Problem 16 - Calculus for Biology and Medicine

Question

The average daily temperature (measured in Fahrenheit) in New York city can be approximated by the following function of the time of year \(t .(t\) measures the fraction of the year that has elapsed since January 1.) $$ T(t)=57.5-22.5 \cos (2 \pi t) $$ (a) Sketch the function \(T(t)\) against \(t\). (b) What is the average daily temperature high, averaged over the course of one year? (c) Explain how you could get your answer in part (b) without doing any integrations. (d) What is the average summer temperature? You may assume that summer corresponds to the interval \(0.47 \leq t \leq 0.73 .\) You will need to use a calculator to evaluate your answer.

Step-by-Step Solution

Answer

Yearly average temperature is 57.5°F; summer average is approximately 70.33°F.

Step 1: Sketch the Function

The function given is \[ T(t) = 57.5 - 22.5 \cos(2\pi t) \]This is a cosine function, vertically shifted by 57.5 and with an amplitude of 22.5. The period of the function is 1, which corresponds to one year due to the factor \(2\pi t\). The cosine function oscillates between -1 and 1, so this function oscillates between \[ T(0) = 57.5 - 22.5 \times 1 = 57.5 - 22.5 = 35 \] and \[ T(0.5) = 57.5 - 22.5 \times (-1) = 57.5 + 22.5 = 80 \].The sketch will show a smooth wave starting at 35 (minimum temperature in winter), reaching a maximum of 80 (maximum in summer) at around mid-year, and returning back to 35 by end of the year.

Step 2: Compute Yearly Average Temperature

The average temperature over one cycle of a cosine function can be found by averaging the maximum and minimum values of the function. For the function given, the maximum value is 80, and the minimum value is 35. The average over a year is:\[ \text{Average temperature} = \frac{80 + 35}{2} = \frac{115}{2} = 57.5 \]

Step 3: Explain Yearly Average without Integration

The yearly average temperature can be obtained from the mathematical properties of sine and cosine functions. These functions are symmetric around their midlines. Since the function is \[ T(t) = 57.5 + a\cos(2\pi t) \]Here, 57.5 is the midline crossing value, and it is also the mean of the cosine wave over a complete cycle. Thus, the average yearly temperature is simply the vertical shift from the average of the cosine wave.

Step 4: Calculate Average Summer Temperature

Suited to the summer interval \( 0.47 \leq t \leq 0.73 \), the average temperature can be approximated by calculating the midpoint of the temperatures at these two points:1. Calculate \( T(0.47) \) and \( T(0.73) \):- \( T(0.47) = 57.5 - 22.5 \cos(2\pi \times 0.47) \approx 57.5 - 22.5(-0.154) \approx 60.97 \)- \( T(0.73) = 57.5 - 22.5 \cos(2\pi \times 0.73) \approx 57.5 - 22.5(-0.987) \approx 79.68 \)2. Average these values:\[ \text{Average summer temperature} = \frac{60.97 + 79.68}{2} \approx 70.33 \]

Key Concepts

Cosine Function

The cosine function plays a pivotal role in modeling periodic phenomena, such as the temperature changes in New York City throughout the year. In this scenario, the function is expressed as \[ T(t) = 57.5 - 22.5 \cos(2\pi t) \]

• **Amplitude**: This is the measure of the maximum deviation from the average value. In our model, the amplitude is 22.5, which signifies the range of temperature changes from the average.
• **Vertical Shift (Midline)**: The constant 57.5 is the vertical shift, meaning that the entire cosine wave is centered around this value. This indicates the overall average temperature level over the year.
• **Period**: The function completes one cycle when \(2\pi t = 2\pi\), thus the period is 1 year. This is why it aptly models the temperature over the course of a year.
• **Phase Shift**: The absence of a horizontal shift parameter in the function simplifies the model, starting the cycle at \(t=0\).

These properties help us visualize how the cosine wave fluctuates high and low, depicting the highs in summer and lows in winter.

Average Temperature Calculation

In temperature modeling, calculating the yearly average temperature from a periodic function like the cosine is insightful and straightforward thanks to the function's symmetry.The simplest technique to find the average of a periodic function over its complete cycle is to compute the mean of its maximum and minimum values. For the cosine function \[ T(t) = 57.5 - 22.5 \cos(2\pi t) \]

• **Maximum Value**: When the cosine function equals -1, resulting in \[ T(t) = 57.5 + 22.5 = 80 \]
• **Minimum Value**: Achieved when the cosine function is 1, thus \[ T(t) = 57.5 - 22.5 = 35 \]

Thus, the yearly average temperature is calculated as:\[ \text{Average Temperature} = \frac{80 + 35}{2} = 57.5 \] This arithmetic mean works since the cosine function oscillates equally around its midline, providing an effortless calculation of the average without detailed integration.

Periodic Functions in Calculus

Periodic functions are fundamental in calculus, effectively used to describe cyclical behavior in real-world phenomena like temperature changes. These functions possess distinct characteristics:

• **Repetition and Predictability**: They repeat their values in regular intervals called periods. This reliable periodicity makes them ideal for modeling natural cycles.
• **Symmetry**: Take the cosine function as an example, which is symmetric around its midline, simplifying calculations of average values over one complete period.
• **Use in Derivatives and Integrals**: Calculus often employs periodic functions to determine rates of change and accumulations over time. For instance, the sine and cosine derivatives are essential in calculating instantaneous changes in periodic diseases.

With their repetitive nature, periodic functions like our temperature model, allow steady predictions across intervals, benefitting fields like meteorology and engineering. Understanding the basic properties provides a deeper comprehension of natural cycles being studied.