A set of ideas and operations used to manipulate functions. The two most basic manipulations are called derivatives and integrations. A ten-year-old child can imitate calculus using successive difference methods (though the ten-year-old probably wouldn't call it that). See also difference of powers. Successive differences are easiest to use for polynomials with integer-only coefficients (e.g. x1, x2, 3x5, etc.). For f(x)=x2 (x squared), the first n+1 values for f(m) where m=0,1,2,...,n-1,n are as follows:
0
1
4
9
16
.
.
(n-3)*(n-3)=n2-6n+9
(n-2)*(n-2)=n2-4n+4
(n-1)*(n-1)=n2-2n+1
n*n
To use successive differences, subtract any value from the value after it:
1-0=1
4-1=3
9-4=5
16-9=7
.
.
(n-2)*(n-2)-(n-3)*(n-3)=2n-5
(n-1)*(n-1)-(n-2)*(n-2)=2n-3
n*n-(n-1)*(n-1)=2n-1
This operation can be repeated as often as desired. Each successive difference set is a pseudo-derivative of the set prior to it. In this example, the next successive difference would be a series of 2's. Note that increasingly complex functions are increasingly difficult to get nice results from. Also note that whenever a successive difference set has values which are all the same (other than zero), they will very likely be related to the factorial of the order of the original function. This leads to some interesting factorial theory . . .
Seeing as this is a node about calculus, some mention should be made of the fundamental theorem of calculus (two parts):
Let f be a continuously differentiable function on a finite interval [a,b]. Then,
/ b
/
|
\
\ f'(x)dx = f(b)-f(a).
|
/
/ a
Here, f'(x) is the derivative of f with respect to x. The second half is similar:
Let f(x) be a continuous function on a finite interval [a,b] and define F(x) in the following manner:
/ x
/
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\
F(x)= \ f(t)dt.
|
/
/ a
Then F is continuously differentiable on [a,b], and F'(x)=f(x).
On the surface, these may seem to be obvious, but they must be meticulously proven at some point or other before mathematicians trust them.