Find the linear approximation of the function g(x) = 3√(1 + x) at a = 0.

Solution:

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

Given, the function g(x) = 3√(1 + x) = 3 (1 + x)^1/2

We have to find the linearization L(x) of the function at a = 0.

Using the formula,

L(x) = g(a) + g’(a)(x - a)

Now,

g(x) = 3(1 + x)^1/2,

g(a) = f(0) = 3

g’(x) = 1/2 (1 + x) ^{- 1/2}

g’(a) = g’(0) = 1/2.(1 + 0) ^{- 1/2} = 1/2

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

L(x) = 3 + (1/2)(x - 0)

Therefore, the linearization of g(x) = 3√(1 + x) at a = 0 is L(x) = 3 + x/2.

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Find the linear approximation of the function g(x) = 3√(1 + x) at a = 0.

Summary:

The linearization of the function g(x) = 3√(1 + x) at a = 0 is L(x) = 3 + x/2.