1.what is linear approximation used for?
2.why would you use the linear approximation of a function than the actual function
method for finding numerical roots of equations. You start with an estimate
of the root, you then replace the function by its tangent (a linear
approximation) find the zero of the equation of the tangent, to give a
new approximation to the root. Repeat this process as often as needed.
Because the computation with a linear approximation to a function is2.why would you use the linear approximation of a function than the actual function
usually simpler than with the function itself. Or to use known results from
a linear theory to give an approximate result in a non-linear case.
A nice example which is probably beyond your current experience is the
Kalman filter, which is an optimal estimator for certain types of linear
systems. With mildly non-linear systems one linearises about an appropriate
estimate, and then uses the KF equations to give a sub-optimal but good
estimator for the non-linear case (known as the Extended Kalman Filter (EKF)).
With linear approximations, there are also error bounds. For problems that you're unable to get an equality, an approximation is the next best thing. They can be especially useful (more so when you know HOW useful it is); I completely disagree with:
"If you are asking for mathematics, then the answer is no purpose at all."
Consider other approximations, such as Riemann sums or taylor polynomials. Do these serve no purpose at all, for an integral (for instance) that cannot be found exactly?
Serge Lang in Real Analysis proves a Mean Value Theorem of which one form is
|f(z) - f(x)| <= |z - x| sup |f'(v)|
for v on the line segment connecting x and z. That is a linear approximation.
He then uses this theorem to prove the Inverse Mapping Theorem, Implicit Mapping Theorem and Existence Theorem for Differential Equations. Theses theorems all require functions to be differentiable so that a linear approximation may be used.
In fact, whenever any theorem or definition requires differentiability, it is probably to use a linear approximation. That's almost true by definition: a function is differentiable iff it can approximated locally by linear map. Consider a C1 manifold in R3 in Differential Topology. It is a surface that can be approximated locally by its tangent plane, which is a linear approximation.
What JakeD said about the MVT is exactly what I meant when inequalities are used non-stop in number theory. However, when in physics we drop a term because we assume it does not exist (since it is almost zero) we need to consider in math, which is what the error terms accomplish.