Goetz's online calculus course setup mangoroot logo


Calculus I developed by Dr. Arek Goetz is a fully online course. It is run only through an acredited by Western Association of Schools and Colleges university. Since its inception over 1000 students have successfully completed the class. The course isopen to all students with sufficient background in algebra, notions of functions, and in elementary trigonometry. Welcomed are students from domestic or international universities as well as professionals who are preparing for graduate schools. The course has been particularly popular among students preparing to enter medical schools as many medical schools require at least one semester of calculus, pharmacy, and MBA programs.

Transcripts & Transfer of credit units

Transcripts, whether official or unofficial are only available through the university through which the student takes the course.

Prerequisites and Pretest

To qualify for the course students need to take this self pretest based on algebra, notions of functions, and basic trigonometry. Before committing to the class all students should review algebra.

Letters of recommendation to your future school

In the past students competing for admission to top medical schools or MBA programs, requested letters of recommendation. Professor Goetz will gladly write a letter if requested by a student whose contributions to online learning are substantial and meaninful.

How is this program different from other online calculus courses?

The class offers 4 semester units. Dr. Goetz's online calculus course is in 11th semester of existence. Hundreds of students from major US universities and some from other parts of the world successfully complete the class every year. The course is designed and taught by a research mathematician with an National Science Foundation track record. The communication among students is facilitated by an intuitive formula generator - Students focus on exchanging ideas, not on learning how to type set mathematics.

Stimulating social learning experience

The course has 25 new lessons. Students spend on average between 6-9 interactive hours per module. In each of the 25 lessons, students (i) interact with a high quality video lecture and at times take a quiz directly related to the lecture. The lectures can be either streamed in high definition over the internet, or preloaded for an off line viewing on laptops, ipads, and android phones or iphones. At anytime students can pause the lecture and ask questions on the forum monitored by the instructor. (ii) Students interact with the live course notes and read the textbook. (iii) Students begin solving homework problems on first their own and then engage in an online discussion with the instructor and other students by offering hints and asking questions. (iv) Submitted homework assignments are human graded and promptly returned with instructor or instructor teaching assistant's comments. Correct solutions to the assignments are then provided. The course combines the best proven methods in a traditional mathematics class together with a new social online learning environment that links students located at remote locations. The forum is actively moderated by the professor.


Students meet online. There are no required in class meetings. There is one proctored event: a comprehensive final examination. Students take the exam at their home institution or work place, or at other locations with a proctor at their locations. The course offers a flexible schedule. The student does not have to be present at specific hours. However, there are homework deadlines and the student needs to commit to working on the course and participating in discussion at least once every two or three days. Most students need between 5 to 7 hours per lesson - for most lessons. This includes all: interacting with the video lecture, analyzing the notes, thinking and working on homework problems, and sharing ideas on the forum. (The summer session is intensive: there are 3 lessons per week, 2 lessons in the Fall and Spring).


The final course grade is based on homework (including quizzes), the student's online contributions, and the final exam. The computer picks the weights in your favor according to these ranges. (a) homework includingquizzes (50%-60%)(b) a comprehensive final examination (20%-30%) (c) the student's online activity (remaining weight) - responses and hints to other students' questions, discussions, etc.  For example, if the student does not perform well under the time pressure of the final exam, you could minimize the final exam contribution (down to 20%) and instead ensure that your homeworks and your online contributions are meaningful and substantial. In trully exceptional circumstances the professor may apply a slightly different grading scheme. A final course grade can only be assigned if there is a record of all three: a proctored final, a record of homeworks and the student's consistent online activity.

Is this course for you?

Students who succeeded in this course tend to be independent, self-motivated, being able to adhere to a regular schedule, students who were comfortable with algebra. Post baccalaureate students preparing for medical schools and MBA as well as high school students passionate about math also tend to do well. The online course is not easier than a regular class, but it is often more convenient. Students who are looking for a course that offers an easy A with little learning should not apply, nor should students who event remotely consider submitting somebody else's work. In cases of proven cheating, we remove the student from the class without a refund and we notify the student's school. In cases where the final grade needs to be based on a limited or nonuniform track record, we investigate the integrity of the records, for example we schedule a skype/video conference with a student. Please read what students who succeeded in this class before are saying. Also, please note that in the summer the course is quite intensive and there are three lessons per week.

About the author

Dr. Goetz is an active researcher in Dynamical Systems, an academic instructor, and an e-learning entrepreneur. Goetz graduated from the University of Illinois at Chicago in 1996 with a doctoral degree in mathematics. Since then he held visiting positions at Boston University, the University of Exeter, University of Marseille, Institut des Hautes Etudes Scientifiques (IHES) in France, and Instituto Nacional de Matematica Pura e Aplicada (IMPA) in Brazil. Dr.Goetz holds a tenured full professorship at San Francisco State University. He is a recipient of two National Science Foundation Grants for his research in Dynamical Systems with geometric singularities. He delivered over a hundred research talks at conferences and universities worldwide. An experienced academic instructor, Goetz is also the founder of Mangoroot, a multimedia internet platform for communicating and learning mathematics over the internet. The platform was used to successfully teach over a thousand of calculus students located in various locations in the US and abroad. His passion to convey mathematics to his students takes him to remote places on the planet such as volcanoes and glaciers in Ecuador where he shoots multimedia calculus clips which are then incorporated in the teaching content.

Syllabus: Learning outcomes and list of topics

The central object of the study in calculus is the concept of a function. Functions are used to describe the real world around us. Calculus I introduces two fundamental concepts which enable us to describe and investigate functions. These are the derivative and the integral. The derivative describes how a function changes at a particular time. The integral carries information about the history of a function. Both, the derivative and the integral are defined using limits. Calculus I covers: Limits, Continuity, Asymptotes, The tangent problem, Rates of Change, Derivatives (including trigonometric and transcendental derivatives), Graphs, and their shapes, Optimizations, Riemann Sums, Integrals including two parts of the Fundamental Theorem of Calculus.
  • Review of quadratic, exponential and logarithmic functions
  • Two fundamental problems in Calculus and Introduction to limits
  • Limit Laws
  • Continuity
  • Long term behavior and asymptotes
  • Rates of change
  • Derivative
  • Derivatives of Polynomials and the Exponential Function
  • The Product and Quotient Rules
  • Derivatives of Trigonometric Functions
  • The Chain Rule
  • Implicit Differentiation
  • Derivatives of Logarithmic Functions
  • Exponential Growth and Decay
  • Minimum and Maximum Values
  • The Mean Value Theorem and Shapes of a Graph
  • Local Extrema and Inflection Points
  • L'Hospital's Rule
  • Shapes of Graphs
  • Optimization
  • Antiderivatives
  • Estimates of areas under graphs
  • Definite Integrals
  • The Area Function
  • Definite Integrals and Antiderivatives
  • Applications of the Fundamental Theorem of Calculus

    Learning outcomes.

    At the conclusion of the course, students will be able to :


    The studnet be provided with a free set of interactive multimedia notes and the student will to print the pdf lesson notes (about the total of 120 pages). The notes are the base material, not a full textbook. In expensive used major textbooks should be used to supplement the notes.