- #1
karush
Gold Member
MHB
- 3,269
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Ok Just have trouble getting this without a function..
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Mean Value Theorem: Showing Change in a Function is Bounded
In summary, the Mean Value Theorem is a mathematical concept that states for a continuous and 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. It is used to show that the change in a function over a given interval is bounded and can only be applied to continuous and differentiable functions. The Mean Value Theorem can also be used to prove the existence of extrema, but it cannot provide the exact value of the maximum or minimum. Other methods are needed for this.
Ok Just have trouble getting this without a function..
- #2
skeeter
- 1,103
- 1
average rate of change of $f(x)$ on the interval $[2,8]$ is $\dfrac{f(8)-f(2)}{8-2}$
the MVT states there exists at least one value of $x \in (2,8)$ where $f'(x) = \dfrac{f(8)-f(2)}{8-2}$
$3 \le f'(x) \le 5 \implies 3 \le \dfrac{f(8)-f(2)}{8-2}\le 5 \implies 18 \le f(8)-f(2) \le 30$
the MVT states there exists at least one value of $x \in (2,8)$ where $f'(x) = \dfrac{f(8)-f(2)}{8-2}$
$3 \le f'(x) \le 5 \implies 3 \le \dfrac{f(8)-f(2)}{8-2}\le 5 \implies 18 \le f(8)-f(2) \le 30$
1. What is the Mean Value Theorem?
The Mean Value Theorem is a mathematical concept that states that if a function is continuous on a closed interval, then there exists at least one point within that interval where the slope of the tangent line is equal to the average rate of change of the function over that interval.
2. How is the Mean Value Theorem used to show change in a function is bounded?
The Mean Value Theorem can be used to show that the change in a function over a given interval is bounded by finding the maximum and minimum values of the slope of the tangent line within that interval. If the slope is always between these two values, then the change in the function is bounded.
3. What is the significance of the Mean Value Theorem?
The Mean Value Theorem is significant because it provides a way to connect the concepts of slope and average rate of change in a mathematical proof. It also has many practical applications in fields such as physics, engineering, and economics.
4. Can the Mean Value Theorem be applied to all functions?
No, the Mean Value Theorem can only be applied to continuous functions on a closed interval. If a function is not continuous or the interval is not closed, then the theorem cannot be used.
5. How is the Mean Value Theorem related to the Intermediate Value Theorem?
The Mean Value Theorem is a special case of the Intermediate Value Theorem. The Intermediate Value Theorem states that if a function is continuous on a closed interval, then it must take on every value between the minimum and maximum values of the function. The Mean Value Theorem is a specific application of this concept, where the function's average rate of change is equal to the slope of the tangent line at a specific point.
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