How do you find the linear approximation of the function g(x) = 5√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) = 5√1 + x = (1 + x)^1/5

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) = (1 + x)^1/5

g(a) = f(0) = 1

g’(x) = 1/5 x^-4/5

g’(a) = g’(0) = 1/5 . 0^-½  = 1/5

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

L(x) = 1 + (1/5)(x - 0)

L(x) = 1 + x/5

Therefore, the linearization of g(x) = 5√1 + x at a = 0 is L(x) = 1 + x/5.

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How do you find the linear approximation of the function g(x) = 5√1 + x at a = 0?

Summary:

The linearization of the function g(x) = 5√1 + x at a = 0 is L(x) = 1 + x/5.