In Example 4 in Section 1.3 we arrived at a model for the...

In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the $t$ th day of the year: $$ L(t)=12+2.8 \sin \left[\frac{2 \pi}{365}(t-80)\right] $$ Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21

In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the $t$ th day of the year:
$$
L(t)=12+2.8 \sin \left[\frac{2 \pi}{365}(t-80)\right]
$$
Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21

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Key Concepts

Interpretation of Derivatives in Context

When applying derivatives to physical models, the result reveals the rate at which the modeled quantity is changing. For instance, evaluating the derivative of the daylight model at particular days helps compare the speed of daylight increase on those days, which is essential for understanding seasonal variations.

Phase Shift and Frequency

In sinusoidal functions, the phase shift adjusts the horizontal positioning of the graph, aligning the model with specific time frames or events in the cycle, while the frequency determines the number of cycles over a fixed period. Both parameters are crucial for accurately modeling real-world periodic phenomena.

Sinusoidal Models for Periodic Phenomena

Sinusoidal models use sine and cosine functions to describe phenomena that exhibit repeating or cyclical behavior over time. This approach is particularly useful for modeling natural cycles, such as temperature fluctuations, tides, or daylight duration throughout the year.

Derivatives of Trigonometric Functions

The derivative of a trigonometric function, like that of the sine function in the model, provides the instantaneous rate of change at any point. In the context of a daylight model, differentiating the sine function helps determine how quickly daylight hours are increasing or decreasing on any given day.

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