Find the linearization l(x) of the function at a. f(x) = x^1/2, a = 9

Solution:

Linearization is a mathematical process of determining the linear approximation of inputs and corresponding outputs.

Given, the function f(x) = x^1/2

We have to find the linearization L(x) of the function at a = 9.

Using the formula,

L(x) = f(a) + f’(a)(x - a)

Now,

f(x) = x^1/2,

f(a) = f(9) = 3

f’(x) = 1/2 x ^{- 1/2}

f’(a) = f’(9) = 1/2 . 9 ^{- 1/2} = 1/2 . 1/3 = 1/6

Substituting the values of f(a) and f’(a), the function becomes

L(x) = 3 + (1/6)(x - 9)

Therefore, the linearization of f(x) = x^1/2 at a = 9 is L(x) = 3 + (1/6)(x - 9).

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Find the linearization l(x) of the function at a. f(x) = x^1/2, a = 9

Summary:

The linearization of the function f(x) = x^1/2 at a = 9 is L(x) = 3 + (1/6)(x - 9).