Linear Approximation Table

Introduction to linear approximation table:

In mathematics, the procedure for determining the straight line is that closely fit curvature at various sites. Linear approximation equation is given as y = ax + b, the values of a and b are chosen that the line meet the curve at the preferred site, or value of x, and the evaluation of the line equals the rate of change of the curve at that site.

The majority curvature, linear approximation are first-rate only particularly close to the preferred x. The table consist both rows and columns. We can put the values in the table. It should be very easy for the student to study.

Formula-linear Approximation Table:

Linear approximation of the domain function represented as  f(x). Digression line to the table of f(x) at the point (`x_0, y_0` ) where `y_0` is given as `y_0` = f(`x_0` ). It is represented as

` y-y_0= f(x_0)(x-x_0)`

If x is closed the `x_0` , the formula can be written as `x_1 = x_0Deltax `

In mathematics, a linear approximation is an approximation of a universal function. The linear approximation uses a linear function. The linear function is more accurate and affine functions .They are widely used in the method of finite differences to create first order methods for solving or determining solutions to equations. It is given by,

f(x )= f(a) + f'(a)(x-a) + R2
f(x) = f(a) + f'(a)(x-a)

Linear approximation function:

A linear approximation functions to a function at a point can be computed by striking the primary expression in the Taylor series,

`F(x_0+Delta)`  = `f(x_0)+f(x_0)Delta+....`

The Newton’s method linear approximation can be estimated using linear approximation.

Example Problems- Linear Approximation Table:

Example 1- Using linear approximation table

Consider the linear function y = f(x) = `5x^2`

Solution:

Let` Deltax ` is an increment of x.

Therefore, `Deltay ` is also an increment of y.

Hence, we have

= f(x + `Deltax` )-f(x)

= 5(x + `Deltax` )`^2` – `5x^2`

= 25x( `Deltax` ) + 4 `(Deltax)^2`

The differential equation can be determined by using the given linear function.

Therefore `(dy)/(dx)` =25x

Hence, dy = 10x dx

Example 2- Using linear approximation table

Consider the linear function y = f(x) = `3x^2`

Solution:

Let` Deltax ` is an increment of x.

Therefore, `Deltay` is also an increment of y.

Hence, we have

= f(x + `Deltax` )-f(x)

= 3(x + `Deltax` )`^2` – `3x^2`

= 9x( `Deltax` ) + 2` (Deltax)^2`

The differential equation can be determined by using the given linear function.

Therefore `(dy)/(dx)` =9x

Hence, dy = 6x dx.

These all are the above explanation and examples make clear about this linear approximation.