I have tried to be somewhat rigorous about proving. In effect, the table indicates that the universally quantified statement is true provided that the truth set of the predicate equals the universal set, and the existentially quantified statement is true provided that the truth set of the predicate contains at least one element. The paper describes a personal experience with using counterexamples as a pedagogical strategy in the teaching and learning of an introductory calculus course at a university of technology. The students reported that it the students reported that it helped them to understa nd concepts better, prevent mistake s, deve lop logi. In manufacturing, it is often desirable to minimize the amount of material used to package a product. Einstein in the previous chapter, we studied propositional logic.
Full version experience with using counterexamples in an. Even if they know they cannot prove a theorem by providingonly examples, it is hard for some students to accept the fact that a single counterexample disproves a statement. Students have a hard time understanding the squeeze theorem. It is not the case that all linear functions in one variable are perpendicular to one another. For example, companies often want to minimize production costs or maximize revenue. Many students have limited example spaces available to them for seeking inspiration, for testing out conjectures, or for use in applications. Product and quotient rule in this section we will took at differentiating products and quotients of functions. Incorrect statements are grouped topically with sections devoted to functions, limits, continuity, differential calculus and integral calculus. Instead of using variables and abstract notations, you can use numeric examples to disprove an argument. Counterexample concept geometry video by brightstorm. They dont really need to be associated with math or even philosophy to be applicable.
Using counterexamples in calculus times higher education the. The notes were written by sigurd angenent, starting from an extensive collection of notes and problems compiled by joel robbin. Predicates and quanti ers nested quanti ers using predicate calculus predicate logic lucia moura winter 2010. Pdf this book makes accessible to calculus students in high school, college and university a range of counterexamples to conjectures that. Pdf using counterexamples in calculus researchgate. Counterexamples in calculus mathematical association of america. In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
The videos, which include reallife examples to illustrate the concepts, are ideal for high school students, college students. Richstein, verifying the goldbach conjecture up to 41014, mathematics of. In the following exercises, justify your answer with a proof or a counterexample. This book covers calculus in two and three variables. The project gutenberg ebook of calculus made easy, by silvanus thompson this ebook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. Direct proof and counterexample 1 in this chapter, we introduce the notion of proof in mathematics. There may or may not be challenging exercises in the exam. Functions, limits, continuity, differential calculus and i ntegral calculus. If the statement is false, provide a speci c counterexample. Counterexamples are helpful because they make it easier for mathematicians to quickly show that certain conjectures, or ideas. A sequent calculus for integer arithmetic with counterexample. The prerequisites are the standard courses in singlevariable calculus a. Also, your comment on the question is very misleading. Simplifying conditional expressions rules of boolean algebra equational reasoning proofs using truth tables tautologies and automatic verification of tautologies arguments, satisfiability and truth trees.
Students quite often fail to see a single counterexample as disproving a conjecture. Find materials for this course in the pages linked along the left. Leibniz provided a counterexample, using v x and y x. The answer is that the chain rule is being applied in a wrong manner, not that the chain rule will lead to missing definitions. Predicates and quanti ers nested quanti ers using predicate calculus predicate logic lucia moura winter 2010 csi2101 discrete structures winter 2010. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the example spaces to which they have access, and to. Series convergence tests math 122 calculus iii d joyce, fall 2012 some series converge, some diverge. Calculus blends the best aspects of calculus reform along with the goals and methodology of traditional calculus. Counterexamples in calculus provides all students a foundation on which to build an upward spiral of understanding of the calculus, and an appreciation forthe mathematics behindit. For example, the length of time a person waits in line at a checkout counter or the life span of a light bulb. Disproof by counterexample is the technique in mathematics where a statement is shown to be wrong by finding a single example whereby it is not satisfied. For example, to prove that \not all triangles are obtuse, we give the following counter example.
Counterexamples in calculus classroom resource materials. To prove that this is true, we need to nd a pair of linear functions in one variable that are not perpendicular. Give an example of a function that has the same property as the one displayed here by f. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn calculus i or needing a refresher in some of the early topics in calculus. Counterexamples on jumaries two basic fractional calculus. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. The reason why this is important is because if you can find a counterexample for a definition, lets say a teacher asks you to. A sequent calculus for integer arithmetic with counterexample generation philipp ru. Student manual for mathematics for business decisions part 2. Counterexamples aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. Here are my online notes for my calculus i course that i teach here at lamar university.
You may copy it, give it away or reuse it under the terms of the project gutenberg license included with this ebook or online at. Analytic geometry i exercises for exam 1 solutions exam 1 will of course be much shorter than this set of exercises. In 2003 my colleague and i conducted an international study of more than 600 students from 10 universities in different countries looking at their attitudes to using counterexamples in teaching. Some students believe that a particular counterexample is just an exception to the rule at hand, and that no other pathological cases exist. Chapter 12 greens theorem we are now going to begin at last to connect di. This chapter is dedicated to another type of logic, called predicate logic. Methods of producing these incorrect statements vary. Counterexamples in first year calculus stack exchange. Since both a and b are in each of these i n, this single point of intersection is a b. If you dont know the difference between 15,000 and 150,000, you probably shouldnt try to buy a new car. We would like to show you a description here but the site wont allow us. Math 221 1st semester calculus lecture notes version 2. Ten days later, in a manuscript dated november 21, 1675, leibniz provided the correct product and. Not surprisingly, disproof is the opposite of proof so instead of showing that something is true, we must show that it is false.
Counterexamples are often used in math to prove the boundaries of possible theorems. Then every sequence of distinct points of xconverges to every member of x. In algebra, geometry, and other branches of mathematics, a theorem is a rule expressed by symbols or a formula. Mit professor gilbert strang has created a series of videos to show ways in which calculus is important in our lives. One common application of calculus is calculating the minimum or maximum value of a function. Using counterexamples in teachinglearning of calculus can be beneficial for. Counterexamples in calculus pdf free download epdf. Many quantities can be described with probability density functions. Rewrite the equation above 1 x in terms of the function f. The riemann integral i know of some universities in england where the lebesgue integral is taught in the.
The book is in use at whitman college and is occasionally updated to correct errors and add new material. In addition to all our standard integration techniques, such as fubinis theorem and the jacobian formula for changing variables, we now add the fundamental theorem of calculus to the scene. The function fx is continuous at x a if the left and righthand limits of fx as x a exist and are equal. For example, to prove that ot all triangles are obtuse, we give the following counter example. The following counterexample shows a function where f x and f y are continuous everywhere and yet f xy0, 0.
Counterexamples in teachinglearning of calculus core. Arithmetic college calculus abstract mathematics in elementary school you largely learn arithmetic and the basic notions of shape. A short introduction to the lambda calculus achim jung. Dec 03, 2009 using counterexamples in calculus provides the serious or, alternatively, mildly sadistic mathematics lecturer with a cornucopia of counterexamples to falsify conjectures and expose sham statements in firstyear undergraduate calculus.
A counterexample is an example that disproves a universal for all. We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculusa generalization of multivariable calculus to infinitedimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. If false, provide a counterexample to the statement. It was submitted to the free digital textbook initiative in california and will remain unchanged for at least two years. Pdf on jan 1, 2010, sergiy klymchuk and others published counterexamples in calculus find, read and cite all the research you need on researchgate. None of these quantities are fixed values and will depend on a variety of factors. Our algorithm will produce a counterexample if the model does not satisfy the given speci. Shorser this proof structure allows us to prove that a property is not true by providing an example where it does not hold. Functions, limits, continuity, differential calculus and integral calculus. Textbook calculus online textbook mit opencourseware. We start using these in geometry because thats the first course that really teaches proving things. The notes were written by sigurd angenent, starting. This is the mathematics all of us need in order to function in the real world.
In fact, many standard calculus texts enable professors to challenge di erential calculus students by including several such proofs as exercises for example, \prove that the equation cosx x has a solution in 0. Derivatives of trig functions well give the derivatives of the trig functions in this section. We know when a geometric series converges and what it converges to. In english, the predicate is the part of the sentence that tells you something about the subject. A counterexample is a special kind of example that disproves a statement. I think it depends on your goals with your calculus class. March 18, 2004 abstract the lambda calculus can appear arcane on. Set up and solve optimization problems in several applied fields.
This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. This document models those four di erent approaches by proving the same proposition four times over using each fundamental method. Using the graph, find each limit or explain why the limit does not exist. These questions have been designed to help you gain deep understanding of the concept of continuity. Incorrect statements are grouped topically with sections devoted to. Counterexamples in calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. All major mathematical results you have considered. Chapter 3 predicate logic \logic will get you from a to b. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion is true under the assumption that the premisses are true. And if your goal is to filter, then perhaps you want to hit them with difficult stuff for no particular reason, including counterexamples. Conceptual understanding through verbalization mathematical communication cooperative learning group research projects integration of. The complete textbook is also available as a single file. Counterexample for mixed partial derivative for most textbook examples in multivariate calculus f xy a, b f yx a, b. The format of this text is enhanced, but is not dominated by new technology.
Improve your math knowledge with free questions in counterexamples and thousands of other math skills. Using counterexamples in calculus world scientific. The divisions into chapters in these notes, the order of the chapters, and the order of items within a chapter is in no way intended to re ect opinions i have about the way in which or even if calculus should be taught. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. Counterexample we know that the limits of sequences in hausdor spaces are unique. Derivatives of exponential and logarithm functions in this section we will. By the above two counterexamples, we show that jumaries two basic fractional calculus formulae are not true, and then all results obtained in,, by using jumaries formulae are incorrect. We need logic laws that work for statements involving quantities like some and all. Counterexamples aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool. A counterexample is a special kind of example that disproves a statement or proposition. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the. In algebra, most counterexamples involve manipulation using different positive and negative or odd and even numbers, extreme cases and special numbers like 0 and 1.
Calculus this is the free digital calculus text by david r. The issues to solve show up mainly in the handling of assignment, loops and calls, and they are the same independent of the wp calculus variant. If the domain is empty, 8xpx is true for any propositional function. In this section we will look at probability density functions and computing the mean think average wait in line or average life span. A geometric series x1 n0 arn converges when its ratio rlies in the interval 1. Math 221 first semester calculus fall 2009 typeset. If your goal is primarily to prepare students to apply calculus in other subjects, then perhaps counterexamples are not so important. Erdman portland state university version august 1, 20.
Benjamin filippenko, zhengyi zhou, katrin wehrheim download pdf. An elephant is a mammal but is not a dolphin inverse negating the clauses contrapositive reversing and negating the clauses if an animal is not a dolphin, then it is not a mammal. A whale is not a dolphin but is still a mammal if an animal is not a mammal, then it is not a dolphin. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool.
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