A model for the length of daylight (in hours) in...

A model for the length of daylight (in hours) in Philadelphia on the $t$ h day of the vear is given by the function $$L(t)=12+2.8 \sin \left[\frac{2 \pi}{365}(t-80)\right]$$ Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May $21 .$

A model for the length of daylight (in hours) in Philadelphia on the $t$ h day of the vear is given by the function
$$L(t)=12+2.8 \sin \left[\frac{2 \pi}{365}(t-80)\right]$$
Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May $21 .$

Key Concepts

Chain Rule in Differentiation

When differentiating composite functions, such as those in sinusoidal models where the argument of the sine function itself is a linear function of time, the chain rule is an essential technique. It allows one to correctly account for the inner function’s rate of change while computing the overall derivative, thereby accurately assessing how changes in time affect the modeled quantity.

Rate of Change and Differentiation

The rate of change of a function is measured by its derivative, which indicates how quickly the function's output is varying with respect to changes in the input variable. In the context of a daylight model, differentiating the sinusoidal function provides the instantaneous increase or decrease in hours of daylight, enabling comparisons at different times of the year.

Periodic Functions

Periodic functions are mathematical functions that repeat their values in regular intervals or periods. They are used to model phenomena that have cyclic or seasonal behavior, making them ideal for describing variations in natural processes such as the length of daylight over the course of a year.

Sinusoidal Models

Sinusoidal models employ sine (or cosine) functions to capture the oscillatory nature of periodic phenomena. These models use parameters like amplitude (which determines the extent of fluctuation), period (which defines how long one complete cycle lasts), and phase shift (which adjusts the timing of the peaks and troughs) to fit real-world data accurately.

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