Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$

Step-by-Step Solution

Answer

The values of \(x^n e^{-ax}\) for respective \(x\) values 1, 10, 25, and 50 are 0.606, 673.8, 30.67, and 0.0035.

Step 1: Compute the expression for x = 1

Plug \(x = 1\), \(a = 0.5\), and \(n = 5\) into the expression \(x^n e^{-ax}\) and calculate. The result is \(1^5 * e^{-(0.5)*1} = 1 * e^{-0.5} = 0.606.\)

Step 2: Compute the expression for x = 10

Plug \(x = 10\), \(a = 0.5\) and \(n = 5\) into the expression \(x^n e^{-ax}\) and calculate. The result is \(10^5 * e^{-(0.5)*10} = 100000 * e^{-5} = 673.8.\)

Step 3: Compute the expression for x = 25

Plug \(x = 25\), \(a = 0.5\), and \(n = 5\) into the expression \(x^n e^{-ax}\) and calculate. The result is \(25^5 * e^{-(0.5)*25} = 9765625 * e^{-12.5} = 30.67.\)

Step 4: Compute the expression for x = 50

Plug \(x = 50\), \(a = 0.5\), and \(n = 5\) into the expression \(x^n e^{-ax}\) and calculate. The result is \(50^5 * e^{-(0.5)*50} = 312500000 * e^{-25} = 0.0035.\)

Step 5: Analyze the results

Comparing the results, we can see that the value of \(x^n e^{-ax}\) decreases as \(x\) increases, which verifies the limit \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0\).

Key Concepts

Exponential Functions

Exponential functions are a critical part of the mathematical landscape, and they appear in many different forms. An exponential function is expressed as \(f(x) = a^{x}\), where the base \(a\) is a constant. These functions are characterized by their rapid growth or decay. The most important thing to remember about exponential functions with a base greater than 1, such as \(e^x\) where \(e\) is approximately 2.718, is that they grow extremely fast as \(x\) increases.
In the context of the given problem, we use an exponential decay function, \(e^{-ax}\), where \(a > 0\). As \(x\) becomes larger, \(e^{-ax}\) approaches zero since the negative exponent implies a decreasing function. This characteristic makes exponential functions especially useful in modeling situations like radioactive decay, loan interest, and more. Their ability to model real-world situations where change happens rapidly cannot be overstated.

Polynomials

Polynomials play a big role in mathematics and are an expression involving a finite sum of powers in one or more variables. These look like \(x^n + x^{n-1} + \,\ldots\,+ C\), where \(C\) are coefficients. The degree of the polynomial is determined by the highest power of \(x\). In our exercise, the polynomial function \(x^n\) highlights this concept of power and growth.
Unlike exponential functions, polynomial functions grow at a slower rate when \(x\) becomes large. For instance, \(x^5\) is a polynomial where the highest degree, 5, influences how steeply the function climbs or drops as \(x\) increases or decreases. Understanding polynomials is fundamental to grasp the limits and behavior of combined functions, like in our exercise which involves both a polynomial term \(x^n\) and an exponential term \(e^{-ax}\).
In our example, we can see that despite the polynomial \(x^n\) growing rather large, the exponential decay of \(e^{-ax}\) eventually dominates as \(x\) tends towards infinity, driving the overall expression to zero.

Spreadsheet Calculations

Spreadsheets are versatile tools used to perform calculations easily. In mathematics, they can help visualize how functions behave. For tasks like calculating \(x^n e^{-ax}\), spreadsheets can speed up the calculations by quickly changing variables like \(x\), and automatically updating the outputs in tabular form.
When setting up the calculation on a spreadsheet, you can use columns to represent different values of \(x\), like in the exercise (1, 10, 25, 50). By filling in the expression \(=x^n * EXP(-a*x)\) into a cell, spreadsheets can compute results like \(e^{-0.5}\) at \(x = 1\) or \(e^{-12.5}\) at \(x = 25\) quickly and with repeated accuracy.
Spreadsheets don't just perform fast calculations; they help spot patterns and verify results visually. For instance, you see the trend as values decrease with each increasing \(x\), directly illustrating how the limit \(\lim_{x \rightarrow \infty} x^{n} e^{-a x}=0\) is achieved, confirming the function's behavior as intended.