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How to properly study for Calculus? Any tips?

  1. Sep 8, 2013 #1
    Hi, I think this belongs in this thread?

    I have a very weak math foundation stemming from high school. My Algebra II teacher was pretty bad, but it was my fault that I did not put the effort in to learn and teach myself Algebra II. Then my geometry teacher was an easy A because she was new to the job. Precalculus was an absolute disaster where I would constantly not understand material because I slept through lectures (plus my high school had a pretty bad math faculty in general).

    Now it's come to bite me in the butt in university. The only classes I've ever failed (two, to be exact) are math classes. I got an F in precalculus (and found it entirely useless in actual calculus) and I believe I have also failed Calc II--grades have not been curved yet, but I am very sure it's not passing. My Calc II course was a mistake for me to take because, instead of the normal quarter session of 10 weeks, I took it in summer school and it went at a faster pace since the session was only five weeks. I was too confident that I could pass.

    I think my problems are:
    1) I do not have much of a foundation for any higher math.
    2) I stupidly jumped into calculus after not having taken math for two years and failing precalculus in high school.
    3) I only memorize instead of applying conceptually.

    I can put the effort into going to office hours and doing practice problems, but how do I get away from the mindset of memorizing to actually being able to apply it conceptually? Any help/resources would be nice and please be honest. :) Thanks!

    P.S. To clarify, I did pass Calc I and Calculus classes are the only ones giving me problems. My professors teach physics algebraically.
  2. jcsd
  3. Sep 8, 2013 #2
    I would go through the derivation of the formulas step by step. Get a note book, write down the derivation in detail. Write it out step by step force you to follow the thinking logically instead of trying to remember. Writing it out is like teaching yourself the subject and you'll find out whether you really understand.

    You cannot take Cal II in the summer class, it's too condense and short. You fall asleep once, you'll get lost and you have no time to catch up. Calculus takes time to absorb and sink it.

    Depends on your major, if it is physics or engineering, I strongly suggest you to review Cal I and make sure you understand the theory and not by memorizing. Those are foundations that you need in the future. If you really fail Cal II, make sure you review Cal I before re-taking Cal II again. You want to go slow to make sure you really get it. Particularly if you are weak in algebra and geometry, you are going to have to stop and catch up those along the way.
    Last edited: Sep 8, 2013
  4. Sep 8, 2013 #3
    Hey, thanks for replying!

    Writing the derivations step-by-step is a really good idea; do you happen to have any resources, though? Like an online book or tutorial? My math textbook is really terrible with this. I use Calculus, fifth edition, by Deborah Hughes-Hallett, et. al.; published by John Wiley & Sons, Inc.; 2009 for the whole calculus series. My TA would skip around while explaining homework problems (he was kind of a jumpy one) step-by-step.

    Yeah, I realized that now after making my mistake, unfortunately. At least I learned! I initially only took it because it was a concurrent requirement for the first physics class and I was lagging behind on my major requisites.

    My major is human bio and I'm taking the "regular" calculus instead of the one for engineering/science majors, since that more difficult series isn't required. Calc III would be my last required math class for my major. I'll review algebra because I'm pretty good with geometry since it's more visual, but I was also wondering if I should teach myself trigonometry? I was actually supposed to take that in HS instead of precalculus, but that class got cut so I had to go into precalculus. Also, how different is Calc III from Calc II and I? EDIT: I did some research before and saw that Calc III was more like 3D Calc I?

    Sorry for all the questions! Thank you so much for the time; it really means a lot.
  5. Sep 8, 2013 #4
    This is a good book:https://www.amazon.com/Calculus-Analytic-Geometry-George-Thomas/dp/0201531747/ref=sr_1_1?s=books&ie=UTF8&qid=1378625028&sr=1-1&keywords=thomas+and+finney+calculus. Get the used one for less than $15. I have very good luck with used books on Amazon. You should be able to study on your own.

    Cal III is multi-variables calculus, which is 3D. I don't know exactly what Human Bio major is. But it does not sound like you need a whole lot of calculus. But Calculus is the language of science like English for everything else. Upper level physics and electromagnetics in EE are explained in calculus. Completing Cal I II and III will open the door for you in a lot of majors in case you change your mind.

    You should review trigonometry as you should see Cal II involve quite a bit of trig., So is Cal III. You should enroll in the real calculus, those non science calculus might be even harder to learn as they try to water down too much.
  6. Sep 8, 2013 #5


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    I think it is extremely important to review algebra and to learn trigonometry. It's a pity you have limited time because that non-science-major calculus is going to be about memorizing and looking up formulas and I think that makes calculus more difficult to learn, there are just too many formulas to take in in a short time.

    Search for past posts of mine in the Book section, you will find recommendations for algebra and trig books. Definitely that is where you must focus now, even if it means taking a slight timeout from your studies, usually you are allowed to take an extra 2 years to complete your degree. The reason I say this is because I foresee that you will struggle to get through calculus and will eventually drop to a humanities-only degree instead. I think that can be avoided.

    This is the Guidance section and this is my guidance, learn algebra and trigonometry (not precalculus, as you know it is irrelevant stuff). Get to know the syllabus, you won't need complex numbers or matrices but the rest, learn it. Take the time to do that, either by enrolling in a college algebra class or through self study with help here and using online videos/sites as well as books like I have recommended before.

    PS. If you look at reviews on this site, you'll see that that Hughes-Hallett book is not highly regarded.
    Last edited: Sep 8, 2013
  7. Sep 8, 2013 #6


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    One of my sons had an experience like yours ... then he had an epiphany and went back to the beginnings (algebra I) and re-taught himself everything through pre-calc/trig. It took about 6-8 weeks in the spring of his junior year in high school. I helped him a bit with calculus over the summer, but he mostly taught himself.

    He mostly used this online site:

    Pauls Online Math Notes: http://tutorial.math.lamar.edu/

    He then aced AP calculus in his senior year, and went on to become a math major in college; he is now a junior actuary. Prior to his epiphany about math he had planned to be a fireman - he thought he was a poor student; he then realized that learning is something that _you_ do. Teachers are just a guide.
  8. Sep 8, 2013 #7
    Paul's online calculus is very good.

    As I said, re-study Cal I is important. I did badly when I was in college because my heart was not in it. A few years ago, I re-studied start at page 1 of the calculus book, it is very quick. Foundation is very important, you cannot skip trig and basic calculus and hope you can do good in Cal II and III.
  9. Sep 8, 2013 #8
    I'll go get that book then; thanks!

    Human Biology is just heavy biology/chemistry. I only take physics and calculus for one year each and then the rest of it is chemistry and biology. I'm pretty sure I'll need calculus for future physics classes and maybe some of population biology and physical chemistry, but that's about it.

    Yeah, I learned my lesson when I went through Calc II.

    The thing is, if I'm already struggling in normal calculus, I can't make it in scientific calculus. My friend, who passed her AP Calculus BC exams with a five and did well in the normal calculus classes is already struggling with the first part of the scientific calculus series. I'm not qualified to take it either (I only passed the qualifying exam to take regular Calc I).

    I actually understood why we used limits more than why we used integrals and then my whole mindset went down in the dumps, as in I kept thinking, "why is this useful?" I think part of my troubles are due to how I approach math; I only hate it because I don't do well in it, which is not a good excuse at all.

    Skipping trig was not something I could help in high school because it got cut (my HS was very poor), but I'll be sure to study it now.

    Thanks for your help and your honesty!
    Last edited: Sep 8, 2013
  10. Sep 8, 2013 #9
    I actually have pretty good memory so I can memorize formulas or write them on the allowed cheat sheets--it's just that I have no idea how to apply them. I can do pretty standard differentiation just fine, but deviate from that a little or force me to manipulate an equation, and I'll have no idea how.

    Definitely! I dropped physics and calc III to replace them with easier classes in order to allow myself time to study math and get a tutor. I am determined NOT to drop to a humanities-only degree because I need calculus to get into med school, which is my ultimate goal. The thing is that, as you said, I don't have too much time since I'm already behind. I only need to get through calc II and calc III and I'll be finished with math classes for the rest of my degree.

    Algebra and trig, got it! Geometry, too, I'm assuming? I can't afford a college algebra class and my uni doesn't offer one (lowest math is precalculus), but tutoring and self-study, I will do.

    P.S. I realized that my math book was really bad from Calc I, but it's the default book for all of the calculus classes here, sigh, so I can't avoid it. Thanks for your help!
  11. Sep 8, 2013 #10
    Thank you for sharing this with me. I actually felt really alone because everyone around me in university did not really struggle with their math classes like I did, both in HS and university. Hearing somebody else do the self-teaching and succeed means a lot to me!

    I actually use that site quite a bit and it is useful!

    Thank you for the motivation; I'm determined not to give up on my degree and med school, especially since calculus classes are just a relatively small part of it. Maybe, when I see my dad, I'll ask him for math help, haha.
  12. Sep 8, 2013 #11
    Don't be scared, Study the scientific calculus is not necessary harder. They go into more details so you have a better chance to understand it. I hate those so called watered down classes for the non major, they usually end up harder to learn because they skip too much, they don't derive the formulas and just give you the formula, you use it without really understand it.

    I was a bio chem major, I did not even have to have the real physics class that use calculus. I was in the same boat as you. I regret I took the easy way out, cause me a lot of time to catch up with the calculus later on. Take the real calculus class, ask question here and people are happy to help.

    Calculus is an acquired knowledge, it takes a lot of effort to get start on it. Because you only enrolled in the non science calculus class, you don't know your potential. I TA a nursing students' chemistry lab and I had to give lectures every week. I read the book and It was hard......not that the material was hard, the book skip so many steps that I cannot imagine the girls can understand studying that book. Those poor girls were struggling. I had a hard time following the book even though I knew the materials. So just because you did bad does not necessary you are not good, you need to give it another chance. This time study hard, remember the learning curve is very steep.

    Without a good foundation of Cal I, it is not surprising you don't do well in Cal II.
    Last edited: Sep 8, 2013
  13. Sep 8, 2013 #12
    Hm, I'll ask my friends if I can take a look at their textbook and notes and see the differences for myself, if that's okay. Yeah, that's been the pattern so far, but there are so many risks that I have to calculate between switching between maths.

    Yeah, my first physics class didn't use any calculus, but the textbook did. I don't think I can take the science calculus yet because I don't qualify, but I'll see if I can later on. Thanks so much for the help!
  14. Sep 8, 2013 #13
    If you are really that afraid, here is a calculus book that is the easiest to study for Cal I and Cal II:https://www.amazon.com/Calculus-Analytic-Geometry-Stein-Sherman/dp/0070611750/ref=sr_1_1?s=books&ie=UTF8&qid=1378701750&sr=1-1&keywords=sherman+stein

    I used this book to study Cal I and Cal II. This is the very kind of book that is very easy to start, but gets bad towards the end. Typical of books that is written too simple and disintegrated on the difficult part. For cal I and II, it's pretty good. Get it used, you can even buy the solution manual so when you work on the problems, you can check the steps.
  15. Sep 8, 2013 #14


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    Calculus 1 through 3 requires very good to excellent skill and understanding of Algebra. You need to be good with intermediate Algebra.

    PreCalculus is NOT a waste of time. Often you can do well in Calculus without some of it, as long as you are excellent in Intermediate Algebra. PreCalculus often has more advanced topics that can help you with dealing through topics of Calculus. You may also begin a study of limits of functions, designed as a way to give a bridge to Calculus 1. PreCalculus also includes most of the important topics of Trigonometry, so you have the chance to build or rebuild your trigonometry knowledge before you need it in Calculus 1&2.

    You had a passing or acceptable grade in your Calculus 1, but you in fact may NOT have truely passed the course. A most probable reason you did not do well in Calculus 2 & 3 is that you do not know Calculus 1 well enough. The grade you earned in Calculus 1 means very little.

    Relearn Elementary and Intermediate Algebra and become excellent with this intermediate level. Relearn all of Calculus 1; you can learn these things much, much, much, much better the second (or third) time through, as long as you are thorough. Whether you restudy these in class or on your own is whichever way you choose, but you need to restudy them.
  16. Sep 9, 2013 #15


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    brikayyy complains this way:

    That is almost certainly false. You could very well find that "scientific Calculus" is taught more thoroughly, and conceptually better developed. If you have good enough Algebra knowledge, you should be able to study Calculus 1 for scientists and engineers well enough.
  17. Sep 9, 2013 #16
    I'm not sure if your precalculus class differed from my own, but I found that I very rarely ever used anything covered in it in my calculus classes, except for knowing the natural logarithm/base e exists and a few trig functions/unit circle. I feel it can help with trigonometry, but not much else--I could be very wrong, though. I just haven't had it proved otherwise.

    Yeah, you can fly by some of these calculus classes depending on the professor. I think I only passed because more people did worse than me due to the curve. This is why I don't feel like I'm well-equipped for Calc II and Calc III, though I haven't taken the latter one yet.

    Thanks for being specific and giving me help! So most of algebra, trigonometry, and then calculus 1. I'll browse through the book thread for recommendations about these.

    Is it? I don't qualify to enter this scientific calculus, but I do want to ask my friends about it now.
  18. Sep 9, 2013 #17
    I have been saying the exact same thing. You might want to bone up with the trigonometry though and algebra. But go direct to the scientific calculus.

    You said you pass Cal I, is that the scientific calculus? If not, then just study the trig and algebra on your own, then enroll in the real calculus I.
  19. Sep 9, 2013 #18
    Could you please give an example of an exercise or problem that you have trouble solving so that I can try to guess where is the source of your doubts (or confusion)? I mean -- if it is in the concepts,
    ability to work with the formulae, lack of understanding of prerequisite topics, physical and/or geometrical interpretation of concepts, etc.

    Anyway, I think a good understanding of the real number system, elementary functions and the notion of convergence (the latter is central) is what makes a Calculus course be meaningful. With that knowledge acquired, a (basic) Calculus couse should not be too difficult when properly taught and understood.

    Just another tip: you don't need to memorize all the formulas that turn up in your book. Follow these steps:

    1) Understand the concepts and reread/ask questions until they are meaningful for you. Writing down derivations won't help if you don't understand what's going on.
    2) Solve easy problems (purely mathematical immediate applications) attentively and do not feel satisfied until you grasp why and how you did all steps.
    3) Progress in the level of difficulty, not escaping from hard problems (usally long calculations or proofs). They will make you feel confident when you finish them, even though they will take a longer time and thus seem a waste of time at first.
    4) Clearly understand physical and geometrical interpretations of the concepts (and sometimes of problems and their results). Always ask yourself: can I apply what I learned in Calculus to something I'm learning in another subject? The answer will often be yes.
    5) Only memorize the most important formulae. In fact you'd better memorize and understand theorems rather than formulae. Other formulae that appear in your book will sometimes be easily deduced from the most important ones. For example, instead of memorizing the derivative of an exponential of any base, memorize only that of base [itex]e[/itex], namely [itex](e^x)'=e^x[/itex] and use the definition of exponential and the chain rule (a theorem) to obtain that [itex](c^x)'=c^x\ln c[/itex].

    I hope that helps.
  20. Sep 9, 2013 #19


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    I would argue that the only thing making calculus difficult is the algebra and trig knowledge required and the number of formulas. Otherwise it is contained and, while difficult in itself for the algebra skill required, is progressive in that what one learns gets used later when integrating. Let's not build it up into something huge and scary.

    Although, multivariable calculus is huge and scary. It is a big obstacle in the way of Brikayy's progress right now. Ideally he/she must get the algebra/trig knowledge and then learn calculus again, with help here and MIT's video lectures, to get a good knowledge of calculus so that Calc III will feel like a follow on.
  21. Sep 9, 2013 #20
    When I picked up calculus after being away for like 25 years, I had not been using algebra or trigonometry for long long time. I was not particular good at it. I started from page 1 of the calculus book, when I encounter trig and others, I just stop and learn or review those. As long as OP is willing to spend more time, budget the time that he/she need to stop and review the basic stuff, I think he/she can go to Cal I, just don't go with the summer class that is short. You can't rush calculus.

    Advantage is he/she took the basic calculus before, something much rubbed off from that. Also, there is a pretty good supporting system here that you can ask questions.

    I actually heard a few people consider Cal II is the hardest because of the series. I feel only the last part of Cal III is harder, the line integrals, Stoke's and Green's theorem. The first 3/4 are piece of cake.
    Last edited: Sep 9, 2013
  22. Sep 9, 2013 #21
    I've never really thought it is such a monster. In fact, usually multivariate calculus is much about repeating univariate concepts [itex]n[/itex] times. The only problem here is that it demands other prerequisites. In fact, besides having properly learned univariate calculus, one should also have a deeper knowledge of analytical geometry and linear algebra. But some calculus programs don't even mention linear algebra and the result is that students can't apply the chain rule and get scared of Jacobian and Hessian matrices. Besides, a little more physics should help interpret line and surface integrals.

    So it is not calculus what makes multivariate calculus more difficult, but the fact that it is a part of a bigger mathematics that should be learned together.

    Again, I don't think emphasis sholud be given on "algebra skill", but on understanding the concepts.
    All the so called "algebra" involved in a basic calculus course is just a mishmash of a dozen basic properties (axioms) of the real number system and their consequences. The only "grown-up people" algebra that shows up occasionally is polynomial division. But that is very analogous to integer division, which is very basic stuff (although sometimes not properly taught and understood).

    So don't be afraid of calculus. Besides, you can review prerequisites while you study it.
  23. Sep 10, 2013 #22
    On an unrelated subject. I thought ODE is so so much hard the the Cal I, II and III combined. That was just shocking. This is the only one that I actually took the class. I got the first in the class, BUT I spent 5 to 6 hours a day, 6 days a week studying that. A few other students said they spent more time on this single class than the rest of the classes in the semester combined together. I since studied PDE on my own, yes it was hard but nothing shocking like the ODE. I actually studied the chapter before the class, sat in the first row, did all the homework!!!! That was just shocking. Lucky for OP, he never mentioned this is the requirement!!! He only need 3 classes instead of 4!!!
  24. Sep 10, 2013 #23
    Basic ODE and PDE themselves are not that hard because really hard equations just cannot be solved explictly and it is just a matter of either resorting to numerical methods or qualitative theory. Again, I think people get stuck because they try to memorize rather than understand.

    More advanced DE courses are difficult because they require functional analysis, Lebesgue integration and spectral theory. But don't worry: it all starts with basic stuff which is not too hard and, depending on your carreer, you won't need to go so deep in studies.
  25. Sep 10, 2013 #24
    I have been around on this board for quite long, seeing that many schools in the US bring up a whole lot of wonderful programs for a plethora of courses of different disciplines, I am really amazed. The course work for an engineering major I took 8-10 years ago in a Japanese school didn't go into much detail to categorize or subdivide maths related subjects engineer students needed to take as the OP explained. Right before I graduated, thinking that I would be leaving the school soon I was eager to learn a lot myself and discovered many new branches I wasn't heard about, e.g Algebra (applied in vector space, kernels for SVM etc), Calculus (series analysis for signal/sound/speech processing etc), and even a great deal of necessary parts in statistics and probability (decision making based on Bayesian method, different distance methods for data processing and comparing), etc. I was disappointed and sad.
    That is lucky of you, I should probably hang out with them during office hours over as many questions and issues I run into as I can if I were you.
    Last edited: Sep 10, 2013
  26. Sep 10, 2013 #25
    I think it's opposite, ODE was so hard because I tried to study the derivation of all the different ways to solve the ODE. This is one that you almost have to look up the formulas and find the one that fit. Then the series solution.............Particular our ODE was a 4 unit class, we had Fourier and Laplace transforms in ODE class. It even has basic PDE about separation of variables. That's why I said it has so much stuffs.

    Numerical stuff are the hard ones. I spent half the time studying Bessel's and Legendre function which are all numerical methods. I don't know how's the other colleges, but San Jose State spend only 2 weeks or so on these topics. I think they are skipping a lot of the hard stuff. I spent a lot more time on these. Getting through these, PDE is easier than ODE for sure. PDE does not have all the different varieties as ODE. Fourier and Laplace transform are not that bad particularly it was taught in ODE class already.
    Last edited: Sep 10, 2013
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