## Thursday, July 10, 2014

### The Instantaneous Intercept

The Instantaneous Intercept

Definition

Consider two points on the graph of f(x):  (x1, f(x1)) and (x2, f(x2)). What is the equation of the line joining them? If it is y = Ax+B, then
Ax1 + B   =  f(x1)
Ax2 + B   =  f(x2)

Then by Cramer’s Rule:
A       =       (f(x1) – f(x2))  /  (x1 – x2)           :      Newton’s Ratio
B       =       (x1 f(x2) – x2 f(x1))  /  (x1 – x2)   :    Hellerstein’s Ratio.

As x2 approaches x1:
A approaches f '(x), pronounced “f-prime”: it is the ‘derivative’, or ‘instantaneous slope’, or ‘fluxion’.
B approaches f #(x), pronounced “f-grid”; it is the ‘instantaneous intercept’, or ‘interceptual’.
The tangent line to the graph of y=f(x) at x=x1 is
y         =      f '(x1) x  + f #(x1)
so      f(x1)    =      f '(x1) x1  + f #(x1)
so      f #(x1)  =        f(x1) -  f '(x1) x1
so      f #       =        f   -   x f '
You can also derive this from the above limit definition.

Table of Intercepts

f #       =        f   -   x f '   =       (f dx – x df) / dx     =   2f – (x f)'
(a f(x) + b g(x)) #             =       a f #(x) + b g#(x)
K#                                   =       K
(xN)#                               =       (1 – N) (xN)
x#                                   =       0
(x2)#                                =        - (x2)
(x3)#                                =       - 2 (x3)
(x-1)#                               =       2 (x-1)
(ln(x))#                            =       ln(x) - 1
( - x ln(x) )#                     =       x
( xN ln(x) )#                     =       xN ( (1-N)ln(x)  - 1 )
( x f(ln(x)) )#          =       - x f '(ln(x))
( x cos(ln(x)) )#               =       x sin(ln(x))
( x sin(ln(x)) )#                 =       - x cos(ln(x))
(ex)#                               =       (1- x) ex
(x ex)#                             =       - x2 ex
(x2 ex)#                            =       (-1 – x) x2 ex
(x3 ex)#                            =       (-2 – x) x3 ex
(xN ex)#                            =       (1 – N – x) xN ex
(sin(x))#                          =       sin(x) – x cos(x)
(cos(x))#                         =       cos(x) + x sin(x)
(tan(x))#                          =       tan(x) – x sec2(x)
(sec(x))#                          =       sec(x)(1 – x tan(x))
(arctan(x))#                      =       arctan(x) –  x/(1+x2)
(f *g)#                             =       f#g + fg# - fg
(f /g)#                              =       (xfg + f#g - fg# ) / g2
(1 /g)#                             =       (xg + g - g# ) / g2
(g2)#                                =       2gg#  -  g2
(g3)#                                =       3g2g#  -  2g3
(gN)#                               =       NgN-1g#  -  (N-1)gN
(ln(g))#                            =       ln(g) – 1 + g#/g
(eg)#                                =       (1- g + g#) eg
(f(g))#                             =       f(g)   (f(g) – f#(g))(g – g#) / x

If F equals the inverse of f, then
(F)#             =       (F)  -  x / (f '(F)     =       (F)  -  x2 / (x - f #(F)

Given an intercept f #, we can calculate the derivative f ':
f '       =       ( f - f #) /  x

Intercept  and Newton’s Method

Newton’s Method of approximating roots of equations is the iteration
xN+1   =       xN      f(xN) / f '(xN)
This is the same as:
xN+1   =       -  f#(xN) / f '(xN)
and also:
xN+1   =       -  xN f#(xN) / (f #(xN) - f(xN) )
and also:
xN+1   =       xN ( 1  {–}  f#(xN) / f(xN) )
where {–} is the reciprocal subtraction operator:
a {–} b  =   1 / (1/a  - 1/b)

The Anti-Intercept

F is an anti-intercept of f if F#  =  f  .  Note that (F + ax)#            =   F#; so any anti-intercept will have an anti-interception term ax.
Denote the general anti-intercept as ai(f). Then:
ai(xN)     =     (1/(1-N)) xN   +   ax
ai(x)       =     - x ln(x)   +   ax

For instance:
ai(7x3 – 5x2 + 3)   =  (-7/2) x3 + 5x2  + 3  + ax
ai(5x4 + 5x)   =  (-5/3) x4  -  5xln(x)  + ax

In general, if F  =  ai(f), then:
F –  x F '   =   f
F ' – (1/x)F  =  - f / x
(1/x)F ' – (1/x2)F  =  - f / x2
((1/x)F) '   =  - f / x2
(1/x)F   =  ∫(-f/x2)dx    +  a
F          =    - x  * ∫(f/x2)dx   +  ax
So:     ai(f)     =     - x  * ∫(f/x2)dx   +  ax
Also: ∫(f)dx   =     (-1/x)  * ai(x2f)   +   a

Also: ai(x f(ln(x)))    =   -x (∫f)(ln(x))   +  ax

Interceptual Equations

If   f#   =   r f      then      f   =  a x(1-r)
If   f##   =   f      then      f   =  a x2 + b
If   f##   - 5f#  + 6f  =   0      then      f   =  a x-1 + b x-2
In general, if   a x2 + bx  + c  =  0   has roots r and R, then
a f##  + b f#  + c f  =  0
has solutions    f   =   A x(1-r)  +  B x(1-R)
This extends to complex roots. For instance:
If   f##   =   - f      then
f   =  x ( A cos(lnx)) + B sin(ln(x)) )

Applications?

We can proceed like this, reproducing calculus with derivative replaced by intercept. Even extrema can be redone within this parody of freshman calculus; for at a minimum or maximum value, f#(x) = f(x).
The question is, what specific applications does the interceptual in itself have? What use has an anti-intercept? Or an interceptual equation?
Perhaps for calculating involutes and evolutes?