\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Learning and Study Techniques

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On this page you can find general college learning and study techniques. We give you ideas on how to improve your understanding of information in less time, as well as links to other pages with more specific information. These techniques can be applied to most any college course. But first, this important concept:

studying ≠ learning

Knowing how to study is important but knowing how to learn is what makes studying effective. So what is the difference and why should you care? Because time is your most valuable resource and it is possible to learn more in less time and then have more time for other important things.

One of the most helpful books we have read recently is the book The Practicing Mind: Developing Focus and Discipline in Your Life - Master Any Skill or Challenge by Learning to Love the Process, which we highly recommended to help you focus and be able to learn, not just study.

Differences Between Studying and Learning

Let's look at some of the differences between learning and studying.


mostly done on a regular basis in your early life, up through college and then sporadically through the rest of your life


something you do for your entire life


usually associated with an intense period of conscious concentration


constantly ongoing subconsciously


repeating something a few times


involves doing something repeatedly until it becomes ingrained


understanding it in your head


knowing it as a part of you

If you have ever had one of those 'AHA!' moments, you have experienced understanding. If you have ever had a deep feeling of satisfaction when you have applied your knowledge to a new problem that you have never seen before after lots of work and persistence, you have demonstrated that you learned something. So how do you know when you move from studying to learning?

Free Online Course: Learning How To Learn

There is a great free online course through Coursera that we highly recommend called Learning How to Learn: Powerful mental tools to help you master tough subjects.
One thing we really like about this course is that, not only do they explain why it is important to learn how to learn, but they give practical, realistic techniques on how to do it.
Check out the online class yourself. If there is not a current session, put the course in your watchlist and you will be notified when a section opens up.
Here are several good quotes from the course.

'The retrieval process itself enhances deep learning.’
Barbara Oakley [from the 2-4 Illusions of Competence video]

'Recalling material when you are outside your usual place of study can also help you strengthen your grasp of the material.’
Barbara Oakley [from the 2-4 Illusions of Competence video]

Take Responsibility For Your Own Learning
'Approaching material with the goal of learning it on your own can give you a unique path to mastery. Often, no matter how good your teacher or textbook are, it’s only when you ‘sneak off’ and look at other books or videos that you begin to see what you learn through a single teacher or book is a partial version of the full three dimensional reality of the subject which has links to still other fascinating topics that are of your choosing.'
Barbara Oakley [from the 4-3 Change Your Thoughts video]

Specific Learning Techniques

Okay, so you are now in college. You got through high school, maybe even successfully. You think you know everything there is to know about studying and learning. But I can tell you there is more. You can improve your study techniques, learn more easily in less time with less effort and do better in school than you have up until now. You can also learn to teach yourself and be more independent. You will need to put in a little more effort up front but it will pay off big time later.

The first thing you need to do is to understand and believe that you can learn how to study better. You need to decide that you WANT to. If you think you know everything about studying or you don't want to learn more, you won't, so just stop reading now and go back to what you are doing. HOWEVER, if you really want to improve your study skills and you know that you can, then read on. Here are some videos and comments that we think will help you.

YesReneau - How to Get a Perfect on ANY Test - YesReneau

video by YesReneau

The Greatest Brain Exercise Ever - Kyle Cease

Okay, so . . . I am not a fan of positive thinking. It doesn't work for me and I don't really think it works for most people. However, I do believe that what we think affects our performance.
Here is something that I believe works. It is not positive thinking but it does change the way we can think to improve our learning. This is a short video (only about 7mins long), so take a few minutes and watch it. Then try it.

The Greatest Brain Exercise Ever - Kyle Cease

Think101x, The Science of Everyday Thinking

Okay, so now after the previous topic, you should be ready for some specifics on what to do and how to learn more effectively. Let's start with a set of videos from a course I took recently on edx called Think101x, The Science of Everyday Thinking. We recommend you that you look into the course yourself for lots of good material. We have set up a youtube playlist with the videos associated with study and learning techniques. Do not skip these videos. These techniques could change your life.

playlist: Think101x, The Science of Everyday Thinking

What Do Top Students Do Differently?

So have you ever wondered what the 'smart' or 'successful' students do differently that make them so successful? I know I have. They don't really have secrets per se. They just have a couple of techniques that work. Here is a good video that talks about those couple of techniques as well as what doesn't work.

What do top students do differently? [14mins-37secs]

You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)


\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)


\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)


\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)


\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)


\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)


\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)


\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)


\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)


\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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