Section 13: Finding the Roots of an Equation 189
If you have some knowledge of the behavior of the function f(x) as it
varies with different values of x, you are in a position to specify initial
estimates in the general vicinity of a zero of the function. You can also
avoid the more troublesome ranges of x such as those producing a
relatively constant function value or a minimum of the function’s
magnitude.
Example: Using a rectangular piece of
sheet metal 4 decimeters by 8 decimeters,
an open-top box having a volume of 7.5
cubic decimeters is to be formed. How
should the metal be folded? (A taller box is
preferred to a shorter one.)
Solution: You need to find the height of
the box (that is, the amount to be folded
up along each of the four sides) that gives
the specified volume. If x is the height (or
amount folded up), the length of the box is (8 − 2x) and the width is
(4 − 2x). The volume V is given by
V = (8 − 2x)(4 − 2x)x.
By expanding the expression and then using Horner’s method (page 79),
this equation can be rewritten as
V = 4((x − 6)x + 8)x.
To get V = 7.5, find the values of x for which
f (x) = 4((x − 6)x + 8)x − 7.5 = 0.
The following subroutine calculates f (x):
Keystrokes Display
| ¥
000-
Program mode.
´ b 3
001-42,21, 3
Label.
6
002- 6
Assumes stack
loaded with x.
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