Lapse rates in the atmosphere Refer to Example 2. Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is -10.3° C and at an elevation of 3.2km , the temperature is 8.0°C . Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of 7°C/ km at some intermediate elevation? Explain.

Verified step by step guidance

1. First, understand the concept of the Mean Value Theorem (MVT). The MVT states that for a function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that the derivative at c is equal to the average rate of change over [a, b].
2. Identify the function and interval in this problem. The function here is the temperature T as a function of elevation h, and the interval is [3.2, 6.1] km.
3. Calculate the average rate of change of temperature over the interval [3.2, 6.1] km. This is done by finding the difference in temperature divided by the difference in elevation: \( \frac{T(6.1) - T(3.2)}{6.1 - 3.2} \).
4. Substitute the given temperatures into the formula: \( \frac{-10.3 - 8.0}{6.1 - 3.2} \). This will give you the average rate of change of temperature with respect to elevation over the interval.
5. Compare the calculated average rate of change to the threshold value of 7°C/km. If the average rate of change exceeds 7°C/km, then by the Mean Value Theorem, there exists at least one elevation between 3.2 km and 6.1 km where the lapse rate exceeds 7°C/km.

Key Concepts

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative of the function equals the average rate of change over that interval. In the context of lapse rates, it implies that there is at least one elevation where the instantaneous rate of temperature change matches the average rate calculated between two points.

Lapse Rate

Lapse rate refers to the rate at which temperature decreases with an increase in altitude in the atmosphere. It is typically expressed in degrees Celsius per kilometer (°C/km). Understanding lapse rates is crucial for analyzing atmospheric conditions, as they can indicate stability or instability in the atmosphere, affecting weather patterns and phenomena.

Average Rate of Change

The average rate of change of a function over an interval is calculated by taking the difference in the function's values at the endpoints of the interval and dividing it by the difference in the input values. In this case, it helps determine the overall change in temperature between two elevations, which can then be compared to the threshold lapse rate to assess if the instantaneous lapse rate exceeds that value at some point in between.