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More About Solving; How Solve Finds A Root - HP 35s User Manual

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More about Solving

This appendix provides information about the SOLVE operation beyond that given in
chapter 7.

How SOLVE Finds a Root

SOLVE first attempts to solve the equation directly for the unknown variable. If the
attempt fails, SOLVE changes to an iterative(repetitive) procedure. The iterative
operation is to execute repetitively the specified equation. The value returned by the
equation is a function f(x) of the unknown variable x. (f(x) is mathematical shorthand
for a function defined in terms of the unknown variable x.) SOLVE starts with an
estimate for the unknown variable, x, and refines that estimate with each successive
execution of the function, f(x).
If any two successive estimates of the function f(x) have opposite signs, then SOLVE
presumes that the function f(x) crosses the x–axis in at least one place between the
two estimates. This interval is systematically narrowed until a root is found.
For SOLVE to find a root, the root has to exist within the range of numbers of the
calculator, and the function must be mathematically defined where the iterative
search occurs. SOLVE always finds a root, provided one exists (within the overflow
bounds), if one or more of these conditions are met:
Two estimates yield f(x) values with opposite signs, and the function's graph
crosses the x–axis in at least one place between those estimates (figure a,
below).
f(x) always increases or always decreases as x increases (figure b, below).
The graph of f(x) is either concave everywhere or convex everywhere (figure
c, below).
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