Understanding Calculus


What is the biggest mathematical breakthrough since the wonderful discoveries made by the ancient Greeks? Yes, it is calculus! Without calculus the world wouldn't have seen man-made wonders such as space flight, skyscrapers, aircrafts, economic modeling, weather forecasting, many medical technologies, and countless other achievements that have changed our lives today.

Without a doubt, calculus is so awesome and its techniques so flexible and diverse that it trains you to look at problems, no matter how complex, as solvable until proven otherwise. Furthermore, the skill of twisting a problem over in your head, selecting an approach, and then working through a solution develops in you the ability to think clearly. As a result, the study of calculus is considered by many to be very important for improving your cognitive skills and is also a requirement for admission to most top universities.

A Physicist’s View of Calculus: What topics in calculus would physicists like to see their students learn? Here are some highlights - some areas of special concern to the study of physics.

How Calculus Became the Most Popular Class on Campus: A calculus teacher named Mr. Winn has made calculus the most popular class at the Crawford High Educational Complex in San Diego, California. Read on to find out how.

Calculus at MIT: Here is the selected relevant material from MIT's introductory courses to support students as they study and educators as they teach the AP* Calculus curriculum. This section is organized by the topics that you’ll see on the calculus exam at MIT.

What I’ve Learned from Many Years of Teaching Calculus to First-Year College Students: Joseph G. Rosenstein shares his experience of teaching calculus to students in their freshman years.

Visual Calculus: This resource contains a collection of modules that can be used in the studying or teaching of calculus.

Teaching Calculus Coherently: The Oregon State University’s guide to improving the teacher’s ability to teach calculus in a more effective manner.

Pitfalls in Teaching and Learning Calculus: Students studying calculus often make the same mistakes. Similarly, teachers teaching calculus have patterns of mistakes. This guide by Harvard University points out the patterns of these mistakes and is designed to help teachers and students eliminate these habitual errors one by one.

How to Teach and Develop Lesson Plans

The following is a brief guideline for developing calculus lesson plans and to make sure the students learn effectively:

  • Background - Make sure the student is ready. This means that the student has to have a background in basic algebra, geometry and trigonometry. A number of high schools teach basic preparatory courses that should be taken as well. A pre-calculus course (college level) may also be required. When you’re certain that the student is ready to learn calculus, each of the following steps must be completely understood before you allow them to advance to the next level.
  • Limits - Begin by determining the slope of a tangent. This is a very good way to explain the concept of limits, which requires the student to comprehend what happens when a number approaches but does not exactly reach some value.
  • Derivation - Establish the derivation of a function as the slope of a tangent, thus, allowing the development of the concept that the derivative is a representation of instantaneous change.
  • Infinite Series -Discuss the idea of infinite series next. This provides the basis for demonstrating how integrals are a means of measuring infinite sums.
  • Integration - Finally, study integration, which should be the last step of a 1st semester calculus course, and the basic theorem of calculus can now be brought forward. At this stage, the integral of a function should be the result of the area under that function when plotted on a graph.

 

Fun Ideas for Teaching Calculus: A set of fun ideas for teaching calculus to students so that a possibly dry subject can be made interesting and entertaining.

Calculus Connections - The Chain Rule in You: The Teaching Channel presents this wonderful video, showcasing a teacher’s modern and fun methods to teach students the chain rule of calculus.

Video Calculus â€" University of Houston: Explore a variety of key calculus concepts with the help of advanced video learning, simplifying the learning process.

Calculus Quizzes: Reinforce your learning with the help of these online quizzes, carefully crafted by the department of mathematics, university of Houston.

Teaching Calculus by Finger painting: This is an interesting interview with Vanderbilt mathematics professor Sal Froipol on his innovative use of finger painting to teach calculus. Read on and find out if the method works for you.

What’s Wrong with Calculus?: Jeff Knisley, along with Kevin Shirley writes about the common difficulties in calculus education and its long term impact.

Calculus in Context: A complete calculus course with a downloadable text book in parts, as well as in a single PDF file.

Calculus Facts

Calculus is among the most powerful and astonishing man-made inventions ever, yet it is a skill that can be acquired by anyone with an understanding of intermediary mathematics.

Among its many uses, calculus teaches you to:

  • Analyze a variety of scenarios involving instantaneous change, be it an accelerating rocket, the growth of a bacterial colony, or fluctuating stock prices.
  • Measure optimum values, for example, the maximum volume for a box with a given surface area or the maximum feasible net profit that can be generated on the sales of a given item.
  • Calculate irregular shapes; such as the volume of a donut-shaped object or the area of a piece of land surrounded by a body of water, such as a river.

Paul’s Online Math Notes: Here are the online notes from a teacher’s calculus I course that is being taught at Lamar University.

World Web Math: A nicely organized resource on calculus, describing different concepts in detail by breaking down the information in the form of easy lessons.

History of Calculus: Beginning from Isaac Newton to the advent of the most modern uses of calculus, this document provides a good summary of how calculus was first invented and how it got us where we are today.

Math Fun Facts: This archive is designed as a resource for enriching your math courses and nurturing your interest and talent in mathematics.

Applications of Calculus: Calculus.org is a wonderful resource to see what’s happening in the world of calculus and to learn from some very useful links provided there.

Why Calculus?: This essay by Eric Schecter provides a brief look at part of the history of mathematics and calculus.

 

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