The graph in Figure 4.6 shows the variation in average daily...

The graph in Figure 4.6 shows the variation in average daily temperature in Tempe, Arizona, during a typical 365 -day year. The equation that approximates the Fahrenheit temperature on day $x$ is $$y=19.3 \sin \left[\frac{2 \pi}{365}(x-101)\right]+70$$ (a) On approximately what day of the year does the daily temperature show the greatest increase from the previous day? (b) About how many degrees per day is the temperature increasing at that time of the year?

The graph in Figure 4.6 shows the variation in average daily temperature in Tempe, Arizona, during a typical 365 -day year. The equation that approximates the Fahrenheit temperature on day $x$ is
$$y=19.3 \sin \left[\frac{2 \pi}{365}(x-101)\right]+70$$
(a) On approximately what day of the year does the daily temperature show the greatest increase from the previous day?
(b) About how many degrees per day is the temperature increasing at that time of the year?

Key Concepts

Derivative of Trigonometric Functions

The derivative of a trigonometric function provides information about the rate of change of the function at any given point. By differentiating a sinusoidal function, one can determine the rate at which the modeled quantity (such as temperature) increases or decreases, and locate the point of maximum rate of increase.

Frequency and Period

Frequency in a sinusoidal function represents the number of cycles that occur in a fixed interval, while the period is the duration of one complete cycle. In the context of daily temperature variations, these concepts allow determination of how quickly the temperature oscillates throughout the year.

Application of the Derivative to Identify Maximum Increase

By applying the derivative to a sinusoidal model, one can find the time at which the rate of change is at its peak. This involves identifying where the derivative reaches its maximum value, which corresponds to the moment of greatest increase in the modeled phenomenon.

Amplitude

Amplitude refers to half the difference between the maximum and minimum values of a periodic function. It defines the magnitude of oscillation from the function's central value, indicating the extent of variation in the data being modeled.

Sinusoidal Function

A sinusoidal function, typically described by the sine or cosine function, is used to model periodic or oscillatory phenomena. In many natural processes, such as temperature variation over a year, these functions effectively capture the repetitive cycles and provide insights into key characteristics like peaks, troughs, and mean values.

Phase Shift

Phase shift is the horizontal translation applied to a periodic function. It indicates how the function is shifted along the time axis, affecting the position of its peaks and troughs. This is crucial when aligning the model with the actual timing of observed events, such as the point of highest temperature increase.

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