Plus42 Solver: Derivatives
Introduction
The Technical Applications book for the HP 27S and HP 19B (and can apply to the HP 17B outside of trigonometry) shows the numerical first and second derivative can be calculated by the formulas:
f ' (x) = (f(x+h) - f(x-h)) / (2 * h)
f ' ' (x) = (f(x + h) - 2 * f(x) + f(x + h)) / h^2
where h is sufficiently small, like 10^-5 to 10^-12.
Legacy Formulas vs. Plus42 Formulas
The formulas suggested by the Technical Applications Book are:
First Derivative:
Second Derivative:
Depending on the function FX, this above can turn the above into long equations. With the ability of user functions, this allows us to use the original definitions.
FX(X): f(x) (insert f(x)
First Derivative:
F'X=(FX(X+H)-FX(X-H))÷(2×H)
Second Derivative:
F''X=(FX(X+H)-2×FX(X)+FX(X-H))÷SQ(H)
SQ: press by the key sequence [(shift)] (x^2)
': (ALPHA) [ ↓ ] (PUNC) [ ↓ ] ( ' )
Note: Radians mode
Examples
FIX 5 mode is set.
Example 1:
f(x) = 0.5 * cos(3*x)
x = Ļ/4
FX(X):0.5×COS(3×X)
f'(x) ≈ -1.06066
f''(x) ≈ 3.18198
Example 2:
f(x) = (x^2 + 3*x + 5) / (4*x - 1)
x = 2
FX(X):(X^2+3×X+5)÷(4×X-1)
f'(x) ≈ -0.22449
f''(x) ≈ 0.54227
Functions with Variable Constants
It is easy to expand the user function FX to include variable constants. For example:
f(x) = -ln(cos(√(a*x)))
Calculate the value and first derivative at x = 0.11 and a = 0.46
Attach variable constants at the end of FX:
FX(X:A):-LN(COS(SQRT(A×X)))
F'X=(FX(X+H:A)-FX(X-H:A))÷(2×H)
f(x:a) ≈ 0.02552
f'(x:a) ≈ 0.23396
The user function makes the calculating numerical derivatives easier.
Source:
Technical Applications: Step-by-Step Solutions for Your HP-27S or HP-19B Calculator Hewlett Packard. Edition 2. Corvallis, OR. November 1988. pg. 44
Have any Halloween plans? Wishing you a great day,
Eddie
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