Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

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Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b], which is given by:

\frac{1}{(b - a)} \int f_{x} d x

. Here, a = 0 and b = ln(2).

Step 2: Substitute the given function ƒ(𝓍) = e²ˣ into the formula. The integral becomes:

\frac{1}{(\ln (2) - 0)} \int e^{2 x} d x

Step 3: Compute the integral of e²ˣ with respect to 𝓍. Use the rule for integrating exponential functions:

\int e^{k x} d x = \frac{e^{k x}}{k}

, where k is a constant. Here, k = 2.

Step 4: Evaluate the definite integral from 𝓍 = 0 to 𝓍 = ln(2). Substitute the limits of integration into the antiderivative obtained in Step 3.

Step 5: Multiply the result of the definite integral by

\frac{1}{\ln}

to find the average value of the function on the interval [0, ln(2)].

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주요 개념

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

Average Value of a Function

The average value of a continuous function f(x) over the interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is essential for determining how the function behaves on the specified interval, providing a single representative value that summarizes the function's output.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. It is denoted as ∫[a to b] f(x) dx and is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration, allowing us to evaluate the integral using antiderivatives.

Exponential Functions

Exponential functions, such as f(x) = e^(kx), where e is Euler's number, are characterized by their constant growth rate proportional to their value. In this case, the function e^(2x) grows rapidly as x increases, and understanding its properties is crucial for evaluating integrals involving exponential terms.