Problem 73 - Calculus Single Variable

Question

In a particular regional climate, the temperature varies between \(-22^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\), averaging \(\mu=11^{\circ} \mathrm{C}\). The number of days \(F(T)\) in the year on which the temperature remains below \(T\) degrees centigrade is given (approximately) by $$F(T)=\int_{-22}^{T} f(x) d x \quad(-22 \leq T \leq 40)$$ where $$f(x)=12.72 \exp \left(-\frac{(x-11)^{2}}{266.4}\right) $$ Notice that \(F(T)\) is the sort of area integral that we studied in Section 5.4 . a. Use Simpson's Rule with \(N=20\) to approximate \(F(40) .\) What should the exact value of \(F(40)\) be? b. Heat alerts are issued when the daily high temperature is \(36^{\circ} \mathrm{C}\) or more. On about how many days a year are heat alerts issued? c. Suppose that global warming raises the average temperature by \(1^{\circ} \mathrm{C}\), shifting the graph of \(f\) by 1 unit to the right. The new model may be obtained by simply replacing \(\mu\) with 12 and using [-21,41] as the domain (see Figure 13). What is the percentage increase in heat alerts that will result from this \(1^{\circ} \mathrm{C}\) shift in temperature?

Step-by-Step Solution

Answer

F(40) is approximately 1; heat alerts occur on fewer than 20 days; with the temperature increase, heat alerts rise significantly.

Step 1: Approximate F(40) using Simpson's Rule

For part (a), we must approximate the integral \( F(40) = \int_{-22}^{40} f(x) \, dx \) using Simpson's Rule with \( N = 20 \). The interval \([-22, 40]\) is divided into 20 subintervals, each of width \( h = \frac{40 - (-22)}{20} = 3.1 \). The approximation is calculated as:\[F(40) \approx \frac{h}{3} \left[f(-22) + 4 \sum_{i=1,3,5\ldots,19} f(x_i) + 2 \sum_{i=2,4,6\ldots,18} f(x_i) + f(40)\right]\]Substitute the function \( f(x) = 12.72 \exp \left(-\frac{(x-11)^2}{266.4}\right) \) into this formula and compute each term using the given \( x_i \) values for the approximation.

Step 2: Calculate the Exact Value of F(40)

The function \( f(x) \) is a probability density function for a normal distribution with mean \( \mu = 11 \) and variance \( \sigma^2 = 133.2 \) (since \( 266.4 = 2\sigma^2 \)). The value of \( F(40) \) for the full range represents the entire probability mass, which is 1, hence the exact value of \( F(40) \) will be 1.

Step 3: Determine Days with Heat Alerts (T ≥ 36°C)

For part (b), we need to calculate \( F(36) \) using Simpson's Rule, similar to step 1. With \( F(36) = \int_{-22}^{36} f(x) \, dx \), perform the approximation with \( N = 20 \). Then, compute the number of days with alerts as \( 365 - 365 \times F(36) \), which is the complement.

Step 4: Adjust Function for Temperature Increase Due to Global Warming

For part (c), update the function by shifting the mean to \( \mu = 12 \) while adjusting the domain to \([-21, 41]\). Compute the new \( F(36) \) using the modified function and employ Simpson's Rule for approximation. Determine the change in the number of heat alert days. Then calculate the percentage increase as:\[\frac{\text{New Alerts} - \text{Old Alerts}}{\text{Old Alerts}} \times 100\%\]

Step 5: Calculate New Heat Alert Days and Percentage Increase

Using the recalculated \( F(36) \) from Step 4, calculate the new number of heat alert days. Compare the new total with the initial total calculated in Step 3, and find the percentage increase using the formula derived in Step 4.

Key Concepts

Definite Integral

A definite integral provides the accumulated value or area under a curve within a specific interval. In our exercise, it helps us understand how often temperatures remain below a given value over a year.

• Accumulation Function: The integral of a function from point A to B accumulates all the individual values of the function between these two bounds. It sums these function values and is bound by the curve of the function and the x-axis.
• Integral Notation: The integral can be represented by \( \int_{a}^{b} f(x) \, dx \), where \( a \) is the lower limit, and \( b \) is the upper limit of the interval.
• Simpson's Rule: For numerical approximation, Simpson's Rule is often employed. It segments the interval into smaller segments and calculates the "area" by applying a quadratic polynomial for each segment, facilitating approximation without exact integration.

When evaluating real-world problems where exact values are crucial—like climate analysis—the definite integral provides a powerful tool to assess distributions and make sense of climatic data variations.

Probability Density Function

A probability density function (PDF) characterizes the relative likelihood of a random variable to take on a given value.

• Function Nature: PDFs encapsulate how probabilities are distributed across possible values. For continuous data, the area under a PDF curve between two points equates to the probability that a random variable falls within that interval.
• Properties: A PDF is non-negative for all x, and the total area under the curve is equal to 1, indicating total probability.
• Role in Climate Analysis: In climatology, PDFs help model temperature variations. For the given exercise, it indicates the probability distribution of temperature values over a year.

By understanding and applying PDFs, we can interpret how likely it is for temperatures to reach certain thresholds, assisting in predictions that are vital for planning and mitigating climate impacts.

Normal Distribution

The normal distribution, or Gaussian distribution, appears amusingly bell-shaped and is characterized by its symmetry about the mean. It's used extensively in statistics and climatology because it simplifies the analysis of temperature datasets, among others.

• Parameters: Defined by its mean \(\mu\) and standard deviation \(\sigma\). These parameters shape the curve, determining its position and spread.
• Significance in Measurements: Many natural factors, like temperature, follow a normal distribution, which makes this model crucial for evaluating the frequency of extreme weather conditions.
• Application to Exercise: In this exercise, temperatures are modeled using a normal distribution, allowing us to infer the distribution of temperatures over a year and see how often extreme values, like high temperatures, occur.

Understanding normal distributions aids in analyzing the central tendency and variability of data, which is fundamental in studying climate patterns effectively.

Global Warming Impact on Climate

Global warming refers to the progressive increase in Earth's average temperature, attributed largely to human activities such as burning fossil fuels.

• Shifting Averages: Global warming results in shifting temperature averages, affecting distributions—like the one in our exercise—by shifting it to the right, hence affecting the probability of extreme weather conditions.
• Increased Frequency of Extremes: As models predict, even a 1°C increase can lead to disproportionate increases in the number of heat alerts, because of the non-linearities in the distribution tails.
• Climatological Impacts: These shifts impact ecosystems, sea levels, and weather patterns globally, making accurate climate modeling more crucial. Our exercise models how such changes influence heat alerts.

By linking academic exercises to real-world scenarios, understanding global warming becomes crucial for formulating proactive solutions to mitigate its effects and prepare for anticipated changes.