# Linear Approximation Equation

**Linear Approximation: **We have just seen how derivatives allow us to compare related quantities that are changing over time. In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher-degree polynomials Introduction to Power Series and Functions.

This lesson shows how to find a linearization of a function and how to use it to make a linear approximation. This method is used quite often in many fields of science, and it requires knowing a bit about calculus, specifically, how to find a derivative.

## Linear Approximation Calculator

In this section we’re going to take a look at an application not of derivatives but of the tangent line to a function. Of course, to get the tangent line we do need to take derivatives, so in some way this is an application of derivatives as well.

Take a look at the following graph of a function and its tangent line.

From this graph we can see that near \(x = a\) the tangent line and the function have nearly the same graph. On occasion we will use the tangent line, \(L\left( x \right)\), as an approximation to the function, \(f\left( x \right)\), near \(x = a\). In these cases we call the tangent line the **linear approximation** to the function at \(x = a\).

**Example.** Consider the function *y* = *f*(*x*) = 5*x*^{2}. Let be an increment of *x*. Then, if *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 *dy*(replacing *dx* by

## What is the linear approximation formula?

**Linear approximation** is a method of estimating the value of a function, f(x), near a point, x = a, using the following **formula**: The **formula** we’re looking at is known as the linearization of f at x = a, but this **formula** is identical to the **equation** of the tangent line to f at x = a.

## Why do we use linear approximation?

**Linear approximation**, or linearization, **is** a method **we can use** to **approximate** the value of a function at a particular point. The reason liner **approximation is** useful **is** because it **can** be difficult to find the value of a function at a particular point. Square roots **are** a great example of this.

## Is linearization the same as linear approximation?

**linear approximation**,

**linearization**, and tangent line

**approximation**all refer to the

**same**thing. There are other

**linear approximations**used in mathematics besides this one. For instance, in statistics, regression analysis is used to fit the “best”

**linear**function to a set of data.

## Linear Approximation Formula

So, how do you find the linearization of a function *f* at a point *x* = *a*? Remember that the equation of a line can be determined if you know two things:

- The slope of the line,
*m* - Any single point that the line goes through, (
*a*,*b*).

We plug these pieces of info into the point-slope form, and this gives us the equation of the line. (This is just algebra, folks; no calculus yet.)

*y* – *b* = *m*(*x*–*a*)

But, in problems like these, you will not be given values for *b* or *m*. Instead, you have to find them yourself. Firstly *m* = *f *‘(*a*), because the derivative measures the slope, and secondly, *b* = *f*(*a*), because the original function measures *y*-values.

Let *x*_{0} be in the domain of the function *f*(*x*). The equation of the tangent line to the graph of *f*(*x*) at the point (*x*_{0},*y*_{0}), where *y*_{0} = *f*(*x*_{0}), is

If *x*_{1} is close to *x*_{0}, we will write *x*_{1},*y*_{1}) on the tangent line given by

If we write

In fact, one way to remember this formula is to write *f*‘(*x*) as *d* by *x* is close to *x*_{0}, we have

## Local Linear Approximation

Given a twice continuously differentiable function {\displaystyle f}

- {\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}\ }

where {\displaystyle R_{2}}

linear approximation is obtained by dropping the remainder:

- {\displaystyle f(x)\approx f(a)+f'(a)(x-a)}.

This is a good approximation for {\displaystyle x}**tangent line approximation**.

If {\displaystyle f}^{}

Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function {\displaystyle f(x,y)}

The right-hand side is the equation of the plane tangent to the graph of {\displaystyle z=f(x,y)}

In the more general case of Banach spaces, one has

- {\displaystyle f(x)\approx f(a)+Df(a)(x-a)}

where {\displaystyle Df(a)}