This document discusses linear approximations, which are used in numerical techniques like Euler's method for solving differential equations. It explains that the linear approximation rests on the closeness of the tangent line to the function graph near a point. The formula for the linear approximation of a function value near x0 using the tangent line at (x0, f(x0)) is presented, along with an example demonstrating its use in estimating a function value.

1. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Linear Approximations
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
If x1 is close to x0, we will write , and we will approximate
by the point (x1,y1) on the tangent line given by
If we write , we have
In fact, one way to remember this formula is to write f'(x) as and then
replace d by . Recall that, when x is close to x0, we have
Example. Estimate .
Let . We have . Using the above approximation, we
get
We have

2. TARUN GEHLOT (B.E, CIVIL, HONOURS)
So . Hence
or . Check with your calculator and you'll see that this is a pretty good
approximation for .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
On the other hand, we obtain for the differential dy:
In this example we are lucky in that we are able to compute exactly, but in general
this might be impossible. The error in the approximation, the difference
between dy(replacing dx by ) and , is , which is small compared
to .