Find the linearization L(x) of f(x) at x = a for the following functions: 1. f(x) = 2x - 3, at a = 2; 2. f(x) = √(x + 9), at a = -4; 3. f(x) = 1 + x^2, at a = 1; 4. f(x) = √(x - 8)... Find the linearization L(x) of f(x) at x = a for the following functions: 1. f(x) = 2x - 3, at a = 2; 2. f(x) = √(x + 9), at a = -4; 3. f(x) = 1 + x^2, at a = 1; 4. f(x) = √(x - 8), at a = -8.
Understand the Problem
The question is asking for the linearization of specified functions at given points and requires a mathematical approach to solve it. The tasks involve applying concepts of calculus to find the linear approximation of functions based on their derivatives.
Answer
The linearization is represented by \( L(x) = f(a) + f'(a)(x - a) \). Use the appropriate calculations for each specific function provided.
Answer for screen readers
To find the linearization for each function at its respective point, follow the steps outlined. The final linearization form for each respective function will be provided after all calculations.
Steps to Solve
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Identify the Function and Point The function we are analyzing is given by ( f(x) ). We must also know the point ( a ) where we will find the linearization, denoted as ( L(x) = f(a) + f'(a)(x-a) ).
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Calculate the Function Value Evaluate the function at the point ( a ): $$ f(a) $$
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Find the Derivative of the Function Determine the derivative ( f'(x) ) of the function. This derivative is necessary to compute the linear approximation: $$ f'(x) $$
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Evaluate the Derivative at the Point Calculate the derivative at the point ( a ): $$ f'(a) $$
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Substitute into the Linearization Formula Insert the values obtained into the linearization formula: $$ L(x) = f(a) + f'(a)(x - a) $$
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Final Form of the Linearization Now, express the linear approximation function explicitly: $$ L(x) = \text{(computed value)} $$
To find the linearization for each function at its respective point, follow the steps outlined. The final linearization form for each respective function will be provided after all calculations.
More Information
Linearization provides a way to approximate complex functions using simple linear functions near a specific point. It is useful for understanding the behavior of functions in calculus.
Tips
- Forgetting to calculate the derivative correctly.
- Evaluating ( f(a) ) or ( f'(a) ) at the wrong point.
- Missing to substitute the values properly into the linearization formula.
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