Mean Value Theorem Applications

Use the function $f(x) = -x^2 + 3x + 10$ to answer the following: a. On the interval [2,6], what is the average rate of change?

b. On the interval (2,6), when does the instantaneous rate of change equal the average rate of change?

The Mean Value Theorem is a key concept in calculus that connects the average rate of change of a function over an interval to the instantaneous rate of change at a specific point within that interval. In simpler terms, it tells us that for a smooth, continuous function, there is at least one point on the curve where the slope of the tangent line (instantaneous rate of change) is exactly equal to the average slope between the two endpoints of the interval.

For part (a), the average rate of change is calculated by looking at how much the function’s value changes between the two endpoints of the interval and dividing that by the length of the interval. This gives us a measure of how the function behaves, on average, between those two points. It's similar to finding the overall speed of a car trip by looking at the total distance covered and the total time taken, without worrying about speed at any specific moment.

For part (b), the Mean Value Theorem tells us that at some point within the interval, the instantaneous rate of change, or the slope of the function at that point, must be exactly the same as the average rate of change calculated in part (a). To find when this occurs, we would need to calculate the derivative of the function (which gives us the instantaneous rate of change at any point) and set it equal to the average rate of change. The solution to this equation will tell us the specific point on the curve where the instantaneous rate of change matches the average rate of change. This application is important in understanding how rates of change are distributed across an interval, and it’s often used in practical scenarios like motion problems, where the average speed over a time period must be equal to the speed at some moment in that period.

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