Books/MathForTheLayman/Anti-Derivative and Integral

18. Anti-Derivative and Integral

18A. Introduction

Just as the operator d.1 can be applied to a function f to obtain its derivative, so the operator d._1 can be applied to obtain its anti-derivative, that is, a function whose derivative is f. For example:

      derv=: d.1
      anti=: d._1

      c=: 1 2 3 4 5 6
      c with p.
   1 2 3 4 5 6&p.
      c with p. derv
   2 6 12 20 30&p.
      c with p. anti
   0 1 1 1 1 1 1&p.
      c with p. anti derv
   1 2 3 4 5 6&p.

Exercises

   Experiment with the operators derv and anti on functions such as ^ and 1&o. and 2&o. and 6&o.

   Compare the derivatives of the functions 1 2 3 4 5&p. and 7 2 3 4 5&p. , and explain their agreement.

18B. Area under a graph as a function

A study of the graph of Section 3B (an approximation to a circle) suggests that the area under the graph of a function f from the argument _1 (or any fixed point) to an argument x is itself a function of x (that depends upon the function f).

We will first make a similar graph, using a finer grid (with a spacing 0.05 between points):

      a=:  i:1j40
      PLOT a;cir a

What is the rate of change of the area function?

Consider the point x=: 0.5, the spacing s=: 0.05, and the next plotted point x+s. The change in area is due to the quadrilateral with base s and heights cir x and cir x+s, an area equal to s times the average of the heights cir x and cir x+s.

The rate of change ((cir x+s)-(cir x))%s is therefore the average of cir x and cir x+s. For small values of s, this average approaches cir x; the derivative of the area under the graph of cir is therefore the function cir itself.

In other words, the area under the graph of a function is the anti-derivative of the function. Since this area can be viewed as the aggregation or integration of the component areas, it is also called the integral of the function.

18C. Polynomial approximations

As illustrated in Section 3B, the area under a curve can be computed to give the value of the anti-derivative of a function at a chosen point. But this does not yield a function for the anti-derivative in the sense that the operator d._1 does for the functions to which it applies.

This situation is analogous to the equation-solving of Chapter 9, which gives the inverse of a function for some chosen point, but not the inverse function itself.

A practical solution to the anti-derivative of a function f is provided by finding the coefficients of a polynomial that approximates it, and then using the fact that the anti-derivative (as well as the derivative) of a polynomial is easily obtained.

The expression (f x) %. x^/i.n yields the coefficients of an n-term polynomial that best fits the function f at the points x. For example:

      ]x=: i:1j10
   _1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1

      ]c=: (1&o. x) %. x ^/i.5
   _3.46945e_17 0.997542 9.99201e_16 _0.156378 _7.77156e_16

      c&p. _1 0 1
   _0.841164 _3.46945e_17 0.841164

      1&o. _1 0 1
   _0.841471 0 0.841471

      (c&p. x) ,. (sin x)
      _0.841164 _0.841471
      _0.717968 _0.717356
      _0.564748 _0.564642
      _0.389009 _0.389418
      _0.198257 _0.198669
   _3.46945e_17         0
       0.198257  0.198669
       0.389009  0.389418
       0.564748  0.564642
       0.717968  0.717356
       0.841164  0.841471

The polynomial approximations may, however, be wildly wrong for arguments outside of the list x to which it was fitted.

The function der of Section 6B applied to a list of polynomial coefficients yields the coefficients of the derivative polynomial. A coresponding function for the anti-derivative may be defined as follows:

      der=: 1: |. ] * i. on #
      ant=: 0:,] % next i. on #
      ant c
   0 1 1 1 1 1 1
      der ant c
   1 2 3 4 5 6 0