At 7:00 AM, the temperature was 30 degrees; 3 hours later it was 42 degrees. Assuming the temperature function is differentiable, what can be concluded from the Mean Value Theorem? Don't assume the temperature increased at a constant rate.

I know that Mean Value Theorem is f'(c)=f(b)-f(a)/b-a, but I'm not sure how to apply it to this problem.

The temperature rose 12 degrees in 3 hours.

That is a mean rate of change of 4°/hr
So, somewhere in that interval the tangent to the temperature curve has a slope of 4

To apply the Mean Value Theorem to this problem, we need to identify a function that represents the temperature over time. Let's denote the temperature as T(t), where t is the time in hours.

We're given that at 7:00 AM (which we can denote as t = 0), the temperature was 30 degrees. Three hours later (at t = 3), the temperature was 42 degrees.

Now, we can use the Mean Value Theorem to find a specific point in time where the instantaneous rate of change of the temperature equals the average rate of change between these two points.

The Mean Value Theorem states that if a function is differentiable on a closed interval [a, b] and continuous on the open interval (a, b), then there exists at least one value c between a and b such that f'(c) = (f(b) - f(a))/(b - a).

In our case, a is 0 (7:00 AM) and b is 3 hours later. We are interested in finding the value of c.

To apply the Mean Value Theorem, we need to calculate the average rate of change between t = 0 (a) and t = 3 (b) and then find a value of c where the instantaneous rate of change is equal to that average rate of change.

The average rate of change is given by (T(3) - T(0))/(3 - 0) = (42 - 30)/3 = 12/3 = 4 degrees per hour.

Now, we want to find a value of c such that T'(c) = 4 degrees per hour.

It's important to note that we do not have any information about the behavior of the temperature function, so we cannot make any direct conclusions about T'(c) = 4 degrees per hour.

However, according to the Mean Value Theorem, we can conclude that at some point in time between 7:00 AM and 10:00 AM, the temperature function had an instantaneous rate of change of 4 degrees per hour, and this rate of change was equal to the average rate of change between 7:00 AM and 10:00 AM.

In summary, the Mean Value Theorem tells us that there exists a point in time, c, between 7:00 AM and 10:00 AM, where the rate of change of the temperature function is equal to the average rate of change over that time interval.