Section 7.3 Linear approximation
When we introduced the concept of derivative in Chapter 2 and Section 4.1, we studied how the derivative can be understood geometrically as calculating the slope of the tangent line to the graph of a function at a point. Moreover, we mentioned that if we zoom in on the graph of the function near this point, the tangent line becomes closer and closer to the graph of the function. In other words, it provides a fairly good approximation of the function near that point. This is the idea of “linear approximations”.
In fact, calculus could be understood as a theory of approximations: the derivative is a machinery that allows us to construct polynomial approximations of functions. The linear approximation, which corresponds to the tangent line, is the degree one polynomial approximation of a function at a point.
In this section we study the idea of linear approximations in more detail, and explore how it can be applied in various problems in the physical sciences.
Objectives
You should be able to:
- Find the linear approximation of a function at a given point.
- Illustrate the relation between the function and its linear approximation.
- Use the linear approximation of a function at a point to approximate the value of the function near this point.
- Deduce whether a linear approximation over- or under-estimates the exact value of a function from information about concavity of the function.
Subsection 7.3.1 Instructional video
Subsection 7.3.2 Key concepts
Concept 7.3.1. Linear approximation and linearization.
Recall that the tangent line of a differentiable function \(f(x)\) at point \((a,f(a))\) is
For \(x\) values very close to \(a\) the graph of \(f(x)\) is very close to the graph of the tangent line. Then, we can use the tangent line to obtain the approximation
this is called the linear approximation of \(f(x)\) at \(a\text{.}\)
The function
is called the linearization of \(f\) at \(a\).