An automobile travels 4 miles road in 5 minutes. Use the Mean Value theorem to show that the speedometer reads exactly 48 mph at least once during the trip.

MVT states that for an interval [a,b] there is a c such that f'(c) = (f(b)-f(a))/(b-a)

Here letting f(x) be distance at time x minutes,
f(0) = 0
f(5) = 4

(f(5)-f(0))/(5-0) = 4/5 mi/min (that's 48 mph)

MVT states that at some 0<= c <= 5, f'(x) = 4/5

f'(x) is the speedometer reading at x minutes.

Well, well, well. We're getting mathematical here, aren't we? Alright, buckle up! Let's use the Mean Value theorem and hit the road!

The Mean Value theorem states that if a function f(x) is continuous on the interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) where the instantaneous rate of change, also known as the derivative, is equal to the average rate of change over the interval.

So, let's call our position function f(t), where t represents time. We know that the automobile travels 4 miles in 5 minutes, which means the average rate of change, let's say f'(t), over this interval is 4 miles/5 minutes.

Now, the speedometer measures the speed, which is the magnitude of the derivative of the position function. In other words, it measures |f'(t)|.

We have to prove that |f'(t)| = 48 mph at least once during the trip.

Since the average rate of change, f'(t), over the interval is 4 miles/5 minutes, we can say that there exists at least one point c, where t < c < t + 5 minutes, such that f'(c) = 4 miles/5 minutes.

But wait, we need to convert that to miles per hour! This is where the conversion factor comes into play. We know that 1 mile = 1.60934 kilometers and 1 hour = 60 minutes.

So, f'(c) = (4 miles/5 minutes) * (60 minutes/1 hour) * (1.60934 kilometers/1 mile). Crunching the numbers, we get f'(c) ≈ 77.1744 kilometers/hour.

Now let's convert that back to mph. 1 kilometer/hour ≈ 0.621371 mph.

Therefore, f'(c) ≈ 77.1744 kilometers/hour * (0.621371 mph/1 kilometer/hour) ≈ 47.987 mph.

Bingo! We have f'(c) ≈ 47.987 mph, which is pretty darn close to 48 mph. So, we can confidently say that the speedometer reads exactly 48 mph at least once during the trip!

That's one speedy automobile! Enjoy the ride, my friend!

To use the Mean Value Theorem to show that the speedometer reads exactly 48 mph at least once during the trip, we need to verify that the conditions of the theorem are met.

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 open interval (a, b) such that the instantaneous rate of change (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, the function is the distance traveled by the automobile as a function of time. Let's denote this function as f(t), where t represents time in minutes and f(t) represents the distance traveled in miles.

Given that the automobile travels 4 miles of road in 5 minutes, we can write the function as:
f(t) = 4t/5

The conditions for applying the Mean Value Theorem are:
1) The function f(t) is continuous on the closed interval [a, b]. In this case, the interval [a, b] is [0, 5], which includes the entire duration of the trip from the start to the end.
The function f(t) = 4t/5 is a linear function, which is continuous on any interval.

2) The function f(t) is differentiable on the open interval (a, b). In this case, the open interval (a, b) is (0, 5).
The function f(t) = 4t/5 is a simple linear function and is differentiable on any open interval.

Now, let's calculate the derivative of f(t) to determine the instantaneous rate of change.
f'(t) = 4/5

Since the derivative f'(t) is a constant, it means that the instantaneous rate of change of the distance traveled is constant over the entire interval [0, 5].

Therefore, the average rate of change of f(t) over the interval [0, 5] is equal to f'(t), which is 4/5.

According to the Mean Value Theorem, there exists at least one point c in the interval (0, 5) where the derivative f'(c) is equal to the average rate of change 4/5.

Since f'(c) = 4/5, we can interpret this as the instantaneous rate of change (speed) of the automobile at time c is 4/5 miles per minute.

To convert this rate to miles per hour, we need to multiply it by 60 (since there are 60 minutes in an hour):
(4/5) * 60 = 48 mph

Therefore, based on the Mean Value Theorem, we can conclude that the speedometer reads exactly 48 mph at least once during the trip.

To show that the speedometer reads exactly 48 mph at least once during the trip, we can use the Mean Value Theorem.

The Mean Value Theorem states that if a function is continuous over a closed interval [a, b] and differentiable over the open interval (a, b), then there exists at least one point c in the open interval (a, b) where the instantaneous rate of change (the derivative) is equal to the average rate of change over the closed interval [a, b].

In this case, let's consider the function f(t) representing the distance traveled at time t. We know that the automobile travels 4 miles in 5 minutes, so the average rate of change of f(t) over the interval [0, 5] is (f(5) - f(0)) / (5 - 0) = 4 / 5 miles per minute.

Now, we need to check if there is a point c in the interval (0, 5) where the instantaneous rate of change (the derivative) of f(t) is equal to the average rate of change 4/5 miles per minute.

Let's differentiate the function f(t) to find its derivative f'(t). The derivative will give us the instantaneous rate of change of distance with respect to time.

f'(t) = d/dt[f(t)]

Since the distance traveled is given as 4 miles in 5 minutes, we can write the equation for f(t) as:

f(t) = 4t/5

Taking the derivative of f(t), we get:

f'(t) = 4/5

The derivative f'(t) is a constant 4/5, which means the instantaneous rate of change of the distance traveled is constant and equal to the average rate of change over the interval [0, 5].

Therefore, according to the Mean Value Theorem, there exists at least one point c in the interval (0, 5) where the speedometer reads exactly 48 mph, which corresponds to the instantaneous rate of change of distance being equal to 4/5 miles per minute.