This approximation is crucial to many known numerical techniques such as Euler's Method to approximate solutions to ordinary differential equations. The idea to use linear approximations rests in the closeness of the tangent line to the graph of the function around a point.
Let x0 be in the domain of the function f(x). The equation
of the tangent line to the graph of f(x) at the point
(x0,y0), where
y0 = f(x0), is
Example. Estimate
.
Let
.
We have
.
Using the
above approximation, we get
Remark. For a function f(x), we define the differential df of f(x) by
Example. Consider the function
y = f(x) = 5x2. Let
be an increment of x. Then, if
is the
resulting increment of y, we have
Exercise 1. Use linear approximation to approximate
Exercise 2. Use linear approximation to approximate
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