1. ## Functions

Alright, I'm trying to teach myself Calculus. It's an interesting development. But I guess I'm missing some mathematical background or I have the worst memory in the universe.

Here are a couple problems I'm having with the introductory topic in my Calculus book I bought for myself to learn from:

1. If f(x)=x-1/x show that a) f(-x)=-f(x), (b) f(1/x)=-f(x).
2. Let g(x)=x^3. Show that g(-x)=-g(x).

This one's kind of the same problem but yeah.

3. Let g(x)=x^4+2x^2+1. Show that g(x)=g(-x).

4. We learn that in trigonometry that sin x (3 bar symbol)sin(pi-x). Hence f(sin x)=f sin[pi-x]). Now let f(x)=x sin x. Then x sin x = (pi-x), or x=pi-x. Hence pi=2x and since x is any value we choose, so is pi. What is wrong?

I don't know why, but I don't understand certain parts of a problem. It's like steps are missing in my head. And the textbooks are unforgiving in their explanations because it seems like the parts I'm missing in my head never appear in textbooks, no matter how many prior ones I refer to so I can try to gain an understanding of what I'm missing. I must either be looking in all the wrong places or I have some missing link in my brain. It might be that I have a big problem trying to understand what they're exactly saying, because they're not clear enough for me. I don't know. Help?

2. It appears that you need an additional textbook in simple algebra.
These are just algebra problems. I will do the first.
$\displaystyle f(x) = x - \frac{1}{x},\quad f( - x) = ( - x) - \frac{1}{{( - x)}} = - \left[ {x - \frac{1}{x}} \right] = - f(x)$.

$\displaystyle f\left( {\frac{1}{x}} \right) = (\frac{1}{x}) - \frac{1}{{\left( {\frac{1}{x}} \right)}} = - \left[ {x - \frac{1}{x}} \right] = - f(x)$

3. Great. Apparently I don't remember Algebra then.

4. ## What are you using for a book?

The first three questions are probably drills to help the student remeber the difference between ODD and EVEN functions. It will be useful at some point later in the text, especially if the author writes something like, "and since f(x) is an even function, it follows that...."

I won't touch the third problem as I am much too slow on trig to be of help to you.

For what purpose are you studying calculus?

If you are in a hurry to solve problems that are most elegantly solved by simple calculus, there are many so-called baby calculus books around, usually called business calculus or in college catalogs, Calculus for Business and Social Science Majors. Some of them have sections on trig which you could use for the problems presented in the text.

Any little bit helps.

At this time I am using much simpler texts in order to aquire ideas to teach my daughter such that she sees calculus as an extension of algebra and geometry, and math in general as the language of science.

There is 300 years of calculus already. If you have a lot of free time, you will not be running out of calculus concepts any time soon.

Bye.

5. Hi Beautiful! Good luck with your studies!

Originally Posted by A Beautiful Mind
4. We learn that in trigonometry that sin x (3 bar symbol)sin(pi-x). Hence f(sin x)=f sin[pi-x]). Now let f(x)=x sin x. Then x sin x = (pi-x), or x=pi-x. Hence pi=2x and since x is any value we choose, so is pi. What is wrong?
Are you sure you typed this right? The statement in the fourth sentence doesn't make much sense. I mean, obviously the "proof" must be wrong since $\displaystyle \pi$ is constant, but I see not even an attempt to justify that $\displaystyle x\sin x = \pi - x$. Am I missing something?

Originally Posted by A Beautiful Mind
I don't know why, but I don't understand certain parts of a problem. It's like steps are missing in my head. And the textbooks are unforgiving in their explanations because it seems like the parts I'm missing in my head never appear in textbooks, no matter how many prior ones I refer to so I can try to gain an understanding of what I'm missing. I must either be looking in all the wrong places or I have some missing link in my brain. It might be that I have a big problem trying to understand what they're exactly saying, because they're not clear enough for me. I don't know. Help?
I suggest you go through a high school algebra or precalculus text first. Be sure to understand that there is no shame in it. The decision to self-study mathematics probably means you have all the necessary motivation and discipline to do well, but if you don't have the basics down you will not get anywhere. When learning calculus, it is absolutely essential to have a strong background in algebra, trigonometry, and some geometry.

You may be in a hurry to move on with calculus, but don't be discouraged: refreshing your precalc should not take very long if you are mature, disciplined, and focused.

6. ## So it really depends on what you want in math.

There is over 3,000 years of mathematics, and most of it of any significance was done in the last five hundred years (at least from my point of view).

Others may say it was mostly all done in the last hundred years, but I think that is due to the increase in population and availability of education. We still do things today that were conceived several hundred years ago, even if we do them in what we consider a more elegant manner.

There is so much mathematics already recorded that it is probably impossible for anyone to understand but a small fraction of the myriad (that's a number, you know, or at least it used to be) of maths currently being persued or applied. And much of it is being done on computers, which makes it even less likely that it is widely understood.

In my opinion, you could spend 3 months learning how to think with trigonometry, and probably forget it all a month later. That could be very frustrating if you found that it was needed next year for some other topic, like physics or calculus. Thus it would be good to decide what is your purpose for "learning calculus" and select your texts accordingly.

Discipline is necessary, but even more so, direction is necessary. One has to start at the right place with the right information so there is some sense of continuity in the lessons. Jumping all over the place in math is an amusing diversion, but it will no do for calculus. There is a definite chain of reasoning leading from algebra and geometry to calculus.

By the way, if you are interested in just how math comes about in the human mind, I suggest a fascinating approach from cognitive science, "Where Mathematics Comes From: The Theory of Embodied Mathematics" by Lakoff and Nunez. It is very easy to locate on the Internet.

Bye.

7. I agree with you.

I am also trying to master calculus and have a problem with some of the algebra manipulation techniques required. Can anyone recommend some good texts that I can use? I'm studying for my maths PGCE in Sept and although I'm confient of being able to teach GCE to 11-16 age group I have opted to do the 11-18 course and have just started a maths degree. My degree covered a lot of stats and I studied additional stats and math subsiduary modules. These covered some pure maths but were also predominantly stats based. Having started at level 1 degree standard my previous experience has left gaps in some of the required core knowledge.

Thanks