Problem 50 - Essential Calculus Early Transcendentals

Question

The family of bell-shaped curves $$y=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} /\left(2 \sigma^{2}\right)}$$ occurs in probability and statistics, where it is called the normal density function. The constant \(\mu\) is called the mean and the positive constant \(\sigma\) is called the standard deviation. For simplicity, let's scale the function so as to remove the factor 1\(/(\sigma \sqrt{2 \pi})\) and let's analyze the special case where \(\mu=0 .\) So we study the function $$f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}$$ (a) Find the asymptote, maximum value, and inflection points of \(f .\) (b) What role does \(\sigma\) play in the shape of the curve? (c) Illustrate by graphing four members of this family on the same screen.

Step-by-Step Solution

Answer

The function has no asymptotes, maximum value 1 at \( x = 0 \), and inflection points at \( x = \pm \sigma \). \( \sigma \) determines curve width.

Step 1: Analyzing Asymptotes

The function \( f(x) = e^{-x^2 / (2\sigma^2)} \) has no horizontal or vertical asymptotes. As \( x \) goes to \( \pm \infty \), \( f(x) \) approaches 0 because the exponential function decreases to 0.

Step 2: Finding Maximum Value

To find the maximum value of \( f(x) \), note that the exponential function \( e^{-x^2 / (2\sigma^2)} \) reaches its maximum value of 1 when the exponent is zero, i.e., when \( x = 0 \). Therefore, the maximum value at \( x = 0 \) is \( f(0) = 1 \).

Step 3: Finding Inflection Points

To find inflection points, first compute the second derivative of \( f(x) \). The first derivative is \( f'(x) = \left(-\frac{x}{\sigma^2}\right) e^{-x^2 / (2\sigma^2)} \). The second derivative is \( f''(x) = \left(\frac{x^2 - \sigma^2}{\sigma^4}\right) e^{-x^2 / (2\sigma^2)} \). Setting \( f''(x) = 0 \) gives inflection points at \( x = \pm \sigma \).

Step 4: Understanding \( \sigma \)'s Role

The parameter \( \sigma \) is the standard deviation and affects the width of the bell curve. A larger \( \sigma \) results in a wider and flatter bell curve, whereas a smaller \( \sigma \) results in a narrower, steeper curve.

Step 5: Graphing Family Members

To illustrate the family of curves, graph \( f(x) = e^{-x^2 / (2\sigma^2)} \) for different \( \sigma \) values, such as \( \sigma = 0.5, 1, 2, 3 \). Notice how increasing \( \sigma \) widens the bell curve.

Key Concepts

Asymptote

The concept of asymptotes is about lines that a curve approaches but never quite reaches. In the context of the normal distribution, specifically for the function \( f(x) = e^{-x^2 / (2\sigma^2)} \), there are no traditional horizontal or vertical asymptotes. This might be surprising since we often deal with asymptotes in other mathematical functions.
Instead, what happens here is that as \( x \) moves towards positive or negative infinity, the function \( f(x) \) tends towards 0. This means the curve gets closer and closer to the x-axis but doesn't actually reach it.
- Unlike some functions with asymptotes like hyperbolas, the normal distribution flattens out smoothly as you move away from the mean. - In probability terms, this implies that extreme values (far from the mean) are possible but very unlikely.
The important thing to remember is that the function is heavily influenced by the behavior of the exponential term, which diminishes rapidly as the input value increases.

Maximum Value

In functions, the maximum value is where the curve reaches its highest point. For the normal distribution curve \( f(x) = e^{-x^2 / (2\sigma^2)} \), the maximum value is found in a fairly straightforward way.
The exponential function reaches its maximum value when its exponent is zero. This happens when \( x = 0 \), because the exponent \( -x^2 / (2\sigma^2) \) becomes zero at that point.
- Evaluating \( f(0) \) results in \( e^{0} \) which equals 1. - Thus, the peak or the highest point of the curve is 1, located right at the mean, which in this case, is 0.
This point represents the most probable value according to the distribution, highlighting how likely it is for data to cluster around the mean. Because of this symmetry, the normal distribution is often used to model real-world situations where values are average or typical.

Inflection Point

Inflection points are interesting features in a curve where the curve changes its direction of bending. For our normal distribution function, finding these points requires delving into calculus, particularly the second derivative.
To locate inflection points, you calculate where the second derivative of the function \( f(x) \) is equal to zero. Here, with the normal distribution function, the second derivative \( f''(x) = \left(\frac{x^2 - \sigma^2}{\sigma^4}\right) e^{-x^2 / (2\sigma^2)} \) results in zero at \( x = \pm \sigma \).
- This means inflection points occur at \( x = \sigma \) and \( x = -\sigma \). - At these points, the curve transitions from concave up to concave down (and vice versa).
Inflection points are crucial as they mark changes in the curvature direction, signaling where the rate of increase changes along the curve profile, giving us insights into data dispersion and curve shape.

Standard Deviation

The standard deviation, represented by \( \sigma \), is a valuable statistical measure in this context. It plays a critical role in determining the shape and spread of the normal distribution curve.
A larger \( \sigma \) signifies that the data is more spread out, resulting in a wider and flatter bell curve. Conversely, a smaller \( \sigma \) indicates that the data points are clustered closer to the mean, producing a steeper and narrower bell curve.
- It essentially measures the variability or diversity from the mean. - In practical terms, a larger standard deviation means more variation among the data points.
Understanding \( \sigma \) is essential for interpreting graphs of normal distributions. It characterizes how concentrated the data are around the mean and allows for analysis of probability distributions in fields like finance, research, and engineering.