Midterm examination for a class at MIT covering game theory and its applications to economics. The one-hour-and-twenty-minute open book examination asks open ended theoretical questions. The exam contains questions and solutions.
This week we will examine the concept of a function, a fundamental concept underlying all of modern mathematics. You’re undoubtedly already familiar with functions in an intuitive sense: a function is something which, given
an input, produces an output. But you’ve probably never seen the formal definition of a function as it relates to set theory, which is what we’ll look at this week.
A pre-lecture worksheet for students to preview section 5.5 and find the descriptions for the theorems covered in this section. Also having them try an example for each of the theorems.
This OER explores the basic organization of the Pythagorean Solids. It contains both an activity as well as resources for further exploration. It is a product of the OU Academy of the Lynx, developed in conjunction with the Galileo's World Exhibition at the University of Oklahoma.
This is the online, interactive version of OpenIntro's Advanced High School Statistics (https://www.openintro.org/book/ahss/). It was developed by Emiliano Vega and Ralf Youtz of Portland Community College using PreTeXt.
Advanced High School Statistics covers a first course in statistics, providing an introduction to applied statistics that is clear, concise, and accessible. This book was written to align with the AP© Statistics Course Description, but it's also popular in non-AP courses and community colleges.
We hope readers will take away three ideas from this book in addition to forming a foundation of statistical thinking and methods:
1. Statistics is an applied field with a wide range of practical applications.
2. You don't have to be a math guru to learn from real, interesting data.
3. Data are messy, and statistical tools are imperfect. But, when you understand the strengths and weaknesses of these tools, you can use them to learn about the real world.
This supplemental material is an online resource of OpenIntro Statistics, a textbook available for free in PDF at openintro.org and in paperback for about $10 at amazon.com.
APEX Calculus is a calculus textbook written for traditional college/university calculus courses. It has the look and feel of the calculus book you likely use right now (Stewart, Thomas & Finney, etc.). The explanations of new concepts is clear, written for someone who does not yet know calculus. Each section ends with an exercise set with ample problems to practice & test skills (odd answers are in the back).
Textbook for Portland Community College Calculus sequence.
MTH 251: Includes limits, continuity, derivatives and some applications of derivatives.
MTH 252: Includes antiderivatives, the definite integral, topics of integration, improper integrals, and applications of differentiation and integration.
MTH 253: Includes infinite sequences and series (including Taylor series), vectors, and geometry of space.
MTH 254: Includes multivariate and vector-valued functions from a graphical, numerical, and symbolic perspective. Applies integration and differentiation of both types of functions to solve real world problems.
This text was written as a prequel to the APEXCalculus series, a three–volume series on Calculus. This text is not intended to fully prepare students with all of the mathematical knowledge they need to tackle Calculus, rather it is designed to review mathematical concepts that are often stumbling blocks in the Calculus sequence. It starts basic and builds to more complex topics. This text is written so that each section and topic largely stands on its own, making it a good resource for students in Calculus who are struggling with the supporting mathemathics found in Calculus courses. The topics were chosen based on experience; several instructors in the Applied Mathemathics Department at the Virginia Military Institute (VMI) compiled a list of topics that Calculus students commonly struggle with, giving the focus of this text. This allows for a more focused approach; at first glance one of the obvious differences from a standard Pre-Calculus text is its size.
This module aims to acquaint you with the mathematical aspects of rings and groups and the underlying algebraic structures and when they are looked at as non-empty sets, how their elements are combined by binary operations as well as how those elements behave under transformations such finding inverses. Some non-empty sets, under the operation of addition or multiplication do not include the inverses of their elements as members of the set and they are called semi-groups. The non-empty sets that include the inverses of their elements are full fledged groups. This module fills the gap arising from basic mathematics.
Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.
This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.
Spreadsheets Across the Curriculum module/Geology of National Parks course. Students estimate travel times and costs of a driving/camping trip to visit national parks in Colorado.
- Material Type:
- Science Education Resource Center (SERC) at Carleton College
- Provider Set:
- Pedagogy in Action
- Judy A. McIlrath
- Date Added:
Active Calculus: Our Goals
🔗Several fundamental ideas in calculus are more than 2000 years old. As a formal subdiscipline of mathematics, calculus was first introduced and developed in the late 1600s, with key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz. The subject has been understood rigorously since the work of Augustin Louis Cauchy and Karl Weierstrass in the mid 1800s when the field of modern analysis was developed. As a body of knowledge, calculus has been completely understood for at least 150 years. The discipline is one of our great human intellectual achievements: among many spectacular ideas, calculus models how objects fall under the forces of gravity and wind resistance, explains how to compute areas and volumes of interesting shapes, enables us to work rigorously with infinitely small and infinitely large quantities, and connects the varying rates at which quantities change to the total change in the quantities themselves.
🔗While each author of a calculus textbook certainly offers their own creative perspective on the subject, it is hardly the case that many of the ideas they present are new. Indeed, the mathematics community broadly agrees on what the main ideas of calculus are, as well as their justification and their importance. In the 21st century and the age of the internet, no one should be required to purchase a calculus text to read, to use for a class, or to find a coherent collection of problems to solve. Calculus belongs to humankind, not any individual author or publishing company. Thus, a primary purpose of this work is to present a calculus text that is free. See https://activecalculus.org for links to both the .html and .pdf versions of the text. In addition, instructors who are looking for a calculus text should have the opportunity to download the source files and make modifications that they see fit; thus this text is open-source. See GitHub for the source. Since August 2013, Active Calculus - Single Variable has been endorsed by the American Institute of Mathematics and its Open Textbook Initiative.
🔗In Active Calculus - Single Variable, we actively engage students in learning the subject through an activity-driven approach in which the vast majority of the examples are generated by students. Where many texts present a general theory followed by substantial collections of worked examples, we instead pose problems or situations, consider possibilities, and then ask students to investigate and explore. Following key activities or examples, the presentation normally includes some overall perspective and a brief synopsis of general trends or properties, followed by formal statements of rules or theorems. While we often offer plausibility arguments for such results, rarely do we include formal proofs. It is not the intent of this text for the instructor or author to demonstrate to students that the ideas of calculus are coherent and true, but rather for students to encounter these ideas in a supportive, leading manner that enables them to begin to understand calculus for themselves. This approach is consistent with the scholarly consensus that calls for students to be interactively engaged in class.
🔗Moreover, this approach is consistent with the following goals:
To have students engage in an active, inquiry-driven approach, where learners construct solutions and approaches to ideas, with appropriate support through questions posed, hints, and guidance from the instructor and text.
To build in students intuition for why the main ideas in calculus are natural and true. Often we do this through consideration of the instantaneous position and velocity of a moving object.
To challenge students to acquire deep, personal understanding of calculus through reading the text and completing preview activities on their own, working on activities in small groups in class, and doing substantial exercises outside of class time.
To strengthen students' written and oral communicating skills by having them write about and explain aloud the key ideas of calculus.
Active Calculus is different from most existing calculus texts in at least the following ways: the text is free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the text is open source, and interested instructors can gain access to the original source files upon request; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; the exercises are few in number and challenging in nature.
Active Calculus Multivariable is the continuation of Active Calculus to multivariable functions. The Active Calculus texts are different from most existing calculus texts in at least the following ways: the texts are free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the texts are open source, and interested instructors can gain access to the original source files upon request; the style of the texts requires students to be active learners — there are very few worked examples in the texts, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; the exercises are few in number and challenging in nature.
Effective measurement techniques include the concept of measurement uncertainty. Students may make erroneous conclusions analyzing data using measurements that do not include the uncertainty of the measurement. In this lab, students determine a density range for a metal and identify the material based on this range.
This is a lab activity that allows students to collect data to practice using effective measurement. While other authors have produced similar labs, this version includes uncertainty analysis consistent with effective measurement technique as presented in the module Measurement and Uncertainty.
This resource can be used in providing real-life activity for students by conducting survey. Results of their survey will be organized and presented through text, graphs and tables with research ethics observed.