Instructions: Homework should be prepared as a Microsoft Word document using Equation Editor or Math Type for creation of mathematical notation; Math Type is an upgrade of Equation Editor that you need to pay for. Alternatively you may use LaTex to create a .pdf file. All homework should be sent to the address: math1710@yorku.ca. Homework is due Wednesdays at 11:59 P.M.
Chapter 12 - Systems of Linear Equations and Counting Techniques
Many problems in mathematics reduce to the task of solving simultaneously a system of linear equations. Sometimes it is only 2 equations in two variables and in many others the problem may involve hundreds of equation in hundreds of variables. The case of two equations in two variables is relatively uncomplicated but even for three equations and three variables there are subtleties that can be confusing. As an aide to the understanding there is a brief detour into the topic of 3 dimensional Cartesian geometry in which I introduce an equation in three variables as plane in three space. A unique solution to 3 equations then exists only when the corresponding 3 planes intersect in a unique point. The technique introduced for solving systems of 3 linear equations is the classical Gaussian elimination technique which is applicable to systems of more equations in more variables. You may wish to review the corresponding sections in the book by Stewart et al. sections 9.2-9.4
In the subsection on counting techniques I introduce the standard topics of permutations and combinations.
On-Line Lecture Macromedia Breeze
On-Line Lecture Quicktime
Systems of Linear Equations - Counting Techniques
Exercises
Solutions
Chapter 11 - Trigonometry II - Ideas and Applications
In this segment we continue the study of trigonometry. We begin with a detailed examination of the graphs of the sine and cosine functions and how the equations change when the graphs are altered in various ways. Using this as a foundation we introduce the topic of harmonic motion. Next there is a detailed explanation of the inverse trigonometric functions which are of fundamental importance to applications in which it is necessary to find angular measurements. We conclude with a discussion of the Law of Sines and the Law of Cosines together with some interesting examples. You may wish to consult sections 5.3, 5.4, 6.4, 6.5, and 7.4 of the Stewart, Redlin, and Watson text book for additional examples, discussion, and practice problems.
On-Line Lecture Macromedia Breeze Please note that the audio for slide 14, by error, is a duplicate of the audio for another slide. This is fixed in the Quicktime version.
On-line Lecture Quicktime
Trigonometry II Ideas and Applications
Exercises
Solutions
Chapter 10 - Trigonometry - Basic Notions
Trigonometry has its origins in the efforts of the ancient Babylonians and Greek astrologers and astronomers to understand the motions of the sun, the moon, and the visible planets. On one level trigonometry may be considered as a study of the relationships between the lengths of the sides of a given triangle. This has made trigonometry and essential tool in geometric analysis, and practically surveyors need trigonometry at every stage of their work. From another point of view trigonometry may be thought of as a study of the mathematics of a circle, and in particular a study of circular motion. This has made trigonometry an essential tool for navigation. The earth is roughly spherical so shortest paths between points follow circles called geodesics. From the sixteenth century a ships captains and officers have relied on the techniques of trigonometry. Another fundamental application of trigonometry relies on the fact that any function can be smoothly approximated by sums of trigonometric functions. This is particularly useful in the analysis of wave forms such as audio and video signals. Today trigonometry has immense applications in many areas of technology. In this chapter the basic ideas are developed. You may find it useful to read sections 6.1, 6.2, and 6.3 and parts of 5.2, 5.3, and 5.4 of the Stewart, Redlin, and Watson text book for extra examples and practice exercises. Be advise that the Stewart et al. text has a non traditional approach which is introduced in section 5.1. I have chosen to present the topic from the more traditional point of view. You may also like to visit chapters 16 and 17 of the Schaum book. The exercises will be due Wednesday February 20.
On-line Lecture Macromedia Breeze
On-line Lecture Quicktime
Trigonometry Basic Ideas document
Exercises
Solutions
Chapter 9 - Exponential and Logarithmic Functions
Exponential and logarithmic functions are essential to many applications of mathematics in economics, finance, natural sciences, and of course mathematics itself. Exponential functions have the unique characteristic that the rate of change at any instant is proportional to the value of the function. This means that they are a natural choice for modeling any situation that involves growth or decay. Imagine the situation of the growth of bacteria in a petrie dish. Until the confines of the environment play a role the bacterial culture will grow at a rate directly proportional to the amount of bacteria present. Thus, the exponential function is then a natural choice for modeling this growth.
Logarithm functions are simply the inverse of exponential functions. We are able to derive many of their properties by considering corresponding properties of exponential functions. Logarithm functions provide essential techniques for working with exponential functions and are used extensively to model such physical phenomena such as sound intensity and earth quake intensity.
For supplemental material, extra examples, and practice exercises you may wish to take a look at Chapter 4 of the Stewart, Redlin, and Watson textbook. The topic is also covered in chapters 25 and 26 of the Schaum text - but here the emphasis is on the finding derivatives and integrals - the latter being a calculus topic that is postponed to the next course.
The assignment posted below is due February 6
On-line Lecture
Exponential and Logarithmic Functions document
Exercises
Solutions
Chapter 8 - Optimization - Calculus III
In this section I introduce techniques for finding the maximal and minimal values for functions that are defined on a certain interval. These techniques are fundamental to all applications of differential calculus. As we have seen, finding the points x where the first derivative f '(x) is zero tells us that at the point (x, f(x)) the tangent to the graph of the function is horizontal, and in many cases, but not all, the function has a local maximum or minimum at the point x. The goal is to develop analytic methods that allow us to determine roughly what the graph of the function looks like in the vicinity of the so -called critical points - a critical point being one where f '(x) = 0 or one where the derivative does not exist and which may correspond to a cusp. Now clearly, if x is a critical point and the graph is increasing on one side of x and decreasing on the other, then the function must have a local maximum or minimum at this point. Using this idea the First Derivative Test provides a method for determining whether or not there is a maximum or minimum at critical point. As this can sometimes be a clumsy method, a simpler criterion for determining maximums and minimums exists. It is called the Second Derivative Test, but it depends on the second derivative of the function being defined in some interval containing the critical point. Discussion of The Second Derivative Test involves a discussion of the shape of the curve where we introduce criteria for determining if the graph is concave down or concave up. And related is the notion of an inflection point as a point where the graph changes from one form of concavity to another.
The assignment to be handed in is posted below will be due Wednesday January 23.
In addition read and work through the examples in Chapters 13 and 14 of the Schaum's Outline Calculus.
The fact that it is possible to apply simple analytic ideas to gain precise knowledge of the shape of a graph is one of the great achievements of differential calculus.
On-line Lecture Macromedia
On-line Lecture Quicktime
Calculus III document
Exercises
Solutions
Chapter 7 - Introduction to Applications - Calculus II
In this section I talk about three types of applications. I call them margin analysis, related rates, and the task of finding maximum and minimum values. Margin analysis has to do with extending the graph of a function and is used in predicting a future state of the function. For instance, if you have a function that describes some time dependent quantity, and the data available only allows the function to be defined up to the present moment, margin analysis allows you to predict future values of the function. Related rate applications occur in situations where one needs to find the the rate of change of a component of a system which is dependent of the rate of change of other components. A classic example is when two cars are moving in different directions at different speeds and you are asked to calculate the rate at which the distance between the cars is changing. Finding maximums and minimum of function is perhaps the most important application of differential calculus. The surrounding theory is a masterpiece of human ingenuity. I will not go into full details, but the ideas all have an intuitive geometric base that is easy to communicate. As usual there is some terminology that needs to be introduced. Getting this under your belt requires careful reading. The section ends with some examples where I show an easy technique for finding maximums and minimums. The completion of the topic along with further examples of applications is put off to the next segment. The on-line lecture and the the full text are below as well as the exercises that are due Wednesday January 9. Please get an early start on the exercises. You may find them challenging. And if this is the case you will need time to talk between yourselves. I of course will be available for questions and hints. If there are typos or things that seem to be completely nonsensical, please contact me immediately so that confusion can be quickly dealt with.
Also - it is essential that you to gain facility with differentiation procedure. Since there is more than a month before the next assignment is due, I am also assigning work from the text Schaum's Outline Calculus. You need not hand any of this work in, but if you have questions I am as usual available for help.
On-line Lecture Macromedia Breeze There is a small error in the audio on slide 12 and a larger error on slide 15 - see if you can find them.
On-line Lecture Quick Time corrections are made here
Calculus II document
Exercises
In addition, it is very important that you finish the Schaum assignment from Chapter 6. To help with the related rates problems read carefully Schaum's Chapter 20 - after having read the Solved Problems do problems 9 through 20 in the Supplementary Problems section.
Solutions
Chapter 6 - The Mathematics of Change - Calculus I
This is the first of a planned three segments in which I introduce beginning notions of differential calculus. I begin at the beginning, always a good idea, and introduce the derivative first as the slope of a line tangent to a curve. Later after a discussion of the limit of a function, I give the full definition as a limit of approximations to the tangent. This segment ends with a section on differentiation technique together with some examples. At this point you should consider supplementing this treatment with additional examples and exercises from the appropriate portions of Schaum's Outline of Calculus or any other a standard calculus book. Learning the tricks of differentiation is not hard - really quite mechanical - but it needs practice. From one point of view, this aspect of calculus is simply an extension of the algebraic technique that one learns in high school. The on-line lecture and the accompanying document appear below as well as the exercises due Wednesday November 28.
On-line Lecture Macromedia Breeze
Calculus I document
Exercises In addition: read the solutions to problems 5 through 20 on pages 91-93 of the Schaum Calculus book. Then complete problems 27- 46 on pages 94 - 96, but DO NOT hand in the solutions. A selection of these problems will occur on the next test.
Solutions
Chapter 5 - Functions II - the Sequel
This is the second segment on functions. Again you may wish to consult the text for additional examples and exercises. The appropriate portions are: all of chapter 3 plus section 2.4. It will be important for you to review how to divide polynomials as it appears in section 3.2. The topic of synthetic division is interesting, but we will not be treating it in this course. Your assignment is due Tuesday November 14 and is posted below. There are 7 problems. Do them all. It would be appreciated if you could have an early look at the assignment, for this allows us time to straighten out misunderstandings. For this assignment it will be expected that you complete some of the problems using a computer graphing utility. Again, please consult the Stewart text for further examples.
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On line Lecture - Macromedia Breeze version Please note there is a typo on the last line of slide #12 - an exponent 2 is missing
On line Lecture - Quicktime version - the error is corrected here
On line text document
Exercises
Solutions
Chapter 4 - Introduction to Functions
The topic of functions forms one of the foundations of the course. Initially we will be simply introducing some language and later we will be turning to applications. There will be two 2 week sessions introducing the ideas. This is the first. You may wish to consult the text for more examples and a different exposition. In particular you may wish to look at sections 1.10, 2.1, 2.2, 2.5,2.6, 2.8. The intervening sections are also good but not directly applicable at this time. Your assignment for Wednesday October 31 is posted. There are 6 problems. Do them all. In addition to the exercises posted, there are additional exercises posted in the Forum. These exercises are not optional, but they are not graded. Instead your performance is judged by your participation - questions of any sort - questions on how to solve the problems, questions about proposed solutions, and of course solutions are all valid forms of participation. Your performance in this aspect of the course will be determined by your proportion of the total number of postings.
Now is the time for you to think about graphing software. For the Mac this is easy - an excellent product called Grapher is included with the operating system. For the PC there are a variety of free options. Here two suggestions: grahpmatica at graphmatica and graph available at graph. Which ever utility you use it must have: (i) the capability of copying the graph into Word, (ii) the ability to graph more than one function at a time, (iii) the ability to zoom in on intersection points. Of the two above, I am partial to graph.
On line lecture - Quicktime
On line Lecture Macromedia Breeze version (there is an error on slide 8 - see if you can find it - the audio is ok)
On line text document
Exercises
Solutions
Chapter 3 - Sequences and Series
This is an important topic. It normally comes towards the end of a pre-calculus course, but since our course contains an introduction to calculus, it is better to introduce the topic early. You will find that the work in logic segment should help with your understanding of a "limit of a sequence". This idea is very closely related to the notion of "derivative" in calculus. You may wish to consult the text, chapter 11.1 -11.3 for further explanations, examples, and practice problems. It is expected that before you start the problems that you listen to the lecture and carefully read the text document. The assignment consists of problems : 1, 2, 3, 4, 5, 6, 7 from the document Exercises, with the link below. Please note that the due dates for assignments is changed to Wednesdays at 11:59 P.M. This assignment is then due Wednesday the 17th of October. Please get an early start on this assignment so that there is ample time to respond to questions. Problems 6 and 7 will need some extra time
On line lecture Part I - Quicktime
On line lecture Part II - Quicktime
On line lecture Macromedia Breeze version
On line text document
Exercises
Solutions
Solution for Problem 7
Chapter 2 - Numbers
I am assuming that you are familiar with basic high school math at the grade 11 level. If you need review you may wish to read carefully the appropriate sections of chapter 1 (1.1thruough 1.5 ) in the Stewart test - laws of exponents, solving quadratic equations, and how to multiply and divide polynomials. These are all essential skills that you will get to practice.
The particular sections of the Stewart text that cover the topics of this segment are: Sections 1.1 and 3.4. The problems to complete for Assignment 3 due October 2 are problems 9, 10, 11 ,12, 13, 14 - from the text document below.
On line lecture Part I - Quicktime
On line lecture Part II - Quicktime
On line lecture - Macromedia Breeze version
Numbers document
Solutions
Chapter 1 - Logic
The first portion of the course consists of a quick introduction to principles of logic. Often difficulties students have can be traced to simple errors in logic that nonetheless are confusing in every day language. As with all other topics, the topic is covered in lecture format and in a text document. In this case the text document goes somewhat further than the lecture and contains some exercises, some of which will be assigned as problems to complete as part of the next homework assignment.
On line lecture-Macromedia Breeze
Logic document
Assignment 1 due Tuesday September 18 no later than 11:59 P.M. - from the Logic text document problems 5, 6, 7, 12, 13.
Solutions
Assignment 2 - due Tuesday September 25 at 11:59 P.M. - from the Logic text document ( which has been updated) exercises 15, 24, 25, 26.
Solutions
