153. The linearization of 2ˣ
a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

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Identify the function to be linearized: \(f(x) = 2^{x}\), and the point of linearization: \(x = 0\).

Recall that the linearization of a function \(f(x)\) at \(x = a\) is given by the formula: \(L(x) = f(a) + f'(a)(x - a)\).

Calculate \(f(0)\) by substituting \(x = 0\) into the function: \(f(0) = 2^{0}\).

Find the derivative of the function: \(f'(x) = 2^{x} \ln(2)\), then evaluate it at \(x = 0\): \(f'(0) = 2^{0} \ln(2)\).

Write the linearization formula using the values found: \(L(x) = f(0) + f'(0)(x - 0)\), then round the coefficients \(f(0)\) and \(f'(0)\) to two decimal places.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Linearization of a Function

Linearization approximates a function near a point using the tangent line at that point. It is given by L(x) = f(a) + f'(a)(x - a), where a is the point of approximation. This simplifies complex functions to linear ones for easier calculations near a.

Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x is f'(x) = a^x ln(a). This rule helps find the slope of the tangent line at any point, which is essential for constructing the linearization.

Evaluating Functions and Derivatives at a Point

To find the linearization at x = 0, you must compute both f(0) and f'(0). These values provide the y-intercept and slope of the tangent line, respectively, which are then used in the linear approximation formula.

153. The linearization of 2ˣa. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.