Linear Approximation

Linear Approximation | Formula & Example

Linear Approximation | Formula & Example

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first-order methods for solving or approximating solutions to equations.

Given a twice continuously differentiable function {\displaystyle f} of one real variable, Taylor’s theorem for the case {\displaystyle n=1} states that

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

where {\displaystyle R_{2}} is the remainder term. The 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} when it is close enough to {\displaystyle a}; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of {\displaystyle f} at {\displaystyle (a,f(a))}. For this reason, this process is also called the tangent line approximation.

Linear Approximation
Linear Approximation

If {\displaystyle f} is concave down in the interval between {\displaystyle x} and {\displaystyle a}, the approximation will be an overestimate (since the derivative is decreasing in that interval). If {\displaystyle f} is concave up, the approximation will be an underestimate.

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)} with real values, one can approximate {\displaystyle f(x,y)} for {\displaystyle (x,y)} close to {\displaystyle (a,b)} by the formula

{\displaystyle f\left(x,y\right)\approx f\left(a,b\right)+{\frac {\partial f}{\partial x}}\left(a,b\right)\left(x-a\right)+{\frac {\partial f}{\partial y}}\left(a,b\right)\left(y-b\right).}

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

In the more general case of Banach spaces, one has

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

where {\displaystyle Df(a)} is the Fréchet derivative of {\displaystyle f} at {\displaystyle a}.

Linear Approximation Calculator

Linear approximation is a method of estimating the value of a function, f(x), near a point, x = a, using the following formula:

linear approximation formula
linear approximation formula

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. 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.

Tangent line approximation
Tangent line approximation

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:

  1. The slope of the line, m
  2. Any single point that the line goes through, (ab).

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(xa)

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

Putting it all together and solving for y:

Formula for Linearization
Formula for Linearization

Local Linear Approximation

Linear Approximation/Linearization. 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.

Leave a Reply

Your email address will not be published. Required fields are marked *

Releated

5 Helpful Exam Tips For College Students

There is nothing that makes someone feel great in college more than passing an exam excellently. But for you to achieve this, you need to make a lot of sacrifices. You have to sacrifice your time and energy while revising. Remember, a sitting exam is not like an assignment that you can get assistance from […]

CA – Why Many Fail

A Rich And Diverse Course Chartered Accountant is one of the most diverse courses in India which provides rich exposure in various sectors in different geographical locations and all kinds of domains. Also, it is one of the courses which has the highest failure rate. More than 90% of students fail to clear the exam […]