Alzheimer's Disease A study on the circadian rhythms of...

Alzheimer's Disease A study on the circadian rhythms of patients with Alzheimer's disease found the body temperature of patients could be described by a function of the form $$T=T_{0}+a \cos \left(\frac{2 \pi(t-k)}{24}\right),$$ where $t$ is the time in hours since midnight. For the patients without Alzheimer's, the average values of $T_{0}$ (the MESOR), $a$ (the amplitude), and $k$ (the acrophase) were $36.91^{\circ} \mathrm{C}, 0.32^{\circ} \mathrm{C},$ and 14.92 hours, while for the patients with the disease, the values were $37.29^{\circ} \mathrm{C}$ $0.46^{\circ} \mathrm{C},$ and 16.37 hours. Source: American Journal of Psychiatry. (a) Graph the functions giving the temperature for each of the two groups using a graphing calculator. Do these two functions ever cross? (b) At what time is the temperature highest for the patients without Alzheimer's? (c) At what time is the temperature highest for the patients with Alzheimer's?

Alzheimer's Disease A study on the circadian rhythms of patients with Alzheimer's disease found the body temperature of patients could be described by a function of the form
$$T=T_{0}+a \cos \left(\frac{2 \pi(t-k)}{24}\right),$$
where $t$ is the time in hours since midnight. For the patients without Alzheimer's, the average values of $T_{0}$ (the MESOR), $a$ (the amplitude), and $k$ (the acrophase) were $36.91^{\circ} \mathrm{C}, 0.32^{\circ} \mathrm{C},$ and 14.92 hours, while for the patients with the disease, the values were $37.29^{\circ} \mathrm{C}$ $0.46^{\circ} \mathrm{C},$ and 16.37 hours. Source: American Journal of Psychiatry.
(a) Graph the functions giving the temperature for each of the two groups using a graphing calculator. Do these two functions ever cross?
(b) At what time is the temperature highest for the patients without Alzheimer's?
(c) At what time is the temperature highest for the patients with Alzheimer's?

Key Concepts

Amplitude

Amplitude indicates the extent of oscillation from the MESOR, representing the magnitude of variation in the periodic function. It defines how much the temperature (or any cyclical measurement) deviates from the average value at its peak, and is important for understanding the intensity of the rhythmic oscillation.

MESOR

MESOR, or Midline Estimating Statistic Of Rhythm, represents the mean level around which a periodic variable oscillates. In the context of modeling circadian rhythms, MESOR serves as the baseline temperature or average value, providing a reference point for analyzing deviations due to the cosine oscillation.

Acrophase

Acrophase represents the phase shift in a periodic function, specifically indicating the time at which the maximum value occurs. It is a key parameter in analyzing the timing of peaks in cyclic phenomena such as body temperature rhythms, and it helps in identifying shifts in biological patterns.

Circadian Rhythms

Circadian rhythms refer to the natural, approximately 24-hour cycles in the physiological processes of living organisms. They play a crucial role in regulating sleep-wake cycles, hormone release, and other bodily functions. In mathematical modeling, these rhythms are often analyzed using periodic functions to capture the regular variations in measurements such as body temperature.

Cosine Function

The cosine function is a trigonometric function that models periodic phenomena due to its inherent oscillatory behavior. When used in modeling, it describes cycles with specific amplitude, period, and phase shift, which makes it suitable for representing cyclical patterns observed in biological systems like circadian rhythms.

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