first principles

[person]

i wanted to get a head start in math, so i started looking at a book that introduces its readers to calculus

as i was reading through the book, i noticed that it mentioned something called "first principles"

Quote:

Slope of the tangent to y=f(x) and (x1,y1) can be found through what are known as first principles.

i could not find any definition for this term, it would be greatly appreciated if someone would explain this term to me

[Cervantes]

First principles is a pretty vague term. It applies to lots of different things.

Here, it's referring to the first principle idea of differential calculus (which means finding the slope of a function at any given point).

We do this by picking our point P at the given point, and then picking another point Q somewhere else along the curve, a little bit away. The slope of the tangent at point P can be approximated by the slope of the secant from point P to point Q. Of course, if Q is a stretch away from P, it won't be a very good approximation. So what we do is we slide Q over closer and closer to P. In fact, we slide it infinitely close.

This is where the concept of limits come into play. You can work through the math from this first principle, using limits, to find the slope of a function at any given point.

Of course, where there are "first principles" to be heard of, there are faster ways of doing this. Instead of working through the math for all this, we can use what is called the Power Rule.

As an example, the Power Rule says that the slope of the curve f(x) = x^2 at a point x is equal to 2x. (We get this by dropping the exponent down in front of the x and reducing the exponent by 1.) This is much faster, but not first principles.

[rdrake]

Something like this... Cervantes?

code:

x = lim     f(x + h) - f(x)     h-> 0  ----------------                    h

At least, that's how I recall it being.

[mod:9ad579c5a0="Delos"]
Repeat after me:
"I shall hereby cease to refer to Cervantes by any other names other than Cervantes upon these forums."
He's very touchy about these things, and though nicks like "Clevernuts" might fly, the others shall not. Believe me, I've faced his wrath myself...just sparring you the pain, s'all.
[/mod:9ad579c5a0]

[Cervantes]

Yeah, that's the one. If you can work your algebraic magic long enough before you apply the limit, you can avoid a 0/0 answer and get something sensible, something spectacular.

Edit: Except it's not x=. It's slope=, or m=, or dy/dx=, or f'(x)=.

[person]

Quote:

As an example, the Power Rule says that the slope of the curve f(x) = x^2 at a point x is equal to 2x. (We get this by dropping the exponent down in front of the x and reducing the exponent by 1.) This is much faster, but not first principles.

how does the power rule work?

code:

x = lim     f(x + h) - f(x)     h-> 0  ----------------                    h

how did this equation come to be?

[Cervantes]

person wrote:

Quote:

As an example, the Power Rule says that the slope of the curve f(x) = x^2 at a point x is equal to 2x. (We get this by dropping the exponent down in front of the x and reducing the exponent by 1.) This is much faster, but not first principles.

how does the power rule work?

Re-read the bold, underlined, and italicized text.

So if you've got a function like

code:

f(x) = x^2

The derivative, which represents the slope of that function, will be written as

code:

f'(x)

(read: f prime at x)
And to get it, we drop the exponent, 2, in front, and then subtract 1 from the exponent, so we get

code:

f'(x) = 2x^1 = 2x

Or we can do it from first principles.

person wrote:

code:

x = lim     f(x + h) - f(x)     h-> 0  ----------------                    h

how did this equation come to be?

It came from this:

Cervantes wrote:

We do this by picking our point P at the given point, and then picking another point Q somewhere else along the curve, a little bit away. The slope of the tangent at point P can be approximated by the slope of the secant from point P to point Q. Of course, if Q is a stretch away from P, it won't be a very good approximation. So what we do is we slide Q over closer and closer to P. In fact, we slide it infinitely close.

We're calculating the slope between points P and Q. Plot a graph of f(x) vs. x. Give some values to P, and let Q be h units away from P, horizontally. For Q to still be on the line, we've got to adjust the vertical component of Q accordingly.

code:

P = (x_1, f(x_1)) Q = (x_1 + h, f(x_1 + h))

So to slide Q closer and closer to P, we apply the limit,

code:

lim h->0

So if slope is just ΔY/ΔX, then

code:

m = lim    f(x+h) - f(x)     h-> 0  -------------              (x+h) - x

And we can cancel the x's on the bottom, to get a denominator of h.