# 34 MVT - Application of the mean value theorem

• MHB
• karush
In summary, the average acceleration is 120 mi/h^2 and the Mean Value Theorem can be used to find the specific value of c where the acceleration is equal to 120.
karush
Gold Member
MHB

$10 min = \dfrac{h}{6}$
So
$a(t)=v'(t) =\dfrac{\dfrac{(50-30)mi}{h}}{\dfrac{h}{6}} =\dfrac{20 mi}{h}\cdot\dfrac{6}{h}=\dfrac{120 mi}{h^2}$
Hopefully đź•¶

Last edited by a moderator:
karush said:

$10 min = \dfrac{h}{6}$
So
$a(t)=v'(t) =\dfrac{\dfrac{(50-30)mi}{h}}{\dfrac{h}{6}} =\dfrac{20 mi}{h}\cdot\dfrac{6}{h}=\dfrac{120 mi}{h^2}$
Hopefully đź•¶

The average acceleration is $\displaystyle \frac{20}{\frac{1}{6}} = 120 \,\textrm{mi}/\textrm{h}^2$

Since the function is continuous and smooth, there must be some value $c \in \left[ 0, \frac{1}{6} \right]$ such that $a\left( c \right) = 120$ by the Mean Value Theorem.

Karush, what you said was simply that the average acceleration was120. In order to correctly answer the question, you have to appeal to the "Mean Value Theorem" as Prove It did.

## What is the mean value theorem?

The mean value theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the average rate of change of the function over the closed interval.

## How is the mean value theorem used in real-world applications?

The mean value theorem is used in a variety of real-world applications, such as in physics to calculate the average velocity of an object over a certain period of time, in economics to determine the average rate of change of a market trend, and in engineering to find the maximum and minimum values of a function.

## What is the difference between the mean value theorem and the intermediate value theorem?

The mean value theorem and the intermediate value theorem are both fundamental theorems in calculus, but they have different applications. The mean value theorem is used to find the average rate of change of a function over a closed interval, while the intermediate value theorem is used to show the existence of a root or solution to a function within a given interval.

## Can the mean value theorem be applied to all functions?

No, the mean value theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval. If a function is not continuous or differentiable, then the mean value theorem cannot be used to find the average rate of change of the function.

## What is the geometric interpretation of the mean value theorem?

The mean value theorem can be interpreted geometrically as stating that for any continuous and differentiable function on a closed interval, there exists at least one point where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval. This point is known as the "mean" or average point.

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