At 2:00pm a car's speedometer reads 60mph, and at 2:10pm it reads 65mph. Use the Mean Value Theorem to find an acceleration the car must achieve.
Answer( in mi/h2):
The line joining those two points has slope
(65-60) / (10/60)
That is the value v'(t) must achieve.
50mph^2
To find the acceleration of the car, we need to use the Mean Value Theorem. 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 the interval (a, b) such that the instantaneous rate of change (the derivative) at that point is equal to the average rate of change of the function over the interval.
In this case, let's let t represent time in minutes, and let f(t) represent the car's speed at time t. We have f(0) = 60 (since the speedometer reads 60mph at 2:00pm) and f(10) = 65 (since the speedometer reads 65mph at 2:10pm).
Now, let's find the average rate of change of the function over the interval [0, 10]. The average rate of change (or average velocity) is given by the formula:
Average rate of change = (f(b) - f(a))/(b - a)
Plugging in the values, we get:
Average rate of change = (f(10) - f(0))/(10 - 0)
= (65 - 60)/10
= 5/10
= 0.5
According to the Mean Value Theorem, there exists at least one point c in the interval (0, 10) such that the derivative of f at that point is equal to the average rate of change of f over the interval. In other words, there exists a point c such that f'(c) = 0.5.
The derivative of f(t) is the rate of change of the car's speed with respect to time, which represents the acceleration. So, we need to find the value of c for which f'(c) = 0.5. This will give us the acceleration of the car.
Unfortunately, we don't have enough information to find the exact value of c without additional data or assumptions. However, we can conclude that the car must achieve an acceleration of at least 0.5 mi/h2 in order to go from 60mph to 65mph in a span of 10 minutes.
To find the acceleration of the car using the Mean Value Theorem, we need to first understand the concept behind it. The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on an open interval (a, b), then there exists a point c in the interval (a, b) such that the derivative of the function at c is equal to the average rate of change of the function over the interval [a, b].
In this case, we want to find the acceleration of the car, which describes how the car's speed is changing over time. The average acceleration over a given time interval can be calculated by dividing the change in velocity (speed) by the change in time. Let's apply this to our problem.
First, we need to find the change in velocity. We are given that at 2:00pm, the speedometer reads 60mph, and at 2:10pm, it reads 65mph. Hence, the change in velocity is 65mph - 60mph = 5mph.
Next, we need to find the change in time. The time interval between 2:00pm and 2:10pm is 10 minutes, which is equivalent to 10/60 = 1/6 hours.
Now, we can calculate the average acceleration by dividing the change in velocity by the change in time:
average acceleration = (change in velocity) / (change in time)
average acceleration = 5mph / (1/6 hours)
average acceleration = 30mph/hour
Therefore, the car must achieve an acceleration of 30 mph/hour.