Linear Approximation Formula

Define Linear Approximation Formula

Linear Approximation Formula | Defination

In mathematics, a linear approximation formula 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.

This approximation is crucial to many known numerical techniques such as Euler’s Method to approximate solutions to ordinary differential equations. The idea to use linear approximations rests in the closeness of the tangent line to the graph of the function around a point.

Linear Approximation Formula
Linear Approximation Formula

Formula

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

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.y=f(a)+f'(a)(x-a)

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.

Tangent Planes And Linear Approximations

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly. Therefore, in a small-enough neighborhood around the point, a tangent plane touches the surface at that point only.

Tangent Lines And Linearization

Let’s review a basic fact about derivatives. The value of the derivative at a specific point, x = a, measures the slope of the curve, y = f(x), at that point. In other words, ‘(a) = slope of the tangent line at a.

Tangent Lines and Linearization
Tangent Lines and Linearization

Now, the tangent line is special because it’s the one line that matches the direction of the curve most closely, at the specific x-value you are interested in. Notice how close the y-values of the function and the tangent line are when x is near the point where the tangent line meets the curve.

So, if the curve y = f(x) is way too complicated to work with, and if you’re only interested in values of the function near a particular point, then you could throw away the function and just use the tangent line. Well, don’t actually throw away the function. . . we may need it later!

Formula For Linearization

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 the derivative measures the slope, and secondly, b = f(a), because the original function measures y-values.

Local Linear Approximation Formula

Linear approximation is the process of finding the equation of a line that is the closest estimate of a function for a given value of x. Linear approximation is also known as tangent line approximation, and it is used to simplify the formulas associated with trigonometric functions, especially in optics. At infinitesimally close observation, a curve begins to resemble a straight line, so linear approximation can very closely imitate the function. For a twice differentiable, real-valued function f(x), , where R2 is the remainder term. The linear approximation, then, is given by . This approximation is equivalent to the equation for the tangent line at a.

Applications Of Linear Approximations

Optics

Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.

Period of oscillation

The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude.[3] It is independent of the mass of the bob. The true period T of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms.

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