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Calculus: Mean Value Theorem (Rolles, Lagrange & Cauchy) Video Lecture | Engineering Mathematics - Civil Engineering (CE)
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FAQs on Calculus: Mean Value Theorem (Rolles, Lagrange & Cauchy) Video Lecture - Engineering Mathematics - Civil Engineering (CE)
| 1. What is the Mean Value Theorem in calculus? |  |
Ans. The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over [a, b].
| 2. What is the significance of the Mean Value Theorem in calculus? | |
Ans. The Mean Value Theorem is significant in calculus as it provides a fundamental result that guarantees the existence of a point where the instantaneous rate of change of a function equals its average rate of change over a given interval. It serves as a cornerstone for many other theorems and techniques in calculus.
| 3. How is the Mean Value Theorem related to Rolle's Theorem? | |
Ans. The Mean Value Theorem is an extension of Rolle's Theorem. Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are equal, then there exists at least one point c in (a, b) where the derivative of the function is zero. The Mean Value Theorem is a generalized version of Rolle's Theorem, where the condition of equal function values at the endpoints is relaxed.
| 4. How is the Mean Value Theorem related to Lagrange's Mean Value Theorem? | |
Ans. Lagrange's Mean Value Theorem is a specific case of the Mean Value Theorem. It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function is equal to the slope of the secant line connecting the endpoints of the interval. The Mean Value Theorem is a more general statement that encompasses Lagrange's Mean Value Theorem.
| 5. What is Cauchy's Mean Value Theorem in calculus? | |
Ans. Cauchy's Mean Value Theorem is a variation of the Mean Value Theorem that applies to the ratio of two functions. It states that if two functions, f(x) and g(x), are continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and g(x) ≠ 0 for all x in (a, b), then there exists at least one point c in (a, b) where the derivative of f(x) divided by the derivative of g(x) is equal to the ratio of their values at c. Cauchy's Mean Value Theorem provides a way to relate the derivatives of two functions.
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