Find the linear approximation of the function f(x) = √(1 - x) at a = 0.

Solution:

Given, the function f(x) = √(1 - x)

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

Using the formula,

L(x) = f(a) + f’(a)(x - a)

Now,

f(x) = √(1 - x)

f(a) = f(0) = √(1 - 0)

f(a) = 1

f’(x) = (√(1 - x))’

= -(1/2)(1/√(1 - x))

f’(a) = f’(0) = -(1/2)(1/√(1 - 0))

= -(1/2)

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

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

L(x) = 1 - (1/2)x

Therefore, the linearization of f(x) = √(1 - x) at a = 0 is L(x) = 1 - (1/2)x.

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Find the linear approximation of the function f(x) = √(1 - x) at a = 0.

Summary:

The linearization of the function f(x) = √(1 - x) at a = 0 is L(x) = 1 - (1/2)x.