Proving with mean value theorem

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Start date Jan 4, 2005
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Mean Mean value theorem Theorem Value

In summary, the mean value theorem is a fundamental theorem in calculus that states the existence of a point in an interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints. It is commonly used in proofs to establish the existence of a value or property in a given interval. The conditions for the mean value theorem to hold include continuity and differentiability of the function on the interval, as well as a unique derivative at every point. The mean value theorem is also related to Rolle's theorem, which is a special case where the derivative is equal to 0 at a critical point. However, the mean value theorem can only be applied to differentiable functions that satisfy the conditions mentioned above.

Jan 4, 2005

trap

Suppose that g(a) = g'(a)=0 and |g''(x)| < M for all x in [a, a+h] (for some positive constant M). Show that |g(a+h)| < Mh^2.
(Hint: Let k be any number such that 0<= k <= h and apply Mean Value Theorem to g' on [a,a+k].)

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Jan 4, 2005

matt grime

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Well, what does the mean value theorem say, and what does it imply when you use it as hinted?

Jan 4, 2005

Justin Lazear

I'd also keep in mind the definition of a derivative.

--J

1. What is the mean value theorem?

The mean value theorem is a fundamental theorem in calculus that states that for a differentiable function, there exists at least one point in the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.

2. How is the mean value theorem used in proofs?

The mean value theorem is often used in proofs to show the existence of a certain value or property in a given interval. It allows us to connect a function's derivative with its original function, making it a powerful tool in proving various properties and theorems in calculus.

3. What are the conditions for the mean value theorem to hold?

For the mean value theorem to hold, the function must be continuous on the closed interval and differentiable on the open interval. Additionally, the function must have a unique derivative at every point in the open interval.

4. How is the mean value theorem related to Rolle's theorem?

Rolle's theorem is a special case of the mean value theorem, where the slope of the tangent line is equal to 0 at a certain point in the interval. This point is known as a critical point, and it is the point where the function's derivative is equal to 0.

5. Can the mean value theorem be applied to all functions?

No, the mean value theorem can only be applied to differentiable functions. If a function is not differentiable, the mean value theorem cannot be used to prove properties about it. Additionally, the function must also satisfy the conditions mentioned in question 3 for the mean value theorem to hold.