an automobile starts from rest and travel 4 miles along a straight road in 5 minutes. Use the mean value theorem
what is the question?
The MVT states that on the interval [0,5] at some point the graph of the position will have slope 4/5.
To use the mean value theorem in this scenario, we need to find the average velocity of the automobile over the given time interval.
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 a point c in the open interval (a, b) such that the instantaneous rate of change (the derivative) at c is equal to the average rate of change over the closed interval.
In this case, the position of the automobile is changing over time, so we can consider it as a function of time, denoted by f(t).
Given that the automobile starts from rest, we can set the initial time as t = 0 and the initial position (or displacement) as f(0) = 0. Let's assume the velocity of the automobile at time t is given by v(t).
We are given that the automobile travels 4 miles in 5 minutes, which means the final position is f(5) = 4 miles. So we have the position at two points: f(0) = 0 and f(5) = 4.
Now, we want to find the average velocity of the automobile over the interval [0, 5]. The average velocity is given by the formula:
Average Velocity = (Change in position) / (Change in time)
Applying this formula, we have:
Average Velocity = (f(5) - f(0)) / (5 - 0)
Substituting the given values, we get:
Average Velocity = (4 - 0) / 5
Therefore, the average velocity of the automobile over the interval [0, 5] is 4/5 or 0.8 miles per minute.
Since the mean value theorem guarantees the existence of a point c in (0, 5) where the derivative (instantaneous rate of change) is equal to this average velocity, we can conclude that at some point during the 5-minute journey, the automobile must have had an instantaneous velocity of exactly 0.8 miles per minute.