples of rates of change18 6. com useful for school or work? I built this website and developed the material it contains on my own time, and it's entirely free to use. (1)Pisto des (1988. This page contains PDF's of APEX Calculus, Version 4. 5 – Day 2 Page 117 – 120 # 4, 32, 45, 47, Worksheet Day 9 Section 1. frames and PDF formats. 3 Birth of modern science and of calculus Stage III, around 1680: how to speak the language of mathematics!. If g(x) = 3x2 +x. edu/Classe 141 comments. Use the contour plot of f(x;y) to answer the questions. Your instructor might use some of these in class. notes of my topology class either, but rather my student’s free interpretation of it. However: If y = 2x + 3, dy/dx = 2 If y = 2x + 5, dy/dx = 2 If y = 2x, dy/dx = 2. Multivariable Calculus Archive here I have collected the many solutions, reviews, old tests and Lecture Notes from previous calculus III courses I've taught. Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. 3 through 3. The prerequisites are the standard courses in single-variable calculus (a. Math 231 - Calculus III - Course Materials - Fall, 2011. Save Time Editing Documents. ¾find the vector defined by two points and determine the norm of the vector. 4a The Fundamental Therom of Calculus. ftoc part2_notes 5. Whoops! There was a problem previewing Calculus Notes BC8. I may keep working on this document as the course goes on, so these notes will not be completely. xc= is an absolute. of Jon Rogawski's Calculus. 2014 ISBN-10: 1118791290 384 pages PDF 7 MB. You are expected to follow along in class, then reread them on your own, as necessary. jx yj jy xj. Find the area enclosed by the graph of y x= +sin 3( 6π ), the x-axis and the lines x= 0 and x= 6π. Economics is no exception. Here are a representative student's class notes for Mat 241. an integrated overview of Calculus and, for those who continue, a solid foundation for a rst year graduate course in Real Analysis. PDF | These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. Calculus 3. No Assignment list for this section - make sure you try at least one of each question type. Math 291-3: Intensive Linear Algebra & Multivariable Calculus Northwestern University, Lecture Notes Written by Santiago Ca˜nez These are notes which provide a basic summary of each lecture for Math 291-3, the third quarter of “MENU: Intensive Linear Algebra & Multivariable Calculus”, taught by the author at Northwestern University. Introduction to integration using area under velocity-time graphs. Stolz and G. Calculus Notes BC8. Outline!! CH12. Slides for this week's class. Don't show me this again. A Copy of the Solution of the ODE Part of Project 3, Calculus II An Introduction to Phase Space for Second Order ODE's. 4 Notes A position graph will give the location of an object at a certain time. Mathematical Methods in Engineering and Science 3, Contents I calculus Take the preliminary test. (In fact if I had to choose a subtitle for these notes, it would be 'An Anti-calculus-text Book'. and covariant rank 3 (i. (1)Pisto des (1988. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes. 3 Exponential Functions. Here are some notes on sequences and series, including a list of the tests. Note: the use of ”d” instead of ”@”with the function of only one independent variable. A quantity Aij is said to be antisymmetric in the indices i and j if it satis es the property Aij = Aji. Syllabus Math 241 syllabus. 8 Implicit Differentiation 3. Calculus I or needing a refresher in some of the early topics in calculus. These are notes for a one semester course in the diﬀerential calculus of several variables. An introduction to the Calculus, with an excellent balance between theory and technique. Some ofthe historychapters, chapters 13,18, 21, 28, and 39, could be assigned as reading and material for class discussion. Area; area under a curve, area between two curves. 5 – Even and Odd Functions Chap 3 Math Notes Lesson 3. Coördinate pairs of a function. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some of the topics from the class. A brief history of mathematical. Advanced calculus / Lynn H. When asking for help you need to show all your work, by typing it on the email (better) or by attaching a weblink to a scanned copy of your work. muench@gmail. The only prerequisites are several variable calculus and the rudiments of linear algebra and di erential equations. Our study guides are available online and in book form at barnesandnoble. So, for example, page 73 will have a series of problems and blank space for the students to write in the solutions. Exercises13 Chapter 2. Mathematics Learning Centre, University of Sydney 3 Figure 2: Calculation of the velocity of the motorist is the same as the calculation of the slope of the distance - time graph. DEC is a simple, ﬂexible, and efﬁcient framework which provides a uniﬁed platform for geometry processing. You need to know how to solve each of them without the use of the book, formula sheets and/or calculator. of Statistics UW-Madison 1. 4a The Fundamental Therom of Calculus. Exercises18 Chapter 3. I tend to use mupdf to view the pdfs. Calculus I and II). pdf (1/20) Calculus I: Chapter 3 Notes Fall 2011 These are blank lecture notes for the week of 10 – 14 Oct 2011. calculus student about the generally accepted standards of correct and incorrect mathematical notation. With CALCULUS, Eighth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. Tutorials Precalculus. schoolwires. com/products/grapher/grapher-contourmaps. 115 #3-19 EOO (every other odd), 33-37 odd, 39-43 odd **THIS IS DUE ON TUESDAY!** Remember - there is an opportunity to earn 4 extra points for the Ch 1 Test with the Extra Point Quiz on Monday!. Instantaneous velocity17 4. Jamshidi 3. 3: Completing the Square • Ch 3 Review: Printable Notes • Ti 83 • Section 3a • Section 3. Practice Exam 3 Exam 3 Answers. Can we solve other (non-polynomial) equations? Only when the left hand side involves a composition of functions which we know how to invert individually, for example ln · sin µ 1 1+x2 ¶¸ = −3 should pose no diﬃculty. The timeline is not set in stone and is intended as a guideline only. 1 with Chapter 2 on Exam 1 because it really is just a small extension of the topics of Chapter 2. Lecture Notes Multivariable Calculus II Semester 2015 Department of Mathematics University of the West Indies Kingston,. Example 3: The graph below shows that as x approaches 1 from the left, y = f(x) approaches 2 and this can be written as. For all courses you can download a PDF file with the notes (which you should do if you just want to read them) or you can download the source (which you should do if you want to change the notes. pdf Size : 110. The ﬁrst two chapters are a quick introduction to the derivative as the best aﬃne approximation to a function at a point, calculated via the Jacobian matrix. 2 - Integrals In single variable calculus we presented the integral of a continuous function as the 'opposite' of differentiation and showed how it was related to the area under the curve defined by the graph of the function. 745 Kb Type : pdf Calc 3 Test 1 Review. 3 8, P(1:3 split or 3:1 split) = 8 16 = 1 2, P(4:0 split or 0:4 split) = 2 16 = 1 8. There is on-line information on the following courses: † Calculus I. Real numbers 3 can show that the n-fold addition a 1 + a 2 + + a n is deﬁned unambiguously. Lecture Notes Multivariable Calculus II Semester 2015 Department of Mathematics University of the West Indies Kingston,. MALATI materials: Introductory Calculus, Grade 12 13. Calculus Notes BC8. Honors Trig/PreCalc 5/15 Notes on 11. Powered by Create your own unique website with customizable templates. The axioms 1 1. Calculus Tutoring Center Free Walk-in Tutoring - DU 326 Lecture Notes. Calculus Notes 3 June 01, 2016 Jun 15:24 PM. 4a The Fundamental Therom of Calculus. dvi file of this paper. https://www. I will assign a handful of homework problems throughout the course—on average, probably about one problem for every two or three sections. 9 MB, compared to 12 MB for the scanned image PDF. Therefore a consumer has to maximize his/her satisfaction while not spending more than he/she has, i. However, the last few times I taught undergraduate advanced calculus I decided I would do it this way. 3: Polar Coordinates. 2 Measure Theory. Sample text from Stewart's Calculus book. Contour Map From http://ftp. SLOPEFLD, Problem Set 7-4, Problem 14 (page 343) The program plots a slope field for a given differential equation. Whether you are attending Saddleback College's calculus class (math 3A), taking a calculus class at another school, or need to refresh your math skills for a business or science class, we have tools that can help you. The Department of Mathematical Sciences at the University of Wisconsin-Milwaukee is home to an active community of teachers, scholars, undergraduates, and graduate students. This Calculus Handbook was developed primarily through work with a number of AP Calculus classes, so it contains what most students need to prepare for the AP Calculus Exam (AB or BC) or a first‐year college Calculus course. Take good notes and then use them when working homework or practice problems to make sure that the way that you write agrees with the way your professor writes. Multivariable Calculus Archive here I have collected the many solutions, reviews, old tests and Lecture Notes from previous calculus III courses I've taught. Calculus Maximus Notes: 2. Lectures on Vector Calculus Paul Renteln 3 Vector Calculus II: Other Coordinate Systems 48 3. Unlike many other. pdf file or. The notes below are from a previous textbook and syllabus for this class. Lecture Notes in Macroeconomics John C. The following picture shows the. They cover sections 3. Calculus Notes BC8. Calculus 140, section 3. Calculus 1 Online Lessons (Math 1151) To benefit all Math 1151 students, the MSLC is making these online, interactive lessons available on their website to all students. The review will consist of 3 days separated by concepts. Mathematics Examples, Lecture Notes and Specimen Exam Questions and Natural Sciences Tripos Mathematics examples Details on obtaining and updating the source of DAMTP examples (this is aimed at DAMTP Unix account holders only), and the list of course codes and titles referred to in these pages. So, for example, page 73 will have a series of problems and blank space for the students to write in the solutions. When asking for help you need to show all your work, by typing it on the email (better) or by attaching a weblink to a scanned copy of your work. Calculus 3 Course Notes for. Click for ps file or. Enterprising students use this website to learn AP class material, study for class quizzes and tests, and to brush up on course material before the big exam day. Each from these formats has its own advantages. Integration by substitution. This increment in utility is called marginal utility. Bertsekas and John N. The latest versions may be found by. 3 through 3. calculus student about the generally accepted standards of correct and incorrect mathematical notation. 3 Asymptotes and thou Write your questions ghts here! Notes Find the horizontal asymptote(s) of each function. If you are interested in tutoring Calculus on this campus, contact Alireza Rostamian at 305-919-4048. The calculus of di erential forms give an alternative to vector calculus which is ultimately simpler and more exible. Syllabus Math 241 syllabus. Area; area under a curve, area between two curves. Library of Congress Cataloging-in-Publication Data. This is useful notes for integral calculus. Calc Lesson 64 notes. Math isn’t a court of law, so a “preponderance of the evidence” or “beyond any reasonable doubt” isn’t good enough. Calculus II notes and also put a copy in the Calculus III notes. goldensoftware. A function P(t) is increasing over an interval if =P′(t) >0 dt dP. General Syllabus Calc 3. 4_notes_back_complete 5. Along with GATE it will help in the preparation of BARC, BSNL, DRDO, IES, ISRO and other exams. Hour exams are scheduled on the following dates: September 16 (Friday) , October 7 (Friday), October 28 (Friday), November 18 (Friday). Questions are presented along with solutions. It is specifically designed for students and the teachers to give a better understanding of mathematics. You must enable JavaScript in order to use this site. Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. [ 4 points each, if complete, neat, in homework section of notes and submitted for check by the due date, and not after. Finding extrema of functions of several variables 2 3.