Problem 89 - Calculus Early Transcendentals

Question

An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-\alpha x^{2}}\). a. Graph the Gaussian for \(a=0.5,1,\) and 2 b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.

Step-by-Step Solution

Answer

Based on the solution provided: a. To graph the Gaussian function for different values of alpha (0.5, 1, and 2), we plotted three functions: 1. \(f_1(x) = e^{-0.5x^2}\) 2. \(f_2(x) = e^{-1x^2}\) 3. \(f_3(x) = e^{-2x^2}\) b. We computed the area under the curves using the given integral formula: 1. For \(\alpha = 0.5\), the area is \(\sqrt{2\pi}\). 2. For \(\alpha = 1\), the area is \(\sqrt{\pi}\). 3. For \(\alpha = 2\), the area is \(\sqrt{\frac{\pi}{2}}\). c. We evaluated a definite integral involving a quadratic function in the exponential term by completing the square and found the final answer to be: $$\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x= e^{-\left(c - \frac{b^2}{4a}\right)} \sqrt{\frac{\pi}{a}}$$

Step 1: Plot the Gaussian function

To graph the Gaussian function \(f(x) = e^{-\alpha x^2}\) for \(\alpha = 0.5, 1,\) and \(2\), use a graphing software or calculator to plot each of the three functions: 1. \(f_1(x) = e^{-0.5x^2}\) 2. \(f_2(x) = e^{-1x^2}\) 3. \(f_3(x) = e^{-2x^2}\) You should now have a graph that displays the three Gaussian curves for the given values of \(\alpha\). Notice how the curves change their shape as \(\alpha\) changes. #b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a).#

Step 2: Calculate the area using the given formula

We can use the given formula \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}\) to find the area under the curves of part (a) for each value of \(\alpha\): 1. For \(\alpha = 0.5\), the area is \(\sqrt{\frac{\pi}{0.5}} = \sqrt{2\pi}\). 2. For \(\alpha = 1\), the area is \(\sqrt{\frac{\pi}{1}} = \sqrt{\pi}\). 3. For \(\alpha = 2\), the area is \(\sqrt{\frac{\pi}{2}}\). #c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.#

Step 3: Complete the square

To evaluate the integral \(\int_{-\infty}^{\infty} e^{-\left(ax^2+bx+c\right)} dx\), we need to complete the square for the quadratic function inside the exponential term. In general, we can write: $$a x^2 + b x + c = a \left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$$

Step 4: Rewrite the Integral

Now replace the expression inside the integral with the completed square form: $$\int_{-\infty}^{\infty} e^{-\left(a(x + \frac{b}{2a})^2 + \left(c - \frac{b^2}{4a}\right)\right)} dx$$

Step 5: Evaluate the Integral

Next, we will perform a change of variables to simplify the integral: Let \(u = x + \frac{b}{2a}\). Then, \(x = u - \frac{b}{2a}\), and \(dx = du\). The limits of integration will remain the same as \(u\) goes from \(-\infty\) to \(\infty\). So, we have: $$\int_{-\infty}^{\infty} e^{-\left(a(u - \frac{b}{2a})^2 + \left(c - \frac{b^2}{4a}\right)\right)} du$$ Now, we will separate out the exponential term not dependent on \(u\): $$= e^{-\left(c - \frac{b^2}{4a}\right)} \int_{-\infty}^{\infty} e^{-au^2} du$$ Now, apply the integral formula from part b: $$= e^{-\left(c - \frac{b^2}{4a}\right)} \sqrt{\frac{\pi}{a}}$$ And we have our final answer: $$\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x= e^{-\left(c - \frac{b^2}{4a}\right)} \sqrt{\frac{\pi}{a}}$$

Key Concepts

Complete the Square

Completing the square is a mathematical technique used to solve quadratic equations, integrate functions, or find the maximum or minimum values of a parabola. In the context of a Gaussian function, for example, completing the square allows us to manipulate the expression into a form that makes it much easier to perform advanced operations like integration.

Consider the general quadratic expression in the form of \( ax^2 + bx + c \). When we complete the square, we are essentially rewriting this expression in the form of \( a(x+h)^2 + k \), which reveals the vertex of the parabola represented by the quadratic equation. Here's how the process works:

• Divide the coefficient of \( x \) by \( 2a \) and then square the result.
• Add and subtract this squared result within the parentheses.
• Factor out \( a \) from the squared terms inside the parentheses.
• Rewrite the expression in a perfect square trinomial plus a constant.

The ability to complete the square is crucial when working with Gaussian functions, particularly when we're dealing with integrals that involve a quadratic expression within the exponent of an exponential term.

Area Under a Curve

The area under a curve is a fundamental concept in integral calculus and has a myriad of applications in mathematics, physics, and engineering. When we talk about the area under the curve of a function, we are generally referring to the integral of that function over a specified range.

For the Gaussian function, which is symmetric about the y-axis, the area under the entire curve from \( -\text{infinity} \) to \( +\text{infinity} \) is particularly significant because it encapsulates the total distribution of values. This is commonly seen in probability theory, where the total area under a probability density function must equal 1. For a Gaussian function described by \( f(x) = e^{-\text{alpha} x^2} \), the area under the curve is determined by integrals that have known solutions and can be represented by expressions involving the constant \( \text{pi} \).

The calculation of the area under the curve for different values of \( \text{alpha} \) demonstrates how changes in the parameter of a Gaussian function influence the distribution's spread—a key concept in statistics and probability theory.

Normal Distribution

The normal distribution, also referred to as Gaussian distribution, is a continuous probability distribution that is symmetrical and bell-shaped. A key characteristic of the normal distribution is that it describes how the values of a variable are distributed, with most values clustering around a central mean and fewer occurring as we move away from the mean.

Mathematically, the normal distribution is characterized by two parameters: the mean \( \mu \) and standard deviation \( \sigma \). The Gaussian function \( f(x) = e^{-\text{alpha} x^2} \) is closely related, where \( \alpha \) is a parameter related to the variance of the distribution. In the field of statistics, the normal distribution is ubiquitous due to the Central Limit Theorem, which states that the mean of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

Understanding this distribution is crucial not only for theoretical statistics but also for practical applications across various domains, including social sciences, natural sciences, and financial modeling.

Integral Calculus

Integral calculus is the branch of mathematics dealing with integrals and their properties. It's primarily concerned with the accumulation of quantities, such as areas under curves, total accumulated change, and the determination of antiderivatives. In our example involving the Gaussian function, integral calculus helps us understand the total probability across all outcomes, which is essential in statistical analysis.

The integral of the Gaussian function over all real numbers yields a constant that relates to the properties of the normal distribution. Integral calculus not only enables us to find areas under curves but also aids us in solving differential equations, computing volumes, and working with any phenomenon involving accumulation or distribution of quantities.

Through techniques like 'completing the square' and the use of fundamental integral properties, we can evaluate complicated integrals of Gaussian functions and in turn, solve complex problems in various fields such as physics, engineering, and economics.