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# Single-variable calculus learning recommendations

This page provides subject-specific learning recommendations for the subject single-variable calculus. See all our learning recommendations pages |

This page lists recommendations specific to learning single-variable calculus (called *calculus* for short here). This includes limits, differential calculus, integral calculus, and some auxiliary topics. This combines the calculus-related items in our online mathematics learning resources and the calculus-related items in our mathematics reading recommendations and sorts them based on your educational stage.

## Contents

## First-time learning

You're interested in self-studying calculus (roughly at the level of Advanced Placement BC calculus in the US), but with a deeper conceptual understanding and/or better problem-solving skills. The self-study could be undertaken prior to studying the subject at school, in parallel to school, or in place of studying the subject at school. The following are some high-quality resources that can offer you a relatively complete experience. We haven't made an attempt to be comprehensive here, but have listed the best sources.

If this description fits you ... | ... this might be the best recommendation | Cost |
---|---|---|

You have an interest in solving challenging mathematical problems and discovering mathematical ideas through the problem-solving process. | The AoPS calculus text: Calculus by David Patrick, ISBN 978-1-934124-24-6, paperback. You can also enroll in the AoPS online calculus class, though that costs more money. |
~$45 for the text, ~$10 for the solutions manual. Current price for semester-long AoPS calculus class unavailable, but comparable precalculus class costs $425 ($478 with book). |

You prefer to learn from videos that offer conceptual explanations of ideas, and you don't have an aversion to long videos. | The Center of Math calculus videos (here) and their corresponding textbooks (store). These introduce ideas in a concepts-first fashion. | The Worldwide AP Calculus book (that covers all topics) costs $14.95 digital and $39.95 print. Videos require a decent Internet connection, but we expect the Internet connections of most clients to be adequate. |

You just want something to get you started. | Almost any book will do. Here's the calculus online textbook by Gilbert Strang that's reasonably good. Stewart's book is also good, but expensive. For video-based learning, consider Thinkwell if you're willing to pay money. |
Strang's book (digital version) is free. Thinkwell costs ~$150 for year-long course access. |

## Acquiring greater depth in calculus topics you already learned

You have already learned calculus at the Advanced Placement level or equivalent, but you feel that your understanding of the subject was inadequate. You want to acquire greater depth of understanding in some areas of the subject.

**The first-time learner recommendations can be used for relearning**. As a relearner, however, you have the advantage that you can also rely on resources that are very good for some important parts of the subject, even if they are not comprehensive and/or they are not uniformly good. Some such recommendations are:

If this description fits you ... | ... this might be the best recommendation | Cost |
---|---|---|

You are considering studying higher math and want to relearn calculus in a way that prepares you for higher math. | Calculus, 4th Edition, by Michael Spivak is a good choice. This text is used in Honors Calculus classes in many universities. | Expensive, ~$75-100 |

You like a nice integrated course experience with professionally done videos, quizzes, and lecture materials. | Calculus: Single Variable by Robert Ghrist (University of Pennsylvania) on Coursera | Free |

You are looking for detailed explanations of a few important topics in single-variable calculus, without necessarily getting comprehensive coverage of all topics | Vipul's Classroom has video playlists on some calculus-related topics, including limits, many subtopics of integration, differential equations, some subtopics of sequences and series. You might also be interested in these notes by Eus. | Free |

It's also worth remembering that *many texts have conceptual explanations that may not be emphasized much by the authors* (because they do not expect the majority of students to be interested in these explanations). If you are sufficiently interested, you can and should read these explanations. Some explanations may be deferred to the appendix.

## Testing and honing your knowledge and skill through practice

### Practice at basic computation

- You can use any calculus text. The calculus online textbook by Gilbert Strang, available online for free, is a good start. Answers are available to odd-numbered problems at the back of the text.
- For practice with full-length exams, consider the AP exam practice and this page (many links within the latter page are broken).
- If the problem lists in your calculus text are insufficient, you can also buy The Humongous Book of Calculus Problems (~$16 paperback) (and Schaum's 3000 Solved Problems in Calculus (~$15 digital, ~$23 paperback) to get a
*lot*of practice with calculus problem-solving. These books cost some money but are not too expensive. - Although existing problem lists in books are usually more than enough to give you practice and gauge your skill level, you can also in principle vary the numbers to create your own problem variants, then check your solutions against a calculator or Wolfram Alpha.
- If you have Wolfram Pro, you can try the Wolfram Problem Generator.

### Practice at solving tricky problems that rely on knowing calculus well and being able to recall it quickly

Tournaments conducted by universities such as Harvard, MIT, Stanford, and Rice can be a good source of calculus problems. Typically, these problems:

- require you to know calculus well enough to be able to recall relevant facts quickly based on pattern-matching,
- test for general problem-solving skills,
- do not generally require very messy computations, and
- do not test for very deep conceptual understanding.

Particular tournaments:

- Harvard-MIT Math Tournament (2010 and earlier years)
- Rice Math Tournament
- Stanford Math Tournament

### Testing and practice of deeper conceptual understanding without requiring very advanced general problem-solving skills

- Vipul Naik's teaching: Vipul Naik has put up quizzes for single-variable calculus courses he taught. These are focused on testing and improving conceptual understanding of specific topics, and they do require somewhat more advanced problem-solving skills than typical calculus classes, but not at the level needed for tournaments.
- Many standard calculus texts, such as Stewart's text and the Salas-Hille-Etgen text, have a number of advanced problems in their exercises. Teachers often omit these problems when assigning homework, but if you have a calculus text, do consider looking for the advanced problems in the text.

## Note on the Khan Academy

A number of people choose Khan Academy as a source of supplementary learning for all academic subjects, including calculus. We believe our recommended resources are generally better than Khan Academy, regardless of your goals. You might also be interested in reading our discussion of Khan Academy here.

If, however, our previous suggestions didn't fit your needs well, and you have found Khan Academy to generally work well for you, we recommend considering PatrickJMT (patrickjmt.com) as a free but somewhat superior alternative to Khan Academy that is very similar in style. Another alternative is Calculus One from Coursera. Both of these have the advantage over Khan Academy that the instructors have relevant mathematical background as well as teaching experience, so they are more likely to make accurate statements that are reflective of the deeper logic of the subject.

## Related recommendations

- Somewhat lower: High school mathematics learning recommendations, including algebra learning recommendations and precalculus learning recommendations.
- Somewhat higher: Lower division undergraduate mathematics learning recommendations, including multivariable calculus learning recommendations and linear algebra learning recommendations
- Mathematics learning recommendations (full list)
- Online mathematics learning resources (full list)
- Mathematics reading recommendations (for casual mathematics reading)