Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

문제 5.4.43

5장, 문제 5.4.43

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.

ƒ(𝓍) = 1 ― |𝓍| on [―1, 1]

검증된 단계별 안내

Step 1: Recall the formula for the average value of a function on an interval [a, b]. The average value is given by:

\frac{1}{(b - a)} \int_{a, b} ƒ (𝓍) d 𝓍

. For this problem, the interval is [−1, 1] and ƒ(𝓍) = 1 − |𝓍|.

Step 2: Compute the definite integral of ƒ(𝓍) over [−1, 1]. Break the integral into two parts because the absolute value function |𝓍| behaves differently for 𝓍 < 0 and 𝓍 ≥ 0. Specifically:

\int_{- 1, 1} (1 - | 𝓍 |) d 𝓍

\int_{- 1, 0} (1 - (- 𝓍)) d 𝓍

\int_{0, 1} (1 - 𝓍) d 𝓍

Step 3: Simplify and evaluate each integral. For the first integral,

\int_{- 1, 0} (1 - (- 𝓍)) d 𝓍

, rewrite the integrand as

1 + 𝓍

. For the second integral,

\int_{0, 1} (1 - 𝓍) d 𝓍

, the integrand remains

1 - 𝓍

. Compute these integrals separately.

Step 4: Divide the result of the definite integral by the length of the interval (b − a = 1 − (−1) = 2) to find the average value of the function on [−1, 1]. This gives the average value of ƒ(𝓍).

Step 5: Solve the equation ƒ(𝓍) = average value to find the points where the function equals its average value. Substitute ƒ(𝓍) = 1 − |𝓍| into the equation and solve for 𝓍. Consider both cases for |𝓍|: when 𝓍 ≥ 0 and when 𝓍 < 0.

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주요 개념

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

Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that if a function is continuous on a closed interval [a, b], then there exists at least one point c in (a, b) such that the value of the function at c equals the average value of the function over that interval. This theorem is essential for finding points where a function meets its average value.

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 crucial for determining the specific value that the function must equal at some point within the interval to satisfy the Mean Value Theorem for Integrals.

Continuous Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph over that interval. Continuity is a key requirement for applying the Mean Value Theorem for Integrals, as it ensures that the function behaves predictably and that the average value can be accurately determined.

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.ƒ(𝓍) = 1 ― |𝓍| on [―1, 1]