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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 5.4.39
Capitolo 5, Problema 5.4.39

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) = 8 ― 2𝓍 on [0, 4]

Guida verificata passo dopo passo
1
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. Here, a = 0 and b = 4.
Step 2: Compute the definite integral of ƒ(𝓍) = 8 - 2𝓍 over the interval [0, 4]. Set up the integral: ∫0→4(8-2𝓍)d𝓍. Evaluate this integral step by step.
Step 3: Divide the result of the integral by the length of the interval (b - a = 4 - 0 = 4) to find the average value of the function on [0, 4]. This gives the average value of ƒ(𝓍).
Step 4: Set ƒ(𝓍) equal to the average value found in Step 3. Solve the equation 8-2𝓍=average_value for 𝓍.
Step 5: Verify that the solution(s) for 𝓍 lie within the interval [0, 4]. These are the points at which the function equals its average value on the given interval.

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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 function's value at c equals the average value of the function over that interval. This average value is calculated as (1/(b-a)) * ∫[a to b] f(x) dx.
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Fundamental Theorem of Calculus Part 1

Average Value of a Function

The average value of a function f(x) over the interval [a, b] is defined as (1/(b-a)) * ∫[a to b] f(x) dx. This concept allows us to determine a single representative value of the function across the interval, which can then be compared to the function's actual values at specific points within that interval.
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Average Value of a Function

Finding Points of Intersection

To find points where a function equals its average value, we set the function f(x) equal to the average value calculated from the previous concepts. This involves solving the equation f(x) = average value, which may require algebraic manipulation or numerical methods to identify the specific x-values where this equality holds.
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Critical Points
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