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According to the Mean Value Theorem for a continuous function f(x) in the interval [a,b], there exists a value ξ in this interval such that ; ∫ab f(x)dx =
- a)f(ξ)(b-a)
- b)f(b)(ξ-a)
- c)f(a)(b-ξ)
- d)0
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
According to the Mean Value Theorem for a continuous function f(x) in...
∫ab f(x)dx = b⋅
f(b) − a ⋅ f(a)………(1)
This implies that within interval [a,b] there exists a value ξ such that
f(ξ) = b⋅f(b)−a⋅f(a)/(b−a)
f(ξ)(b−a) = b⋅
f(b) − a ⋅ f(a)………(2
From (1) and (2) , ∫ab f(x)dx = f(ξ) (b − a)
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According to the Mean Value Theorem for a continuous function f(x) in...
Mean Value Theorem Explanation
The Mean Value Theorem states that for a continuous function f(x) in the interval [a,b], there exists a value ξ in this interval such that the average rate of change of f(x) over [a,b] is equal to the instantaneous rate of change at ξ.
Implications of Mean Value Theorem
- The integral of f(x) over [a,b] can be interpreted as the area under the curve of f(x) from a to b.
- According to the Mean Value Theorem, this area is equal to the value of f(ξ) times the length of the interval [a,b].
Application to the Given Choices
- Option A states that the integral of f(x) over [a,b] is equal to f(ξ) times the length of the interval [a,b], which aligns with the Mean Value Theorem.
- Options B, C, and D do not correctly represent the implication of the Mean Value Theorem and do not match the theorem's statement.
Therefore, the correct answer is option A, as it accurately reflects the Mean Value Theorem and its implications for the integral of a continuous function over an interval.
The Mean Value Theorem states that for a continuous function f(x) in the interval [a,b], there exists a value ξ in this interval such that the average rate of change of f(x) over [a,b] is equal to the instantaneous rate of change at ξ.
Implications of Mean Value Theorem
- The integral of f(x) over [a,b] can be interpreted as the area under the curve of f(x) from a to b.
- According to the Mean Value Theorem, this area is equal to the value of f(ξ) times the length of the interval [a,b].
Application to the Given Choices
- Option A states that the integral of f(x) over [a,b] is equal to f(ξ) times the length of the interval [a,b], which aligns with the Mean Value Theorem.
- Options B, C, and D do not correctly represent the implication of the Mean Value Theorem and do not match the theorem's statement.
Therefore, the correct answer is option A, as it accurately reflects the Mean Value Theorem and its implications for the integral of a continuous function over an interval.
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According to the Mean Value Theorem for a continuous function f(x) in the interval [a,b], there exists a value ξ in this interval such that ; ∫ab f(x)dx =a)f(ξ)(b-a)b)f(b)(ξ-a)c)f(a)(b-ξ)d)0Correct answer is option 'A'. Can you explain this answer?
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