What is the linear approximation of a function?
The linear approximation of a function f(x) is the linear function L(x) that looks the most like f(x) at a particular point on the graph y = f(x). This depends on what point (a, f(a)) you want to focus in on. Spoiler Alert: It's the tangent line at that point! The tangent line matches the value of f(x) at x=a, and also the direction at that point.
Let's find the equation of the tangent line, then. The limit definition of the derivative at a point tells us that the difference quotient
meaning that the equation of the tangent line at x=a is
That's your linear approximation, brought to you by calculus \and dansmath!/
P.S. This is the first two terms of the Taylor Series; with more terms you can approximate a function at any point with higher degree polynomials!
One of the reasons to use linear approximations is that we can estimate an answer without the use of a calculator because it is usually a simple multiplication/division and addition. The other thing to note is the approximation is better the closer you are to
Let's look at estimating
We want to choose an
So, we evaluate for the desired value:
#=2.0025#done without a calculator (mathamagics)!
This is a very good estimate because