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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.4.43
Capitolo 5, Problema 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]

Guida verificata passo dopo passo
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Step 1: Recall the formula for the average value of a function on an interval [a, b]. The average value is given by: 1(b-a)∫a,bƒ(𝓍)d𝓍. 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: ∫-1,1(1-|𝓍|)d𝓍 = ∫-1,0(1-(-𝓍))d𝓍 + ∫0,1(1-𝓍)d𝓍.
Step 3: Simplify and evaluate each integral. For the first integral, ∫-1,0(1-(-𝓍))d𝓍, rewrite the integrand as 1+𝓍. For the second integral, ∫0,1(1-𝓍)d𝓍, 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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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.
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Fundamental Theorem of Calculus Part 1

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.
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Average Value of a Function

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.
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Intro to Continuity
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