Prepare for AP Calculus and More with Wolfram U
Today I am proud to announce a free interactive course, Introduction to Calculus, hosted on Wolfram’s learning hub, Wolfram U! The course is designed to give a comprehensive introduction to fundamental concepts in calculus such as limits, derivatives and integrals. It includes 38 video lessons along with interactive notebooks that offer examples in the Wolfram Cloud—all for free. This is the second of Wolfram U’s fully interactive free online courses, powered by our cloud and notebook technology.
This introduction to the profound ideas that underlie calculus will help students and learners of all ages anywhere in the world to master the subject. While the course requires no prior knowledge of the Wolfram Language, the concepts illustrated by the language are geared toward easy reader comprehension due to its human-readable nature. Studying calculus through this course is a good way for high-school students to prepare for AP Calculus AB.
As a former classroom teacher with more than ten years of experience in teaching calculus, I was very excited to have the opportunity to develop this course. My philosophy in teaching calculus is to introduce the basic concepts in a geometrical and intuitive way, and then focus on solving problems that illustrate the applications of these concepts in physics, economics and other fields. The Wolfram Language is ideally suited for this approach, since it has excellent capabilities for graphing functions, as well as for all types of computation.
To create this course, I worked alongside John Clark, a brilliant young mathematician who did his undergraduate studies at Caltech and produced the superb notebooks that constitute the text for the course.
The heart of the course is a set of 38 lessons, beginning with “What is Calculus?”. This introductory lesson includes a discussion of the problems that motivated the early development of calculus, a brief history of the subject and an outline of the course. The following is a short excerpt from the video for this lesson.
Further lessons begin with an overview of the topic (for example, optimization), followed by a discussion of the main concepts and a few examples that illustrate the ideas using Wolfram Language functions for symbolic computation, visualization and dynamic interactivity.
The videos range from 8 to 17 minutes in length, and each video is accompanied by a transcript notebook displayed on the right-hand side of the screen. You can copy and paste Wolfram Language input directly from the transcript notebook to the scratch notebook to try the examples for yourself. If you want to pursue any topic in greater depth, the full text notebooks prepared by John Clark are also provided for further self-study. In this way, the course allows for a variety of learning styles, and I recommend that you combine the different resources (videos, transcripts and full text) for the best results.
Each lesson is accompanied by a small set of (usually five) exercises to reinforce the concepts covered during the lesson. Since this course is designed for independent study, a detailed solution is given for all exercises. In my experience, such solutions often serve as models when students try to write their own for similar problems.
The following shows an exercise from the lesson on volumes of solids:
Like the rest of the course, the notebooks with the exercises are interactive, so students can try variations of each problem in the Wolfram Cloud, and also rotate graphics such as the bowl in the problem shown (in order to view it from all angles).
The calculus course includes 10 problem sessions that are designed to review, clarify and extend the concepts covered during the previous lessons. There is one session at the end of every 3 or 4 lessons, and each session includes around 14 problems.
As in the case of exercises, complete solutions are presented for each problem. Since the Wolfram Language automates the algebraic and numerical calculations, and instantly produces illuminating plots, problems are discussed in rapid succession during the video presentations. The following is an excerpt of the video for Problem Session 1: Limits and Functions:
The problem sessions are similar in spirit to the recitations in a typical college calculus course, and allow the student to focus on applying the facts learned in the lessons.
Each problem session is followed by a short, multiple-choice quiz with five problems. The quiz problems are roughly at the same level as those discussed in the lessons and problem sessions, and a student who reviews this material carefully should have no difficulty in doing well on the quiz.
Students will receive instant feedback about their responses to the quiz questions, and they are encouraged to try any method (hand calculations or computer) to solve them.
The final two sections of the course are devoted to a discussion of sample problems based on the AP Calculus AB exam. The problems increase in difficulty as the sample exam progresses, and some of them require a careful application of algebraic techniques. Complete solutions are provided for each exam problem, and the text for the solutions often includes the steps for hand calculation. The following is an excerpt of the video for part one of the sample calculus exam:
The sample exam serves as a final review of the course, and will also help students to gain confidence in tackling the AP exam or similar exams for calculus courses at the high-school or college level.
I strongly urge students to watch all the lessons and problem sessions and attempt the quizzes in the recommended sequence, since each topic in the course builds on earlier concepts and techniques. You can request a certificate of completion, pictured here, at the end of the course. A course certificate is achieved after watching all the videos and passing all the quizzes. It represents real proficiency in the subject, and teachers and students will find this a useful resource to signify readiness for the AP Calculus AB exam:
The mastery of the fundamental concepts of calculus is a major milestone in a student’s academic career. I hope that Introduction to Calculus will help you to achieve this milestone. I have enjoyed teaching the course, and welcome any comments regarding the current content as well suggestions for the future.