I learnt it the hard way. Since I took up business maths in high school (a whole lot of statistics and ratio analyses), I had to read layman's guides to Principia Mathematica, Cauchy's and Weierstrass' texts and common pop-sci/maths books (Ian Stewart, etc.) to really compensate for the lack of a classroom. Only the could I actually solve the end of chapter problems in the school textbooks.
Wouldn't it be easier to comprehend Calculus if schools start teaching Integration first and then Differentiation. It is difficult and there are lots more formulae to mug up, but wouldn't understanding "area below the curve" be sufficient compensation for all that trouble. And "rate of change of velocity" would so much easier to grasp as a "tangent of the curve"
Oh! And Bon Vivant, I didn't look it up till now, but http://en.wikipedia.org/wiki/Derivative
begins with "In mathematics, a derivative is the rate of change of a quantity. A derivative is an instantaneous rate of change: it is calculated at a specific instant rather than as an average over time." Do note the second and third sentences.
March 5th, 2007 2:30pm
>>> And "rate of change of velocity" would so much easier to grasp as a "tangent of the curve"
That's how we were taught it.
And the derivative of accelleration is "jerque."
March 5th, 2007 2:32pm
Ah yes, Wikipedia seems to be making the same error. I should think about correcting it.
Although people often think in terms of time functions when visualizing rates of change, it is not necessary for time to be the independent variable in a derivative; when differentiating a function like y = x^2 there is clearly no reason at all to presume that x is time.
March 5th, 2007 2:34pm
I'm having difficulty understanding your point, Bon Vivant. How can 'rate' not include time?
March 5th, 2007 2:38pm
I'm studying Calculus at the suggestion of someone else here to pick up Silvanus P. Thompson Calculus Made Easy.
He starts with differentiation as well.
March 5th, 2007 2:40pm
Rate does imply time.
Determining an instantaneous rate of change in a process is an example of differentiation.
Mathematically differentiating a function is more general than determining an instantaneous rate of change in a process.
March 5th, 2007 2:42pm
"layman's guides to Principia Mathematica, Cauchy's and Weierstrass' texts "
what titles are those?
March 5th, 2007 2:47pm
How come. Aren't the identical? Is there a function that is not a process? Conversely, can the there be a process that cannot be plotted, albeit, even partially so?
March 5th, 2007 2:50pm
Off my head,
Mathematical Thought from Ancient to Modern Times - Morris Kline
A History of Mathematics - Florian Cajori
Bertrand Russell - I inherited almost all his works from my granddad and dad. Other than *his* Principia Mathematica (available online now) most of his works on maths I picked from his books on philosophy.
Game, Set & Math - Ian Stewart
From Here to Infinity - Ian Stewart
March 5th, 2007 2:56pm
I don't want to go here, I can see trouble ahead.
Oh well, let me try.
A process is usually considered to be something physical that happens in the real universe of time and space. A process may be dynamic, in which case things change with time and it is meaningful to talk about time derivatives. A process may also be steady-state, in which case nothing changes over time, but things may certainly change with position and it is meaningful to talk about spatial derivatives. Most processes change with time and distance, so that derivatives exist in both the time and space dimensions.
A function is a mathematical construct within the rules of algebra. It is required to have no physical existence and exists only inside the mind of the mathematician. Differentiation then becomes a mathematical procedure applied to that function, again only inside the mind of the mathematician. In pure mathematics there is no concept of time or space, only algebra.
March 5th, 2007 3:04pm
>> I don't want to go here, I can see trouble ahead.
Agreed. That fear is shared.
>> Oh well, let me try.
>> A process is usually considered to be something physical that happens in the real universe of time...
>> A function is a mathematical construct within the rules of algebra...
None here as well.
>> In pure mathematics there is no concept of time or space, only algebra.
And logarithms. Derivatives using algebra cannot avoid logarithms and summing up requires discrete points. So your objection to my using "instant in time" would be withdrawn were I to use "discrete" instead?
March 5th, 2007 3:17pm
Derivatives can easily avoid logarithms. Take the y = x^2 example. Differentiating that involves no logarithms.
Differentiating does not involve any summing up. Differentiating involves finding a limit as something approaches zero.
A good alternative to "instant in time" might be "value at a point". "Discrete" is not right, since differentiation is all about continuous functions, and continuous is the opposite of discrete.
March 5th, 2007 3:23pm
The notion that anything that Bertrand Russell has written about mathematics is useful for learning anything is absurd. Calculus education could be vastly improved if instructors simply used the Silvanus P Thompson book, which Feynman himself used to teach himself calculus. Most calculus textbooks either assume the reader is mentally retarded, or already knows calculus.
March 5th, 2007 4:02pm
Calculus??? What's that? I learned in school to use hand-held calculus.
Math? easy! I make it every day.
I buy product for 2$ a piece and sell it for 4$ a piece and thus make 2% profit on it. Now that's business!!
March 5th, 2007 4:51pm
rate doesn't necessarily imply time.
It means "rate of change" of something, with respect to something else.
like "rate of change of height with respect to length" is a long-winded, but correct, way of talking about "slope".
March 6th, 2007 9:53am