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Linear Approximation Formula

For a function of any given value, the closest estimate of a function is to be calculated for which Linear Approximation formula is used. Also called as the tangent line approximation, the tangent line is is used to approximate the function.

The Linear Approximation formula of function f(x) is:

\[\LARGE f(x)\approx f(x_{0}+f'(x_{0}(x-x_{0})\]

Where,
f(x0) is the value of f(x) at x = x0.
f'(x0) is the derivative value of f(x) at x = x0.

We use Euler’s method for approximation solution for differential equations and Linear Approximation is equally important. At the end, what matters is the closeness of the tangent line and using the formulas to find the tangent around the point.

Solved Examples

Question 1: Calculate the linear approximation of the function f(x) = x2 as the value of x tends to 2 ?

Solution:

Given,
f(x) = x2
x0 = 2

f(x0) = 22 = 4
f'(x) = 2x
f'(x0) = 2(2) = 4

Linear approximation formula is,
$f(x)\approx f(x_{0}+f'(x_{0}(x-x_{0})$
f(x) ≈ 1 + 2(x – 2)
f(x) ≈ 1 + 2x – 2
$x=\frac{-1}{2}$

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