Back to Calculus: Single Variable Part 1 - Functions

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Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
In this first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O....

YG

Jul 2, 2018

Very well structured for a refresher course. Thank you Professor Ghrist for your effort in putting this course together. A little additional outside research was required but well worth the effort.

NL

Feb 9, 2021

Excellent introduction to Calculus, I wanted to review the material to tutor my child but I am very happy that I learned a whole new way of looking at Calculus.

Thank you so much Prof. Ghrist.

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By Andrew P C

•Jun 4, 2020

sucks

By Raghu P

•Apr 19, 2020

This course gives a completely different perspective of calculus, which we thought we had already learnt in our colleges. Professor Robert Ghrist explains the finer aspects of analysis of mathematical series. In fact, this course starts with Taylor Series, which is usually taught at the fag end of the undergraduate courses of Calculus. The capability to expand a function into Taylor Series, and then manipulating just enough number of its items to get our results; expanding Trigonometric, Exponential and Hyperbolic functions to make them amenable to mathematical analysis we want to do, etc. have been the main focus area of this course. It is not an easy course to take if you are not fully mathematics oriented. I was not. And that's why I had to spend hours and hours trying to figure out how the homework problems can be solved. Solving those problems gave different perspectives of the beauty of mathematics, which we might have ignored while studying to acquire just a degree in mathematics. Thank you, Professor Ghrist. :)

By Luoqi W

•Aug 30, 2017

Absolutely phenomenal class. Prof. G leads the students in a tasteful exploration into the world of calculus with a curious heart and methodical mind. Lectures are grouped into weeks as mini-series and usually begin with a thought-provoking question or statement. These are immediately followed up with definitions and principles that lay the ground for further examination. But fear not if abstractions aren't your expertise, an army of concrete examples accompany you every step along the way.

Personally I completed this course on Coursera the summer before I entered college in 2013 as an electrical engineering student. To my amazement, even today in 2017 after all those years' bombardment from college level courses I am still revisiting Single Variable Calculus and find it a fresh breeze of air. Thank you Professor G and your team for creating what will be remembered as not just a great math class but a wonderful intellectual adventure.

By Matthew R P

•Jul 21, 2020

Although I've only taken calculus 1 before, beginner calculus, and many moons ago, I thought the visual pictures as well as examples provided were very helpful. Dr. Ghrist is right in the curriculum introduction to the course, it is very hard, at least for me. I had particular trouble with combined and substitution functions, possibly because I've never had it in a math course. But a friend explained them to me, and that made the difference to me. For me I wish professor Ghrist would have gone over those subjects more. Nevertheless, professor Ghrist visual examples really were good for substitution and combination functions when I understood them better. Overall, I understand why Dr. Ghrist had the order of subjects in the course. The course overall was good, albeit very hard.

By Thairo M T d S

•Jan 9, 2017

É incrível ter a experiência de poder realizar um curso tão rico e bem fundamentado de cálculo. Desde o começo, há uma ligação entre os assuntos, descobrindo não como ocorre em algumas aulas de cálculo que certos teoremas "saem da cartola" como mágica, e sim com a ajuda de algumas equações que, a partir do desenvolvimento algébrico delas, abrem o universo das séries de taylor. O que, sem dúvida, culmina na excelente demonstração do lim x->0 (sen x)/x e de L'Hopital.

Porém, é necessário, como o prof. G diz, já ter visto cálculo e ter uma certa maturidade. Recomendo fortemente um caderno ou um bloco de anotações, será bastante útil tanto como recordação ou como material para revisar. Excelente curso!

By Nachiket D

•Feb 15, 2020

I absolutely loved this course!! Dr. Ghrist is an amazing teacher -- I wish I had him for calculus in college! I am truly grateful for my newfound fundamental understanding of Taylor Series and the key contributions of the exponential function, which underlies so much of mathematics. On a personal note, this course has really prepared me well as I make the transition from aerospace engineering to graduate studies in climate and weather science, renewable energy, and robotics. It will be especially helpful in the areas of numerical/computational fluid-thermal dynamics and control systems engineering. On to Part 2!

By Kumail A H

•May 22, 2017

Excellent course! Prior exposure to Calculus (familiarity with a couple famous derivatives and integrals) is helpful, but you're sure to learn much more than that in this course. I have deepened my understanding of limits (not just the notion of a limit, but working out limits is now more intuitive to me) and stuff like l'hopital's rule seems logical now.

Starting with infinite series, with contrast to leaving it at the end, which is the standard in such courses, was brilliant!

Highly recommended!

By Omar H

•Mar 8, 2017

The perfect course for someone with the basic knowledge of calculus and doesn't want to spend much time on the boring basic concepts (I am grad. student and need some calculus review). It is a genius idea that this course started with convergence and Taylor series, it made my life easier and simplified so many other things. The only part that I struggled with was the big "O" notation part, I feel it needs more elaboration. Other than this, everything is great. Off to the next course !

By Alvaro U M

•Jul 19, 2020

I will write in Spanish.

Me gustó mucho el curso, aunque lo encontré difícil para mí. Hacía más de 40 años que no veía estas materias, por lo que tuve que esforzarme y estudiar mucho. Encontré interesante el enfoque poco ortodoxo de comenzar con Taylor Expansions. Me habría gustado que los apuntes de Penn Calc se hubiesen podido leer mejor ya que aparecen escritos como parte de un programa. Me siento muy satisfecho y en un par de semanas voy a seguir con la segunda parte.

By mrunmoy s

•Jan 10, 2020

This course was quite fun to go through. You need to know basics of differentiation and integration. Some of the key points that I learned are Taylor expansions, l'hopital's rule, limits and orders of growth and how to compare two functions using their orders. I absolutely enjoyed the assignments and challenges. Some of them were quite tough but going through discussion forums helped to get an idea about the approach to solve them.

Thank you for this Course!

By John C

•Jun 3, 2020

A course that is best taken after already learning a fair bit of Calculus. I would suggest not taking this course until you have completed a first-semester course and are already somewhat familiar with integrals and summations. With all that said, despite feeling rather confident with my prior abilities, this course has greatly enriched my understanding of many topics in Calculus. It has given me strong tools that I had not been exposed to previously.

By Muhammad Z

•Aug 8, 2023

It is and always a classic course in single variable calculus. Though, as mentioned by Prof. Ghrist, not a course to brush up your previous knowledge.

It is a course for someone who wants to have a subject knowledge at its depth.

I have some suggestions also:

It would be more handy for students , if we have some extra material like solved problems, some tips etc regarding the interval of convergence in Taylor's series and the material of BIG O.

By Mike R

•Oct 1, 2019

I found it very enjoyable and challenging. A lot of effort has been put into its production and the questions test your understanding properly. Its not a beginners course, but it is doable if you have a reasonable grasp of calculus to begin with. I am looking forward to working through the multi-variable calculus ebooks and videos (aka calculus blue) that have been released (not on coursera). Good luck on this course, it is worth the effort.

By Naveen G

•May 13, 2020

Thanks to Prof. Ghrist and team for this wonderful (and of course challenging) course. This course helped me review calculus differently through the use of Taylor series. The problems composed for the quiz were challenging and fun. Although some concepts like Big O could have been explained better (personal opinion), the course lived up to its promise.

I'm truly grateful to Coursera for making this quality course available on the platform.

By Yakovenko I

•Mar 8, 2022

Very informative and insightful course. Visualisations and overall presentation are outstanding. Course material is provided in an innovative way and can be a perfect addition to a classic rigorous calculus course. Many real-life and beautiful examples are given, which support motivation and interest to move further along the course. However some prerequisite knowledge in calculus is required to fully enjoy the course material.

By Chris F

•Nov 5, 2017

I found this course to be very stimulating. It has been a while since I have studied Calculus. I took it a while ago and also have taught it but this is wonderful and frustrating at the same time. Some things you think you know, you need to go deeper . The use of the Taylor Series was interesting and for me different, but I see how it all comes into play and am looking forward to other sections that w ill hopefully follow.

By Gabrielalejandro Q

•Sep 11, 2018

a great course to look at calculus in a different perspective, prof ghrist became one of my fav teachers as he paint with a subtle beauty the mathemathics he explains i'm thrilled to continue this journey of math thanks to him, and this is probably the go to class to expand and deepen on the wonderfull world of Taylor series and a functions in general, also free quizzes, lots of them!

By Albert Z

•Mar 3, 2020

Very great course! It was nice to be able to refresh AB and BC Calculus material after having not touched it for a year. In addition to reviewing what I already knew, this class featured additional information about calculus that was brand new to me. Every lecture taught me something I had not already known before. I feel prepared for what comes ahead in college mathematics now!

By Gregorio A A P

•Jul 8, 2017

Excelente, felicitaciones , solo que es triste no poder disfrutar al 100% un curso de esta calidad al no estar traducido al español, le agradecería que por favor lo traduzca del ingles al idioma español ya que solo esta parcialmente traducido.

nuevamente felicitaciones por la gran didáctica con la que imparte el curso y sobre todo por la calidad con la que enseña.

By Robert H

•Nov 26, 2018

Great course! A particularly novel and extremely well-done presentation of the basics of calculus, part of a multi-course sequence that covers the whole of single variable calculus. Also worth a serious look is Professor Ghrist's material on multivariate calculus, available on You Tube. See his Penn web site for more details (http://www.math.upenn.edu/~ghrist/).

By Hemant P

•Jul 1, 2017

This course really helped me push my limits(forgive the pun) with highschool math. It was challenging but not so much that it demotivated me! Moreover, I watched many lectures twice and thrice and tried replicating the examples demonstrated in each lecture. It really helped me drill in the fundamentals I needed. I can proudly say I know Taylor Series now!

By Nafis F

•Aug 8, 2020

For people with absolutely no knowledge of calculus : Quite difficult to follow. Not for you.

For people with absolutely some knowledge of calculus: TAKE THIS COURSE, if you have the time to do so. It does take a lot of work. But I enjoyed all of it.

By Макарова Ю А

•Mar 28, 2022

Very nice course. I've been searching for a calculus course for some time as I want to try a completely new sphere for me and I needed something just above school program, but also not too advanced as I graduated from school 10 years ago and don't remember everything. I am really grateful for these lectures and tasks, it was actually fun. Thank you!

By Hoagy C

•Sep 29, 2017

Very well explained, well set out, interesting concepts - highly recommended. Would have liked it if the 'Challenge' question sets were a little more thought provoking rather than (often) just requiring more computation - there were some interesting ones (the ln(lnx) question comes to mind) but didn't add as much depth as they could have.

By Emmanuel A

•Dec 9, 2020

I have really enjoyed this course; calculus with UPenn is the best. Thank you a lot especially to Professor Robert Ghrist. I appreciate your teachings and everything you did for me. Infact, I have really understood the practical aspect of mathematics. To all my coursera friends, you're at the right place with University of Pennsylvania.