The graph in the accompanying figure shows the average...

The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365 -day year. The equation that approximates the temperature on day $x$ is $$y=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25$$ and is graphed in the accompanying figure. a. On what day is the temperature increasing the fastest? b. About how many degrees per day is the temperature increasing when it is increasing at its fastest? (FIGURE CAN'T COPY).

The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365 -day year. The equation that approximates the temperature on day $x$ is
$$y=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25$$
and is graphed in the accompanying figure.
a. On what day is the temperature increasing the fastest?
b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?
(FIGURE CAN'T COPY).

Key Concepts

Sinusoidal Modeling

Sinusoidal modeling involves using sine or cosine functions to represent periodic phenomena, such as daily or yearly temperature variations. This approach captures the repeating and oscillatory nature of measurements over time, allowing one to accurately approximate data with fluctuations around a midline.

Amplitude, Period, and Phase Shift

The amplitude measures the maximum deviation from the midline, indicating the extent of oscillation, while the period is the length of one complete cycle. The phase shift represents the horizontal translation of the curve. Together, these parameters help in shaping the sine function to match the real-world periodic behavior observed in phenomena like temperature changes throughout the year.

Derivative of Trigonometric Functions

The derivative of a trigonometric function such as a sine function provides information about its rate of change. By differentiating the sine function, one obtains a cosine function multiplied by constants which indicates how quickly and in which direction the function is changing at any given point. This is crucial for determining where the function is increasing most rapidly.

Maximum Rate of Change

The maximum rate of change of a sinusoidal function occurs when the derivative reaches its maximum value. Since the cosine function oscillates between -1 and 1, the maximum rate of change is found when the cosine equals 1, allowing one to identify both the day and the magnitude at which the temperature is increasing fastest.

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