Linear
Approximation of a Function of Two Independent Variables
In a Nut Shell: Recall that the
"linear approximation", L(x), of a function of one independent variable, x, provides a
way to find the value of f(x) at a neighboring point of a where
x is a point close to
a. This approximation directly relates
to the slope of f(x) at x = a. |
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The figure below depicts
the linear approximation, L(x), of the function, f(x).
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where f(x)
is the value of the function f at x f(a) is the value of the
function f(x) at x = a; x ˗ a is the interval; it is best to have a small interval f '(a) is the slope of the function f(x) at x = a L(x) is the linear
approximation of f(x) near x = a L(x) =
f(a) + df(a)/dx ( x
˗ a ) |
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One can extend the linear
approximation of a function, f(x,y) of two
independent variables say x and y by noting that
the "surface", f(x,y), may change its
slope in both the x and y-coordinate
directions. In this case the relevant
slopes at (x,y) = (a,b)
are: ∂f(x,y)/∂x evaluated at (x,y) = (a,b) i.e.
change of slope in the x-direction and ∂f(x,y)/∂y
evaluated at (x,y) = (a,b) i.e. change of slope in the y-direction
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Copyright © 2018 Richard C. Coddington
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