Graphing
If you want to have a complete graph of a function f,
then these are all of the things that you should make sure show up:
- x = 0, if f is defined at that point;
- x → −∞,
if f is defined in that direction;
- x → ∞,
if f is defined in that direction;
- x → c−,
if f is defined in that direction,
whenever f is undefined or discontinuous at c;
- x → c+,
if f is defined in that direction,
whenever f is undefined or discontinuous at c;
- x = c, if f is defined at that point,
whenever f is undefined
approaching c from either direction (or both);
- x = c, whenever f(c) = 0;
- x = c,
whenever f′ is undefined or discontinuous at c,
if f is defined there;
- x = c, whenever f′(c) = 0;
- x = c,
whenever f′ is undefined or discontinuous at c,
if f is defined there;
- x = c,
whenever f′(c) = 0;
This should be sufficient
whenever f
is a twice-differentiable function whose domain is an interval,
or more generally whenever f is piecewise twice-differentiable:
a piecewise-defined function
in which the domain of each piece is an interval
and in which each piece is twice-differentiable
except possibly at its endpoints.
(There are weirder functions that can't be put in this form,
but you shouldn't have to deal with them in this class.)
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This web page was written in 2014 and 2015 by Toby Bartels,
last edited on 2015 November 16.
Toby reserves no legal rights to it.
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