Tuesday, September 15, 2009

Calculus in 5 minutes (last part)


I am unable to resist saying a word or two about how we get the standard method. I can't emphasize enough that you don't need to know this to use calculus! But it might be interesting. Here goes...

Remember:

y  = xn
y' = n xn-1


Where does this come from?

Well, what is the slope of the curve at any point x? Imagine that we increase x by a small amount, we'll call that amount h. The slope of the curve is the difference in y [ the value of the function f(x) ], between the two points x and x + h. How much does it change?

Suppose:

y = x3

slope = [ f(x + h) - f(x) ] / h
f(x + h) = (x + h)3 = ?


Well,

(a + b)n is the binomial expansion of a + b
(a + b)n = an + n an-1 b + stuff times b2 + more stuff


So...

(x + h)n = xn + n xn-1 h + stuff times h2 + more stuff


And there are three quick points:

• (1) The first term xn is what we need to subtract, it is f(x)

• (2) The third and following terms involve h2 (and higher powers). We ignore these terms!

 Suppose h = 10-6, then h2 = 10-12
Not small enough?
Suppose h = 10-60, then h2 = 10-120
Not small enough?…


The point is that h2 goes away faster than h as h gets very small, and if you don't think it's getting small fast enough, just choose h to be even smaller...

• (3) What remains is the first term of the binomial expansion:

  n xn-1 h


To get the slope, we need to divide by h:

 slope = n xn-1 h / h


The h / h part is still equal to 1 even when h gets very small. So it goes away!

 slope = n xn-1


There you have it.

As I said, this is for my son. Here is a picture from an earlier time:


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