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Find the linearization of ƒ(x) = 2/ (1 - x) + √1 + x - 3.1 at x = 0.

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Identify the function to be linearized: \( f(x) = \frac{2}{1-x} + \sqrt{1+x} - 3.1 \).

Recall the formula for the linearization of a function at a point \( a \): \( L(x) = f(a) + f'(a)(x-a) \). Here, \( a = 0 \).

Calculate \( f(0) \) by substituting \( x = 0 \) into the function: \( f(0) = \frac{2}{1-0} + \sqrt{1+0} - 3.1 \).

Find the derivative \( f'(x) \). Use the sum rule and differentiate each term separately: \( f'(x) = \frac{d}{dx}\left(\frac{2}{1-x}\right) + \frac{d}{dx}(\sqrt{1+x}) \).

Evaluate \( f'(0) \) by substituting \( x = 0 \) into the derivative. Then, use the linearization formula to find \( L(x) = f(0) + f'(0)(x-0) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linearization

Linearization is the process of approximating a function near a specific point using its tangent line. This involves finding the function's value and its derivative at that point. The linearization formula is given by L(x) = f(a) + f'(a)(x - a), where 'a' is the point of tangency. This technique is useful for simplifying complex functions for easier analysis.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In the context of linearization, the derivative at a point provides the slope of the tangent line, which is essential for constructing the linear approximation.

Function Evaluation

Function evaluation involves calculating the output of a function for a given input. In this case, we need to evaluate the function ƒ(x) at x = 0 to find the point of tangency for linearization. This step is crucial as it provides the y-coordinate of the tangent line, which, along with the slope from the derivative, defines the linear approximation of the function.

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