Help differentiating

J

jwpaine

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Mar 10, 2007
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Hello.

I am teaching myself Calculus.

I am starting with differentiating simple algebraic expressions, and need a check / help, thanks!

\(\displaystyle \L y=x^{13}\)
\(\displaystyle \L y+dy = x+dx^{13}\) neglecting the small quantities of high order I get:
\(\displaystyle \L y + dy = x^{13} + 13x^{12}dx\)

subtract the original \(\displaystyle \L y = x^{13}\):
\(\displaystyle \L dy = 13x^{12}dx\)
divide both sides by dx

\(\displaystyle \L \frac {dy}{dx} = 13x^{12}\)

so, the change in y over the change in \(\displaystyle \L x = 13x^{12}\)

right?



now how would I do it for this: \(\displaystyle \L y = x^{ \frac {-3}{2}}\)

y + dy = 1/((x+dx)^(3/2))

I don't know where to go from here.
 
Hey JW:

Do it the same. Take the exponent up front and subtract 1 from -3/2

You have \(\displaystyle \L\\\frac{-3}{2}x^{\frac{-5}{2}}=\frac{-3}{2x^{\frac{5}{2}}\)

Since you are here and into it, take some time to learn a little LaTex. Your posts will look so much nicer. Just a thought. Click on quote at the upper right hand corner of my post to see the code I used to display what I did.
 
jwpaine said:
I am starting with differentiating simple algebraic expressions, and need a check / help, thanks!

y=x^13
y+dy = (x+dx)^13 neglecting the small quantities of high order I get:
y + dy = x^13 + 13x^12*dx
Click to expand...
May I ask where did you get that quaint language for differentials? It has not been used since these seventeenth century. In fact it is the very language that George Berkeley called “the ghosts of departed numbers” in his devastating attack of Newton’s idea of infinitesimals.

It is admiral that you want to learn calculus. It is not so admiral that you trying to teach it to yourself. You run the risk of getting erroneous ideas such as “neglecting the small quantities of high order” that will cause you real difficulties in the future.

If you do decide to continue teaching yourself, the please get a good standard calculus textbook. That is a book that would not use wrong language when talking about limits. After all derivatives are limits! You do understand that don’t you?

There is even a good book that is free for downloading:
http://www.math.wisc.edu/~keisler/calc.html
 
I am using the book: "Calculus Made Easy"

It was recommended to me by a couple people on this board.

It does, in fact, use this terminology...and I don't see anything wrong with it: for example: with y + dy = (x+dx)^3...it end explains that when dy and dx are both made infinitely small, (dx)^2 and (dx)^3 will become infinitely smaller by comparison..so we regard them as negligible. How is that quaint?
 
I don't remember recommending that, but I do have a copy laying around somewhere. I believe it was written in the early 1900's, which is why it may use some anachronistic terminology. Actually, I would suggest the same as pka and get a more modern text. You know, a calc book I have about worn out is Calculus by Anton, 5th ed.
 
jwpaine said:
I am using the book: "Calculus Made Easy"
Click to expand...
Go out at and burn that book! Continued use will spoil you for calculus forever!

jwpaine said:
It was recommended to me by galactus and soroban.
Click to expand...
No serious mathematician that I know would make such a recommendation. I say that having taught mathematics for forty years at the university level; I have also chaired a division of Mathematical Sciences.

Please. Please, do yourself a favor and do it right the first time. Get rid of that book and get a real calculus text.
 
I am almost done precalculus and I am hungry to start learning Calculus before university starts, this fall.

What do you recommend for a text than, for someone starting on their own? Something I can buy and carry in my backpack.

I'm not about to give up on learning, independently, over the summer..and will take advice on a different book.

"The economics of publishing compels authors to add every topic that anyone might want so that no one can reject the book just because some particular item is omitted. The result is an encyclopaedic compendium of techniques, examples and problems that more resemble and overgrown workbook than an intellectually stimulating introduction to a magnificent subject"

What do you recommend that would be good for a highly motivated and eager to learn, student out of precalculus (called algebra III at my HS)
 
jwpaine ...

Unlike other people on this board, I do not have an extensive "mathematical" pedigree. I only have a Bachelor's degree in Math with a Physics minor. I've spent most of my life applying math and physics flying aircraft and driving ships for the US Navy. I have been teaching calculus and physics for the last 12 years since my retirement from active duty. I read the book in my freshman year of college to gain some insight into the calculus beyond that delivered by an overworked and underpaid graduate teaching assistant. I now recommend the book to my students who seek a different tact from the common "classroom" approach. Many of those same students have gone on to earn various levels of degrees in all varieties of fields from prestigious universities.

Read some editorial comments about the book ... there is also a link to consumer reviews. http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480

Better than that, read the book and form your own opinion. You will not warp any mathematical sense you have by reading it. Feel free to compare it to other books. You'll find that most math textbooks are no substitute for a good teacher.

You sought, and I gave some advice about a book that one may seek calculus knowledge on their own. Yes, the recommendation is my opinion.

Sometimes those that reside (or have resided) in the ivory towers of academia think that their way is best. Yet I fail to see how any enlightened person can ever recommend burning a book.
 
skeeter said:
Better than that, read the book and form your own opinion. You will not warp any mathematical sense you have by reading it. Feel free to compare it to other books. You'll find that most math textbooks are no substitute for a good teacher. You sought, and I gave some advice about a book that one may seek calculus knowledge on their own. Yes, the recommendation is my opinion.
Sometimes those that reside (or have resided) in the ivory towers of academia think that their way is best. Yet I fail to see how any enlightened person can ever recommend burning a book.
Click to expand...
It is the difference in wanting students to learn calculus as opposed to just applying calculus. I have long doubted the wisdom of teaching calculus in high schools for that very reason. It is no doubt the high schools can learn calculus but it is highly doubtful that it is being taught correctly.

I have painful memories of dealing with many a distraught freshman (many times with a parent) who was failing calculus. The story line was the same. “He had an A is calculus in high school. And everybody knows that Mr X is a better teacher than anyone you have here. Mr X cares so much about the student’s seeing how it all works and where it is used.” Yes Mr X taught limits as a simple operation of plugging in a value for the variable. So when asked to think more conceptually about limits the reply was “but Mr X made it so easy”. Now this may well be the ivory tower of academia think, but when such young person fails and fails again because of what he thought he already knew, I think that Mr X is to blame for blocking the person from succeeding. Calculus is the gatekeeper course to most careers in any technical field today. To fail at collegee level calculus is to have that gate forever closed. Therefore, a bad start in calculus at any level is block to success. There is no easy way out: Calculus can not be made easy.
 
As I stated previously ... I gave my opinion, and now you have given yours. Let's end it on that note.
 
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