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

17Calculus Article - Learning Math and Science

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On this page we discuss techniques, tools and ideas to help you learn math and science effectively and efficiently. Most of the discussion refers to math but the techniques can be applied to physics, engineering and most hard sciences as well.
Math and science are not difficult, they just require some different study techniques than for other subjects and some persistence.

Before we give you specific advice, let's answer a few questions that many math and science students have.

Why should I learn math/science when I will never use it?

This is a great question! I mean, why study 'useless' material that you will never use, probably don't even like and is not that interesting?

Well, here is why. When you read, hear, see or learn something, the chemistry of your brain changes. These changes help you process other things in other areas that you may not expect or even ever know. For example, learning to play a musical instrument helps you learn more easily. That's right. We discuss it in more detail on the music and learning page. Your brain is changing constantly, controlled by what you put in it, including music you listen to, movies you watch, friends you hang with and books you read. Most of the time, you will not be aware of the changes but they happen.

Here is a good article by one of our favorite bloggers, Scott H. Young entitled Why Learn 'Useless' Things? that you might enjoy. This article applies to language learning but toward the end he discusses other areas of learning.

Learning math, calculus and science, is training your brain to think in a more structured way. The information may or may not be important. But the activity of learning this type of material changes how you think, not just what you think. You will be able to learn other topics more easily and more deeply if you take the time to learn math now.

None of us likes to waste our time. We want to be doing meaningful, fun and fulfilling things, including activities that will get us somewhere. So, do not fret. Learning math, calculus and science is not a waste of time. You are training your brain and how you think. It's hard but it's worth it.

Can I use my calculator in calculus and higher math?

Calculators are incredibly helpful when you need to do calculus. Most graphing calculators have the ability to not only graph but also do derivatives, integrals, limits and many other operations. And the calculators are very reasonably priced, most between $100 and $150. If you consider the processing power and technology involved, that is incredibly reasonable. I am a big fan of calculators. I prefer the HP RPN calculators. But the TI graphing calculators are easier to use for graphing, in my opinion.

However, using a calculator extensively while you are learning calculus concepts can be detrimental. I am going to depart here from the general consensus about using calculators while learning calculus. Many teachers believe that we should integrate technology in the classroom as much as possible. I agree ... to a point. I think we reach a point where the technology gets in the way of learning concepts.

I believe the use of calculators in calculus should be limited to a graphing calculator that is used only for graphing. All other use should be restricted until you learn the concepts. This usually involves pencil and paper combined with repetition and feedback. So, put down the calculator! I know it's harder that way right now but by exam time you will know the material better than the other students in the class and you might even get an A!

However, if you are required to have a calculator, here are some suggestions.

General Math and Science Study Techniques

Before I give you my suggestions, here is a great video for you to watch.

Prof Leonard - How to Study for Math [13min-29secs]

video by Prof Leonard

Here is a page with some other math study suggestions. SLU Success in Mathematics

In this section, we are going to give you some ideas for how you can learn math and science and get a better grade.

Most people don't know that they need to study math differently than other subjects, like history or English. As you read through this, I think you will see that learning math (especially calculus, higher math and science) requires different study techniques.

1. In general, do not fill your brain with useless special cases. It is MUCH easier to remember the general rule and then apply it to special cases. Unfortunately, mathematics textbooks are filled with special cases. So you will need to weed out what is a special case from what you REALLY need to learn. An added complexity is that most teachers teach special cases since it is easier for them. You need to watch for this and learn the concepts, not the special cases.

2. Consider learning math like you would learn a foreign language. I know, I know, you are already thinking it IS like a foreign language, right? Well, you are correct! Math has a language of its own and learning it is more like learning a foreign language than you might think. If there are words you don't understand, look them up in a math dictionary. Look up terms or concepts online. See how other books and teachers teach the same concept. If you are going to spend a substantial amount time learning math this semester, you may as well do it right.

3. Use reference materials, help labs and other students liberally (but ethically). Ask your teacher lots of questions. If you skip material that you don't understand, it will come back and bite you, I guarantee it. Math builds on itself. (This is one big difference between Math and History or English.) So if you miss one small concept, you will miss the next one that builds on it and on and on it goes. It's kind of like driving a car, going somewhere with directions. If you miss the first turn, the rest of the instructions will not make sense and you will never find your destination. So, get help often.

4. If you are nervous about taking calculus, you are perfectly normal! Most people are nervous. I mean, look at the name of the class CALCULUS. Even the name is scary. However, you can do it. Most of the problems you will have when trying to understand this material are not that the material is difficult. It's that you have decided in your mind that it is difficult. Convince yourself that you can do this, that you will do well in this class. Don't let the past, other people or anything tell you you can't!

5. For some extra help with calculus, get the books How to Ace Calculus: The Streetwise Guide (for Calculus 1) and How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus (for Calculus 2 and 3). These are good books that condense the concepts down to what you need to know and presents ideas in a readable form. I have read them and found them helpful. You may be able to find them at a local library. I know you can get them on amazon for about $10 each. They are two of the books in our recommended booklist.

6. Books to NOT get: I do not recommend books that have the word dummy or idiot in the title. You are not dumb or an idiot. Don't put that message in your brain. I also do not recommend books that talk about making math easy. It's not easy, but it is understandable and you can ace it if you decide to.

7. The best way to learn math is ... are you ready for the secret? ... do you REALLY want to know? ... okay, I will tell you. The best way to learn math is to practice, practice, practice. There are plenty of practice problems in your textbook and on this site.

8. Another key to doing well in math is to decide to learn the material. Now that sounds kind of obvious but think about it. Are you in the class to just get through it so that you can get on with your life? If so, you need to rethink that. Decide to learn the material. Don't just do the minimum and try to figure out the problems you are required to do, so that you can get a good grade. If you focus on the grade, you won't do as well as if you focus on learning the material. If you learn the material, the grade will be there. If you focus on the grade, you may not learn the material and, I guarantee you, your grade will be lower. Now, come on, I'm not that naive. I know you think the goal is the grade. But just try it, okay? What do you have to lose?

9. Cramming for math (especially upper level math) exams doesn't work! Oh, it might have when you were in high school or maybe with algebra. But it won't work here. I have found a lot of similarities between studying math and training for a marathon.

Marathon or Math?

This graph is a visual representation of what preparing well for a math exam looks like. The horizontal axis is time and the vertical axis is intensity of preparation or how much time you spend preparing.

Basically, the raw data, which I downloaded from the Internet, shows how many miles you need to train to prepare for running a marathon. However, studying math is a lot like preparing for a marathon. Some people just work a little bit until right before the exam and then they cram and expect to do well. This works when all you have to do memorize equations or something boring like that. Not so with math. You need to train.

Imagine a marathon runner waiting until the week before the marathon and running 50 miles a day to cram in the training. It doesn't work, does it? (And it's kind of silly to think that he can do well!) It’s the same with math. If you have prepared, done the homework, worked practice problems, worked the practice exam problems, then you can relax, go into the exam with no stress and do great. Sounds good, doesn't it?

Here are some more study suggestions.

10. Work on math 6 days a week and take one full day off to rest your brain. If you can, take the day off from all school, work and responsibilities. Go jet skiing. Lay out in the park and get a tan. Watch movies. Read a novel. It's not about what you do. It's about what you don't do on that day. Again, I do this (on Saturdays). I know it works.

11. Allow larger blocks of time periodically during the week to work on math. A minimum of 1.5 hours will work. However, 2-3 hours at a time is best. You need to study 6 days a week. I suggest you take one day off to relax and refresh yourself. Even though you are not working on math, your mind is still working without you knowing it. So rest is very important. If you workout or do weight training, you know what I mean. Your mind works the same way. If you can't study every day, plan ahead to the next time you can study so that you get the homework and exams done well before the time they are due.

12. Work the assignments and try to get them done at least one day before they are due. When you work hard on something, your mind continues to work on it when you are sleeping or doing something else. Sometimes you will have an insight or realize you did something wrong in your work. Getting it done early gives you time to correct mistakes before turning it in.

13. Keep up on your homework, your studying and your exams. If you don't, you will fall too far behind to catch up. Don't skip any work and if you don't understand something don't skip it! This is critically important to your success in this math.

14. Ask for help often. Unlike other types of courses, math builds on itself. If you don't understand something, chances are that you will need it later to understand something else. If you get lost early, you will be struggling later.

But, no matter what, DON'T GIVE UP!

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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